A Proof of the Binary Goldbach Conjecture for All Sufficiently Large
                          Even Integers
                               Denis Saltykov (ds1678@gmail.com)

                                            29 May 2026


                                               Abstract
         We prove that every sufficiently large even integer is a sum of two primes. The proof
     establishes the weighted asymptotic
                                   ∑︁
                                         Λ(𝑛1 )Λ(𝑛2 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )
                                𝑛1 +𝑛2 =𝑁

         for even 𝑁 , with the classical Goldbach singular series S(𝑁 ). The argument begins with a
     fixed-depth Heath–Brown decomposition and routes every resulting B1-origin atom into one of
     five terminal classes: Edge, CKP, GoodAWACK, LongAP/Local, or LocalDiag. Edge terms are
     estimated by strict savings, CKP terms by a Duke–Friedlander–Iwaniec Kloosterman-fraction
     input after a smooth-weight derivative verification, GoodAWACK terms by a TC1 global testing
     route and a finite-grammar closure, and local terms by explicit tagged Λ𝑄 -local projections. A
     global error-budget lemma verifies simultaneous summability of all terminal errors. Prime-power
     removal and positivity of the singular series then give a genuine prime representation.
         MSC 2020: primary 11P32; secondary 11N05, 11N36, 11L05, 11B30.
         Keywords: binary Goldbach problem, von Mangoldt correlations, Heath–Brown identity,
     Kloosterman fractions, Liouville orthogonality, local density, routing lemma.


Contents
1 Introduction and Statement of Results                                                            8
  1.1 Introduction and Relation to Known Work . . . . . . . . . . . . . . . . . . . . . . . 8
  1.2 Main Theorems and Proof Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
       1.2.1 Weighted Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
       1.2.2 Decomposition and Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
       1.2.3 Terminal Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
       1.2.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
  1.3 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Glossary of Internal Labels and Nonstandard Terms                                                    10
  2.1 Terminal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     10
  2.2 Structural Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     11
  2.3 Root, Assembly, and Bookkeeping Labels . . . . . . . . . . . . . . . . . . . . . . . .           12
  2.4 Decomposition, Routing, Edge, and Local Labels . . . . . . . . . . . . . . . . . . . .           12
  2.5 CKP Branch Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        13
  2.6 TC1, BRS, X16, and GoodAWACK Labels . . . . . . . . . . . . . . . . . . . . . . .                13
  2.7 External and Standard Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       15

                                                   1
3 Dependency Tree and Reading Map                                                                     15

4 Parameters, Notation, and Error Bookkeeping                                                         17
  4.1 Notation, Parameters, and Error Budget . . . . . . . . . . . . . . . . . . . . . . . . .        17
      4.1.1 Fixed Depth and Dyadic Partitions . . . . . . . . . . . . . . . . . . . . . . . .         17
      4.1.2 Terminal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      18
      4.1.3 Global Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       18
  4.2 Parameter Hierarchy and Global Error Budget . . . . . . . . . . . . . . . . . . . . .           19
      4.2.1 Parameter Witness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       19
      4.2.2 Multiplicity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      19
      4.2.3 Global Loss Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       19
      4.2.4 Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       19
  4.3 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      20

5 External Analytic Inputs                                                                            20
  5.1 External Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   20
      5.1.1 X1: Heath–Brown Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . .          20
      5.1.2 X9L-GT: Near-Global Liouville/AP Orthogonality . . . . . . . . . . . . . . .              20
      5.1.3 X10: DFI Kloosterman-Fraction Estimate . . . . . . . . . . . . . . . . . . . .            20
      5.1.4 X16: Shiu/AP Divisor-Average Input . . . . . . . . . . . . . . . . . . . . . . .          20
      5.1.5 Citation Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
  5.2 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      21

6 Heath–Brown Decomposition and Typed Blocks                                                          21
  6.1 Heath–Brown Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         21
      6.1.1 B1 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     21
      6.1.2 Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     21
  6.2 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      21

7 Routing Grammar and Terminal Classes                                                                21
  7.1 Routing Exhaustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      21
      7.1.1 Finite Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       21
      7.1.2 Authoritative Routing Operations . . . . . . . . . . . . . . . . . . . . . . . .          22
      7.1.3 Exhaustion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        22
      7.1.4 Terminal Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     22
  7.2 Routing Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    22
      7.2.1 Terminal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      22
      7.2.2 Canonical Routing Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         23
      7.2.3 Routing Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       23
      7.2.4 Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       24
  7.3 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      24

8 Edge and LongAP/Local Terms                                                                         24
  8.1 Edge Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    24
      8.1.1 Edge Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       24
      8.1.2 Admission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     25
      8.1.3 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       25
  8.2 LongAP/Local Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        25


                                                   2
         8.2.1 Terminal LongAP/Local Atoms . . . . . . . . . . . . . . . . . . . . . . . . . .          25
         8.2.2 Exclusion of Nonlocal Coefficients . . . . . . . . . . . . . . . . . . . . . . . .       25
   8.3   Edge Admission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   26
         8.3.1 Edge Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      26
         8.3.2 Admission Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      26
         8.3.3 Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      26
   8.4   Explicit Local Projection Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    26
         8.4.1 The Finite Local Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       27
         8.4.2 Tagged-Cell Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     27
         8.4.3 H4M Admission Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . .          28
         8.4.4 Linearity Over the B1/F3 Partition . . . . . . . . . . . . . . . . . . . . . . .         28
         8.4.5 No Double Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       28
         8.4.6 Local Factors and the Singular Series . . . . . . . . . . . . . . . . . . . . . .        29
         8.4.7 H4M Local Bridge Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .         29
   8.5   Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     30

9 CKP Branch                                                                                            30
  9.1 The CKP Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        30
      9.1.1 CKP Claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         30
      9.1.2 Internal Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        30
      9.1.3 X10 Match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         30
  9.2 CKP/X10 Smooth-Weight Matching . . . . . . . . . . . . . . . . . . . . . . . . . . .              31
      9.2.1 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      31
      9.2.2 DFI Theorem Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          31
      9.2.3 CKP Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          31
      9.2.4 Smooth Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         31
      9.2.5 Excluded Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         32
  9.3 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        32

10 GoodAWACK Branch                                                                                     32
   10.1 The GoodAWACK Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          32
        10.1.1 Branch B Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       32
        10.1.2 TC1 Global Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       32
        10.1.3 Finite GoodAWACK Grammar . . . . . . . . . . . . . . . . . . . . . . . . . .             33
   10.2 TC1, BRS/TTH, and GoodAWACK Finite Grammar . . . . . . . . . . . . . . . . .                    33
        10.2.1 TC1 Route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      33
        10.2.2 Singular Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   34
        10.2.3 Finite GoodAWACK Grammar . . . . . . . . . . . . . . . . . . . . . . . . . .             34
        10.2.4 Formal Interface Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .        34
   10.3 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      34

11 Local Main Term and Global Assembly                                                                  35
   11.1 Local/Main Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       35
        11.1.1 Explicit H4 Local Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      35
        11.1.2 H4 Admission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     35
        11.1.3 Linearity and No Double Counting . . . . . . . . . . . . . . . . . . . . . . . .         36
        11.1.4 Finite Local Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     36
        11.1.5 Weighted Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      36


                                                    3
   11.2 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

12 Prime-Power Removal and Final Handoff                                                               37
   12.1 Prime-Power Removal and Final Proof . . . . . . . . . . . . . . . . . . . . . . . . . .        37
        12.1.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   37
        12.1.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   37
   12.2 Full-Manuscript Integration Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     38

A Parameter Register and Global Error Budget                                                           38
  A.1 Parameter register . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     38
      A.1.1 PAR. Global Parameter Register . . . . . . . . . . . . . . . . . . . . . . . . .           38
  A.2 Global error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      41
      A.2.1 GEB. Global Error Budget and Parameter Hierarchy . . . . . . . . . . . . . .               41

B External Inputs and Theorem Matching                                                                 45
  B.1 Heath–Brown identity verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      45
      B.1.1 X1. Heath–Brown Identity Input for B1 . . . . . . . . . . . . . . . . . . . . .            45
  B.2 Near-global Davenport/AP verification . . . . . . . . . . . . . . . . . . . . . . . . . .        48
      B.2.1 X9L-GT. Davenport/AP Input for TC1 Testing . . . . . . . . . . . . . . . . .               48
  B.3 DFI/X10 Kloosterman-fraction verification . . . . . . . . . . . . . . . . . . . . . . .          55
      B.3.1 X10. DFI Kloosterman Fraction Input . . . . . . . . . . . . . . . . . . . . . .            55
  B.4 Shiu/AP divisor-average verification . . . . . . . . . . . . . . . . . . . . . . . . . . .       66
      B.4.1 X16. Divisor-Sum Input for BRS . . . . . . . . . . . . . . . . . . . . . . . . .           66

C Heath–Brown Decomposition Details                                                                    69
  C.1 B1 typed Heath–Brown decomposition . . . . . . . . . . . . . . . . . . . . . . . . . .           69
      C.1.1 B1. Typed Heath–Brown Decomposition . . . . . . . . . . . . . . . . . . . . .              69
  C.2 B3 block classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    74
      C.2.1 B3. Block Classification Lemma . . . . . . . . . . . . . . . . . . . . . . . . .           74

D Routing Grammar and Complete Routing Exhaustion                                                   80
  D.1 F3F4M master routing theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
      D.1.1 F3F4M. Master Routing Exhaustion Theorem . . . . . . . . . . . . . . . . . . 80
  D.2 F3P intrinsic terminal predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
      D.2.1 F3P. Intrinsic Terminal Predicate Catalogue . . . . . . . . . . . . . . . . . . . 86
  D.3 F3 routing partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
      D.3.1 F3. Routing Exhaustion / No-Cycle Theorem . . . . . . . . . . . . . . . . . . 90
  D.4 F3 complete routing interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
      D.4.1 F3A. Completeness of the F3.6 Routing Interface . . . . . . . . . . . . . . . . 101
  D.5 F3 complete routing table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
      D.5.1 F3T. Finite Routing Table for B1-Origin Atoms . . . . . . . . . . . . . . . . 105
  D.6 F4 large-divisor and quotient-tag routing . . . . . . . . . . . . . . . . . . . . . . . . . 112
      D.6.1 F4. Large Divisor Routing Lemma . . . . . . . . . . . . . . . . . . . . . . . . 112
  D.7 E5 affine regrouping inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
      D.7.1 E5. Content Stability Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 120




                                                   4
E Edge Admission, LongAP/Local, and Local Projection Algebra                                      125
  E.1 C1P strict Edge predicate catalogue . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
      E.1.1 C1P. Strict Edge Predicate Catalogue . . . . . . . . . . . . . . . . . . . . . . 125
  E.2 C1A Edge admission ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
      E.2.1 C1A. Admission of Terminal Edge Atoms . . . . . . . . . . . . . . . . . . . . 128
  E.3 C1 Edge estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
      E.3.1 C1. Unified Edge Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
  E.4 LPI local projection interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
      E.4.1 LPI. Local Projection Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 139
  E.5 D1 LongAP/Local local-coefficient expansion . . . . . . . . . . . . . . . . . . . . . . 142
      E.5.1 D1. LongAP/Local Normalization Lemma . . . . . . . . . . . . . . . . . . . . 142
  E.6 B1 local-density compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
      E.6.1 B1LD. Local Densities for B1-Inherited CKP Coefficients . . . . . . . . . . . 150
  E.7 H4 local reconstruction and singular series . . . . . . . . . . . . . . . . . . . . . . . . 152
      E.7.1 H4. Local/Main Compatibility Lemma . . . . . . . . . . . . . . . . . . . . . . 152
  E.8 H4M master local bridge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
      E.8.1 H4M. Master Local Bridge Theorem . . . . . . . . . . . . . . . . . . . . . . . 163

F CKP/X10 Package                                                                                166
  F.1 G1a CKP gcd splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
      F.1.1 G1a. CKP GCD Splitting Lemma . . . . . . . . . . . . . . . . . . . . . . . . 166
  F.2 G2a smooth AP Fourier expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
      F.2.1 G2a. Weighted Smooth AP Fourier Expansion for CKP . . . . . . . . . . . . 170
  F.3 G3a CKP-to-DFI conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
      F.3.1 G3a. CKP to Kloosterman-Fraction Reduction . . . . . . . . . . . . . . . . . 175
  F.4 CKP/X10 smooth-weight derivative appendix . . . . . . . . . . . . . . . . . . . . . . 179
      F.4.1 CKPD. CKP/X10 Smooth-Weight Derivative Check . . . . . . . . . . . . . . 179
  F.5 G4a DFI matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
      F.5.1 G4a. Exact Kloosterman Black-Box Matching . . . . . . . . . . . . . . . . . 185
  F.6 CKPX10M master CKP/X10 nonzero-frequency theorem . . . . . . . . . . . . . . . 193
      F.6.1 CKPX10M. Master CKP/X10 Nonzero-Frequency Theorem . . . . . . . . . . 193
  F.7 G8a CKP theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
      F.7.1 G8a. CKP Theorem and Zero-Frequency Normalization . . . . . . . . . . . . 197

G TC1, BRS, TTH, and X16 Package                                                                  204
  G.1 TNGTTHM master TC1 no-rogue-short-interval theorem . . . . . . . . . . . . . . . 204
      G.1.1 TNGTTHM. Master TC1 No-Rogue-Short-Interval Theorem . . . . . . . . . 204
  G.2 TC1 GoodAWACK dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
      G.2.1 TGD. Terminal GoodAWACK True-Complexity Split . . . . . . . . . . . . . 210
  G.3 TC1 global testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
      G.3.1 TGT. Aggregated Testing Route for TC1-GoodAWACK . . . . . . . . . . . . 214
  G.4 TGT-MF measured Fourier transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
      G.4.1 TGT-MF. Measured Fourier Transfer for TC1 Global Testing . . . . . . . . . 219
  G.5 TTH-SC structural coarea closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
      G.5.1 TTH-SC. Structural Coarea Closure and No Artificial Short-Interval Refinement223
  G.6 TC1 near-global chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
      G.6.1 TNG. B1-Origin TC1 Near-Global-or-Routed Theorem . . . . . . . . . . . . . 226
  G.7 TC1 testing dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


                                                  5
         G.7.1 TTD. TC1 Testing Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . 231
    G.8 MRT admissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
         G.8.1 MRT. PACK Interface for TC1 Global Testing . . . . . . . . . . . . . . . . . 236
    G.9 ROC singular-origin routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
         G.9.1 ROC. Range-Origin Lemma for Singular TC1 Testing . . . . . . . . . . . . . 238
    G.10 BRS range/slice closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
         G.10.1 BRS. B1 Range/Slice Closure for Singular TC1 Testing . . . . . . . . . . . . 244
    G.11 X16BRS carrier-slice reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
         G.11.1 X16BRS. Carrier-Slice Divisor Estimate for BRS . . . . . . . . . . . . . . . . 249
    G.12 X16-Core Shiu/AP proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
         G.12.1 X16C. Proof of the BRS Carrier-Slice Estimate . . . . . . . . . . . . . . . . . 251
    G.13 TTH near-global length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
         G.13.1 TTH. Internal Length Lower Bound for B1-Origin TC1 Coarea Tests . . . . . 259

H GoodAWACK Finite Grammar and Rank-Dropping AFF Closure                                           265
  H.1 E5-Clean Interface Imported from Appendix D.7 . . . . . . . . . . . . . . . . . . . . 265
  H.2 E10YMX master GoodAWACK finite-grammar theorem . . . . . . . . . . . . . . . . 266
       H.2.1 E10YMX. Master GoodAWACK Finite-Grammar Theorem . . . . . . . . . . 266
  H.3 E10L Branch B GoodAWACK theorem . . . . . . . . . . . . . . . . . . . . . . . . . 271
       H.3.1 E10L. Branch B GoodAWACK Theorem without X8 . . . . . . . . . . . . . . 271
  H.4 BGS skeleton normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
       H.4.1 BGS. Skeleton Normal Form for Terminal GoodAWACK Descendants . . . . 273
  H.5 HGO2 reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
       H.5.1 HGO2R. Reduction of BGS/HGO.2 to Free-Affine HighTC Exclusion . . . . 281
  H.6 BAOC affine origin catalogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
       H.6.1 BAOC. B1 Affine-Origin Catalogue . . . . . . . . . . . . . . . . . . . . . . . . 286
  H.7 E10G catalogue schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
       H.7.1 E10G. Strong BAOC Catalogue and Reduction . . . . . . . . . . . . . . . . . 293
  H.8 E10H matrix-origin reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
       H.8.1 E10H. Matrix Rigidity Reduction for Strong BAOC . . . . . . . . . . . . . . 300
  H.9 E10I matrix-origin rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
       H.9.1 E10I. Matrix-Origin Rigidity Verification . . . . . . . . . . . . . . . . . . . . 307
  H.10 E10J rank-dropping AFF origin verification . . . . . . . . . . . . . . . . . . . . . . . 312
       H.10.1 E10J. Rank-Dropping AFF Origin Verification . . . . . . . . . . . . . . . . . 312
  H.11 E10Y GoodAWACK routing grammar completeness . . . . . . . . . . . . . . . . . . 317
       H.11.1 E10Y. Completeness of the GoodAWACK Routing Grammar . . . . . . . . . 317
  H.12 E10M no untagged rank-dropping AFF . . . . . . . . . . . . . . . . . . . . . . . . . 327
       H.12.1 E10M. No Untagged Rank-Dropping AFF in Terminal GoodAWACK . . . . 327
  H.13 E10X finite GoodAWACK grammar theorem . . . . . . . . . . . . . . . . . . . . . . 334
       H.13.1 E10X. Finite GoodAWACK Grammar Closure . . . . . . . . . . . . . . . . . 334
  H.14 E10K affine-origin completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
       H.14.1 E10K. AFF-Origin Completeness . . . . . . . . . . . . . . . . . . . . . . . . . 340

I   Final Assembly and Handoff Details                                                              345
    I.1 I1 final weighted assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
        I.1.1 I1. Final Weighted Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
    I.2 G2 prime powers negligible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
        I.2.1 G2. Prime Powers Negligible Lemma . . . . . . . . . . . . . . . . . . . . . . . 349


                                                   6
   I.3   G1 weighted asymptotic to primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
         I.3.1 G1. Passage from Weighted Asymptotic to Strong Goldbach . . . . . . . . . 352
   I.4   G0 final handoff verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
         I.4.1 G0H. Final Handoff from I1/G2 to Strong Goldbach . . . . . . . . . . . . . . 355

J Dependency Ledger and Synchronization Notes                                                      357
  J.1 Proof tree and ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

K Bibliography                                                                                       373

References                                                                                           376




                                                    7
1     Introduction and Statement of Results
1.1    Introduction and Relation to Known Work
The binary Goldbach problem asks whether every even integer greater than 2 is a sum of two primes.
Hardy and Littlewood’s circle-method heuristic predicts the asymptotic

                                               Λ(𝑛1 )Λ(𝑛2 ) ∼ S(𝑁 )𝑁
                                     ∑︁

                                   𝑛1 +𝑛2 =𝑁

with the usual Goldbach singular series. Vinogradov’s theorem proves the corresponding ternary
assertion for all sufficiently large odd integers, Chen’s theorem proves that every sufficiently large
even integer is a sum of a prime and an integer with at most two prime factors, and Helfgott’s work
completes the weak Goldbach theorem. These results provide the historical background, not proof
inputs for the present argument.
    The binary problem is more delicate than the ternary problem because the expected main term
has only one degree of freedom, and the minor-arc or off-diagonal cancellation must be reconciled
with the exact local main term. The present proof therefore does not import a completed circle-
method theorem for binary Goldbach. Its active external inputs are instead limited to the specific
theorem invocations recorded in the external-input proof units: Heath–Brown’s fixed-depth identity,
Davenport’s near-global AP/Liouville estimate, the Duke–Friedlander–Iwaniec bilinear Kloosterman-
fraction estimate, Shiu’s Brun–Titchmarsh theorem for multiplicative functions, and the fixed
divisor-function second moment cited in the X16 proof.
    The present proof is organized around a finite decomposition-and-routing architecture. A fixed-
depth Heath–Brown identity expands the two von Mangoldt factors into finitely many typed product
variables. Each product block is then partitioned by deterministic routing operations. The routing
theorem is designed so that every terminal atom belongs to a class with a specified analytic or local
mechanism.
    The five terminal classes are:

    1. Edge, where strict size, conductor, boundary, or square-divisor savings give 𝑜(𝑁 );

    2. CKP, where a balanced bilinear Kloosterman-fraction structure is present;

    3. GoodAWACK, where the remaining Branch B atoms are handled by the TC1 global testing
       route and the E10Y-certified finite GoodAWACK routing grammar;

    4. LongAP/Local, where no nonlocal arithmetic coefficient survives and the term satisfies the
       explicit LPI/H4M tagged local projection condition;

    5. LocalDiag, where forced local dependence is admitted only when it is the same tagged LPI
       local projection later assembled through H4M.

    Two features distinguish the route from a direct circle-method argument. First, the proof does
not try to estimate all descendants with one universal analytic theorem. It proves a complete finite
classification and uses different tools on the different terminal classes. Second, the proof avoids the
earlier inverse-Gowers route X8. The GoodAWACK branch is instead closed by the combination of
a TC1 global testing dichotomy and a structural finite-grammar theorem for rank-dropping affine
regroupings.
    The most delicate points are the completeness of the terminal routing, the CKP/X10 smooth-
weight match, the X16/Shiu carrier-slice estimate used inside the TC1 route, the GoodAWACK


                                                    8
finite-grammar closure, and the local-main reconstruction. These points are isolated as named
lemmas and appendices rather than hidden in the final assembly.
    The bibliography is correspondingly split into active proof inputs and historical/orientation
references. The latter locate the proof relative to the classical Goldbach literature but are not used
as logical dependencies in the proof ledger.

1.2      Main Theorems and Proof Strategy
1.2.1     Weighted Form
Let

                                        𝑅Λ (𝑁 ) =               Λ(𝑛1 )Λ(𝑛2 ).
                                                         ∑︁

                                                    𝑛1 +𝑛2 =𝑁

      The central theorem is Theorem 1.2:

                                          𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )

      for sufficiently large even 𝑁 .
      The singular series is
                                         ∏︁ 𝑝 − 1                    ∏︁ (︂           1
                                                                                         )︂
                          S(𝑁 ) = 2𝐶2                ,        𝐶2 =           1−            .
                                         𝑝|𝑁
                                               𝑝−2                   𝑝>2
                                                                                  (𝑝 − 1)2
                                         𝑝>2

      Since S(𝑁 ) ≥ 2𝐶2 > 0 for even 𝑁 , the weighted asymptotic is eventually positive.

1.2.2     Decomposition and Routing
Lemma B1 gives an exact finite Heath–Brown decomposition of both factors in 𝑅Λ (𝑁 ). Lemmas
B3, F3, F4, E5, and F3T convert every B1-origin atom into a finite tagged sum of terminal atoms
in exactly the five terminal classes

                     Edge,    CKP,      GoodAWACK,             LongAP/Local,          LocalDiag.

   The tagged partition is exact; overlaps between visual algebraic shapes do not create double
counting because the routing history is part of the terminal tag.

1.2.3     Terminal Estimates
The terminal estimates are:

  1. Edge terms first satisfy one of the strict C1P saving predicates; C1A verifies admission of all
     active Edge inputs, and C1 proves their total contribution is 𝑜(𝑁 ).

  2. CKP terms equal the explicit LPI/H4M tagged local projection plus 𝑜(𝑁 ) by G8a, CKPX10M,
     and B1LD.

  3. GoodAWACK terms are 𝑜(𝑁 ) by E10L, TNG, X9L-GT, X16BRS/X16C, E10Y, E10M, E10X,
     and E10K.


                                                          9
    4. LongAP/Local terms satisfy the LPI/H4M tagged admission condition plus 𝑜(𝑁 ) by D1 and
       H4M.
    5. LocalDiag terms are admitted only when they are tagged LPI local projections later assembled
       through H4M.

   The global error-budget lemma GEB proves that these estimates remain 𝑜(𝑁 ) after all dyadic
and routing summations.

1.2.4    Assembly
The local projections admitted by D1, G8a/B1LD, and LocalDiag are all passed through H4M. H4M
imports the H4 Λ𝑄 -model, tagged linearity over the B1/F3 partition, no double counting, and the
CRT local factor computation to prove

                                     𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Combining this with the terminal error bounds gives Theorem 1.2. Lemmas G2, G1, and G0H
then remove nontrivial prime powers and convert positivity into a prime representation.

1.3     Full-Manuscript Integration Note
The proof-level details behind the weighted asymptotic and the final handoff are placed in Appendix
I. The main body uses the following logical route:

                                       𝐼1 + 𝐺2 =⇒ 𝐺1 =⇒ 𝐺0.

  The terminal decomposition and branch estimates are proved in Sections 4–8 and Appendices
C–H.


2     Glossary of Internal Labels and Nonstandard Terms
This glossary fixes the internal vocabulary used in the proof. It is a reader aid, not an additional
hypothesis and not a substitute for the formal statements in the underlying proof text. Standard
analytic abbreviations such as AP, CRT, DFI, and GVN are included only when they are used as
named proof interfaces.

2.1     Terminal Classes
    • Edge: a terminal class carrying a strict C1P saving predicate. Edge terms are proved to be
      o(N) by C1 after C1A verifies that the relevant C1P certificate is present.
    • CKP: the balanced bilinear Kloosterman-fraction branch. Its zero-frequency part is local and
      its nonzero-frequency part is matched to the DFI/X10 estimate after the CKPD smooth-
      weight verification.
    • GoodAWACK: the Branch B affine/global-testing residual class. It is handled by the TC1 near-
      global-or-routed route and by the finite GoodAWACK grammar closure.
    • LongAP/Local: the terminal long arithmetic-progression/local class. F3P gives the positive
      local-coefficient predicate, D1 expands the resulting local algebra into the tagged LPI projection,
      H4 evaluates the local algebra, and H4M supplies the final local bridge into I1.

                                                   10
  • LocalDiag: the terminal local-diagonal class, where forced equality, proportionality, or local
    dependence is admitted only as a tagged LPI local projection later assembled through H4M.

2.2   Structural Terms
  • B1-originatom: a cell descended from the fixed-depth Heath–Brown decomposition in B1.
    Several later theorems apply only to such actual descendants.

  • Branch B: the branch of the routing tree that contains the GoodAWACK residuals after the
    earlier Edge, CKP, LongAP/Local, and LocalDiag alternatives have been tested.

  • tagged cell: a cell together with its complete B1/F3 routing tag. Tags prevent double
    counting and identify the correct local projection.

  • terminal atom: a routed atom after F3/F4 has assigned one of the terminal classes.

  • coarea test: a TC1 testing family obtained from an actual B1-origin affine range/fibre, not
    an arbitrary shifted interval.

  • near-global: a TC1 test whose image length is at least a polylogarithmic fraction of the
    ambient scale, so that X9L-GT applies.

  • carrier slice: a BRS/X16 slice in which one product carrier is fixed and the remaining
    Goldbach-complement divisor correlation is estimated.

  • HighTC: the higher true-complexity part of the GoodAWACK branch.

  • FreeAffineHighTC: the formal high-complexity affine obstruction that must not survive as
    an untagged actual terminal GoodAWACK skeleton.

  • AFF: affine map or affine regrouping, usually in the GoodAWACK finite-grammar and rank-
    dropping discussion.

  • rank-droppingAFF: an affine operation that lowers the rank or true-complexity data relevant
    to the terminal GoodAWACK skeleton.

  • AFF-OC: affine-origin completeness, the assertion that any rank-dropping affine operation
    appearing in an actual terminal GoodAWACK skeleton has an allowed origin tag.

  • MOR: matrix-origin rigidity, the E10 reduction that isolates which formal matrix-origin witnesses
    can arise from actual descendants.

  • RDA: rank-dropping AFF verification, the E10 reduction delegated to the finite GoodAWACK
    grammar closure.

  • Lambda_Q-localprojection: the finite-modulus local model defined by LPI, evaluated by
    H4, and imported into the final assembly through H4M.

  • H4M: the master local bridge theorem. It packages F3F4M, LPI, D1, B1LD, G8a, and H4 to
    prove that the complete admitted local/main contribution is exactly S(𝑁 )𝑁 + 𝑜(𝑁 ), with no
    independent M_otherlocalclass.

  • WACLE: weak affine-linear coefficient expression; an affine finite-convolution coefficient pattern
    used in the routing grammar.

                                                 11
  • MixedResidual: the would-be residual class left after Edge, CKP, GoodAWACK, LongAP/
    Local, and LocalDiag. F3/F4 prove that this class is empty.

  • NoFAH: no-free-affine-HighTC assertion; in the final proof it is supplied by the E10YMX master
    finite-grammar closure.

  • FixedDiv: routing tag for a fixed-divisor quotient or fixed divisibility restriction.

  • VarQuot: routing tag for a variable quotient residual created by an allowed quotient step.

  • PostTerminalNonGenerator: tag for post-terminal analytic slicing or testing operations that
    estimate a fixed terminal skeleton but do not generate a new terminal GoodAWACK skeleton.

2.3   Root, Assembly, and Bookkeeping Labels
  • G0: the final strong Goldbach target for all sufficiently large even integers.

  • G0H: final handoff verification from the weighted asymptotic and prime-power removal to G0.

  • G1: converts positivity of the genuine prime-pair weighted sum into a prime representation.

  • G2: prime-power removal; it shows that nontrivial prime powers are negligible.

  • I1: final weighted assembly proving the von Mangoldt asymptotic.

  • PAR: parameter register and order of constant choices.

  • GEB: global error budget; it records summability of terminal errors and polylogarithmic losses.

2.4   Decomposition, Routing, Edge, and Local Labels
  • B1: typed Heath–Brown decomposition of the two von Mangoldt factors.

  • B3: finite block classification and product-grouping candidates.

  • F3: finite routing partition and routing operations.

  • F3A: F3 routing-interface completeness check.

  • F3T: complete finite routing table from B1/B3/F3P/F3/F4 descendants to terminal classes.

  • F4: large-divisor, quotient, and quotient-tag routing.

  • E5: content stability under the affine/regrouping transports used in Branch B.

  • E1–E7: the strict Edge predicates defined by C1P.

  • C1P: strict Edge predicate catalogue; it defines Edge independently of X10/BRS/X16 branch
    estimates.

  • C1A: Edge admission ledger; it verifies that every active Edge input has a strict C1P predicate.

  • C1: unified Edge estimate proving C1P-Edge terms are o(N).

  • LPI: local projection interface; it defines Lambda_Q, Loc_Q, H4-admissibility, and proves that
    there is no independent residual local-projection source.

                                                12
  • D1: LongAP/Local theorem expanding F3P-local coefficient atoms into the LPI local projection.

  • H4: local/main assembly theorem reconstructing the tagged local Goldbach model and the
    singular series.

  • B1LD or B1-LD: B1 local-density compatibility for CKP zero-frequency local terms.

  • F3-COMPLETE: the completeness assertion that the F3/F4 routing grammar exhausts the
    relevant terminal possibilities.

2.5   CKP Branch Labels
  • G1a: exact CKP gcd splitting.

  • G2a: smooth AP/Fourier expansion for CKP frequencies.

  • G3a: conversion of central CKP nonzero frequencies to DFI Kloosterman-fraction sums.

  • CKPD: CKP/X10 smooth-weight derivative appendix; it verifies the actual two-variable weights.

  • G4a: application of the DFI/X10 estimate after CKPD supplies the derivative hypotheses.

  • CKPX10M: master CKP/X10 nonzero-frequency theorem packaging G3a, CKPD, G4a, X10,
    and X10ER into o(N) cancellation.

  • G8a: CKP branch theorem, giving local projection plus o(N).

  • X10ER: CKP excluded-range record; it routes high-frequency, small-conductor, large-gcd/
    content, boundary, and short-volume ranges away from X10.

  • DFI-X10: the displayed dyadic form of the Duke–Friedlander–Iwaniec estimate used by X10.

2.6   TC1, BRS, X16, and GoodAWACK Labels
  • E10L: GoodAWACK branch theorem proving the terminal GoodAWACK contribution is o(N).

  • TGD: TC1/HighTC dichotomy for GoodAWACK.

  • TC1: true-complexity-one testing route for GoodAWACK residuals.

  • TGT-MF: measured Fourier transfer from global TC1 (Uˆ2)-obstruction to a probability family
    of Liouville tests.

  • TGT: TC1 global testing theorem.

  • TTH-SC: structural coarea closure barrier preventing rogue short-interval refinements.

  • TNGTTHM: master TC1 no-rogue-short-interval theorem packaging TGT-MF, TGT, TTH-SC,
    MRT, TTD, ROC, BRS, X16BRS, X16C, TTH, and X9L-GT.

  • TNG: near-global-or-routed TC1 theorem package.

  • TNG-A: the main theorem inside TNG: every active unrouted TC1 test is near-global or routed
    away.



                                              13
• TTD: TC1 testing dichotomy separating regular and singular testing measures.

• MRT: admissibility and PACK selection for regular TC1 testing families.

• PACK: the selected polylogarithmic testing family in the regular TC1 route.

• ROC: singular-origin routing check for TC1 tests.

• BRS: B1 range/slice closure for singular or complementary short-image TC1 alternatives.

• X16BRS or X16-BRS: reduction of BRS carrier-slice alternatives to X16-Core.

• X16C: X16-Core Shiu/AP proof for product-carrier divisor correlations.

• X16-SH: Shiu/AP divisor-function estimate used inside X16C.

• X16-LFA: local-factor averaging lemma used in X16C to handle non-coprime AP classes.

• TTH: TC1 theta/near-global length theorem supplying the length needed by X9L-GT.

• BGS: B1-to-GoodAWACK skeleton normal form.

• HGO or HGO.2: HighTC GoodAWACK obstruction theorem targeted by the BGS normal form.

• HGO2R: HighTC GoodAWACK obstruction rerouting reduction.

• BAOC: bounded affine-origin catalogue.

• E10: umbrella label for the GoodAWACK closure family E10G–E10Y.

• E10G: strong BAOC catalogue schema and formal obstruction catalogue.

• E10H: matrix-origin reduction.

• E10I: matrix-origin rigidity reduction.

• E10J: rank-dropping AFF origin verification reduction.

• E10Y: finite GoodAWACK routing-grammar completeness theorem.

• E10M: no-untagged-rank-dropping-AFF theorem for actual terminal skeletons.

• E10X: master finite GoodAWACK grammar closure theorem.

• E10YMX: reader-facing master theorem packaging E10Y, E10M, and E10X into the HighTC
  finite-grammar closure.

• E10K: affine-origin completeness theorem used by E10X and E10L.




                                            14
2.7    External and Standard Interfaces
    • X1: Heath–Brown identity used by B1.

    • X2: smooth partition of unity used in B1/B3.

    • X3: Type I or short-variable estimates used inside strict Edge bounds.

    • X4: CRT and finite local-density algebra.

    • X5: Cauchy–Schwarz/GVN machinery, used only in standard forward form.

    • X6: lattice/content algebra.

    • X9, X9L, or X9L-GT: Davenport/AP near-global Liouville orthogonality input used after TTH.

    • X9L-SI: the unused pointwise shifted-interval formulation; the active proof uses X9L-GT
      instead.

    • X10: Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction estimate used in CKP.

    • X11: smooth Fourier/AP expansion.

    • X12: elementary prime-power bound.

    • X13: Euler-product and singular-series algebra.

    • X14: gcd algebra used in CKP splitting.

    • X15: smooth Fourier decay.

    • X16: Shiu/AP divisor-average input; the BRS-specific product-carrier proof is X16C.

    • AP: arithmetic progression.

    • CRT: Chinese remainder theorem.

    • DFI: Duke–Friedlander–Iwaniec Kloosterman-fraction estimate.

    • GVN: generalized von Neumann inequality or Cauchy–Schwarz/GVN transfer step.


3     Dependency Tree and Reading Map
This diagram is a reader map for the active proof dependencies. An arrow A-->Bmeans that A is
used as an input for proving, routing, or assembling B. The authoritative parent/child table remains
the proof ledger; this diagram is a compact visual index.




                                                  15
                                                                                                                   X12
                                                                   C1
      PAR     GEB                                    C1A
                                                                  Edge                                             G2
                                                                                                              prime powers
                                                                                                                                           G0
                                                                                                                             G1   G0H
                            F3                                                                       I1                                 Goldbach
     X1, X2   B1    B3              F4                D1
                         F3A, F3T                                  H4                        weighted asymptotic
                                               LPI             local/main
                                         local interface
                                                      B1LD

                                                                  CKPD
                                                     G1a          X10
                                                                                   G4a
                                                     G2a
                                                                                   G8a
                                                     G3a
                                                                  X10ER
                                                             excluded ranges
                                                                                                   E10Y
                                                                  BGS
                                                                                                   E10M
                                                      E5         HGO2R          E10G–E10J
                                                                                                   E10X
                                                                  BAOC
                                                                                                   E10K
                                                                                                                 E10L
                                                                                                              GoodAWACK
                                                                                   TNG
16




                                                                                 TGT+MRT
                                                                               TTD+ROC+BRS
                                                                  X16
                                                                                                   TTH
                                                                  X16C
                                                                                                  X9L-GT
                                                                 X16BRS
    Text fallback for PDF readers:

G0
+-- G0H
    +-- G1
    +-- G2
    +-- I1
+-- G1
    +-- G2
    +-- I1
        +-- PAR -> GEB
        +-- Decomposition/routing:
        |   +-- B1/X1/PAR -> B3 -> F3P -> F3/F3A/F3T -> F4
        |   +-- F3F4M master routing theorem packages the F3/F4 partition
        |   +-- E5 stability and transport compatibility
        +-- Edge:
        |   +-- C1P -> C1A -> C1
        +-- Local/Main:
        |   +-- F3F4M + LPI -> D1 + B1LD + G8a + H4 -> H4M
        +-- CKP:
        |   +-- G1a -> G2a -> G3a -> CKPD + G4a/X10 -> CKPX10M
        |   +-- CKPX10M routes excluded nonzero CKP ranges through X10ER -> C1P/C1A/C1
        |   +-- h=0 goes through G8a/LPI
        |   +-- G8a + B1LD -> H4M
        +-- GoodAWACK:
            +-- E10L
                 +-- TGD
                 +-- TC1: TNGTTHM = TGT-MF + TGT + TTH-SC + MRT + TTD + ROC + BRS
                 |        + X16BRS/X16C + TTH + X9L-GT
                 +-- HighTC/grammar:
                          BGS + HGO2R + BAOC + E10G/E10H/E10I/E10J
                          + E10YMX = E10Y + E10M + E10K + E10X
                          + E5-clean interface imported from the E5 master proof


4     Parameters, Notation, and Error Bookkeeping
4.1     Notation, Parameters, and Error Budget
Throughout, 𝑁 is an even integer tending to infinity and

                                           𝐿 = log 𝑁.

   All implicit constants may depend on the fixed decomposition depth and on the fixed smooth
cutoffs, but not on 𝑁 .

4.1.1    Fixed Depth and Dyadic Partitions
The Heath–Brown depth is fixed once and for all. We use a parameter 𝐽0 large enough for the
global hierarchy. The explicit consistency witness used in the parameter register is

                                                1                 1
                               𝐽0 = 20,    𝜂=      ,       𝜃=        .
                                                40              4000



                                               17
   The witness is not optimized. Its role is to show that all inequalities in the parameter hierarchy
can be satisfied simultaneously.
   Every variable produced by the B1 decomposition is smoothly dyadically localized. For fixed 𝐽0 ,
the total number of active dyadic and routing cells is 𝐿𝑂(1) .

4.1.2   Terminal Classes
The proof uses the following terminal classes.


   Class                           Meaning                          Output
   Edge                            strict saving, boundary, short   𝑜(𝑁 )
                                   volume, square-divisor, or con-
                                   ductor loss
   CKP                             balanced Kloosterman-fraction local projection +𝑜(𝑁 )
                                   branch
   GoodAWACK                       Branch B affine/global-testing 𝑜(𝑁 )
                                   branch
   LongAP/Local                    long local arithmetic progres-   local projection +𝑜(𝑁 )
                                   sion branch
   LocalDiag                       forced local dependence or diag- local projection
                                   onal branch




4.1.3   Global Error Budget
GEB records the summability principle used throughout the proof. If a per-terminal estimate
gains either a fixed power of 𝑁 or a sufficiently large negative power of 𝐿, then the 𝐿𝑂(1) terminal
multiplicity is harmless.
    The constants are chosen in the following order:

  1. fix 𝜃 ≪ 𝜂;

  2. choose 𝐽0 ≥ 𝐽* (𝜂);

  3. fix the routing multiplicity constant 𝐶0 (𝐽0 );

  4. fix the Edge and divisor losses;

  5. choose the X16/Shiu logarithmic floor exponent;

  6. choose the TC1 near-global modulus and AP exponents;

  7. choose CKP high-frequency and DFI derivative thresholds.

  With this order, later logarithmic exponents can always dominate earlier polylogarithmic losses.
GEB is invoked only as a bookkeeping lemma; it does not replace the branch estimates.




                                                  18
4.2     Parameter Hierarchy and Global Error Budget
4.2.1    Parameter Witness
The proof fixes parameters in a nonempty hierarchy. One explicit witness is
                                                  1              1
                                𝐽0 = 20,     𝜂=      ,    𝜃=        .
                                                  40           4000
   For this witness 𝐽0 ≥ 𝐽* (𝜂). After 𝐽0 is fixed, all later logarithmic exponents are chosen in the
order recorded by GEB.

4.2.2    Multiplicity Principle
The B1/B3/F3/F4 process creates at most 𝐿𝐶0 terminal cells. Hence:

  1. 𝑂(𝑁 𝐿−𝐶0 −𝐴 ) per cell sums to 𝑂(𝑁 𝐿−𝐴 );

  2. 𝑂(𝑁 1−𝜌 𝐿𝐶 ) per cell sums to 𝑜(𝑁 );

  3. a normalized 𝑜(1) estimate remains 𝑜(1) after polylogarithmic testing-family losses, once the
     relevant logarithmic exponent is chosen large enough.

4.2.3    Global Loss Table

   Source                          Estimate                         Global conclusion
   B1 decomposition                exact identity                   no error
   F3/F4 routing                   exact tagged partition           no error
   Edge                            C1 strict saving                 𝑜(𝑁 )
   LongAP/Local                    LPI local projection plus Edge   local +𝑜(𝑁 )
                                   errors
   CKP excluded ranges             Edge/local/auxiliary routing     𝑜(𝑁 ) or local
   CKP central nonzero range       CKPX10M packages X10 after       𝑜(𝑁 )
                                   CKPD
   GoodAWACK TC1 regular           X9L-GT after TTH                 𝑜(𝑁 )
   GoodAWACK singular              BRS/ROC rerouting                handled by existing branches
   X16/BRS carrier slice           Shiu/AP carrier estimate         Edge or 𝑜(𝑁 )
   HighTC/grammar                  E10YMX consumes E10Y/            no residual
                                   E10X/E10M/E10K
   LPI/H4 local projection         CRT local model                  S(𝑁 )𝑁 + 𝑜(𝑁 )




4.2.4    Consequence
After summing over the complete terminal partition,

                                    𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   No hidden terminal-family summation remains outside GEB.
   The prime-power removal is not a terminal branch summation. It is performed after I1 by G2,
which gives 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑜(𝑁 ).

                                                 19
4.3     Full-Manuscript Integration Note
The complete parameter register and global loss table are proved in Appendix A. The concrete
consistency witness retained throughout the manuscript is

                              𝐽0 = 20,      𝜂 = 1/40,      𝜃 = 1/4000.


5     External Analytic Inputs
5.1     External Inputs
The proof uses four external inputs, each in a fixed stated form.

5.1.1    X1: Heath–Brown Identity
X1 is the fixed-depth Heath–Brown identity used to decompose each von Mangoldt factor. The
active formulation is the one needed by B1: after choosing 𝐽0 , Λ is represented by a finite linear
combination of typed divisor-convolution factors with controlled coefficients and smooth dyadic
localization.

5.1.2    X9L-GT: Near-Global Liouville/AP Orthogonality
X9L-GT is the Davenport/AP input used only after the TC1 route has produced near-global active
B1-origin coarea tests. It is not invoked on arbitrary shifted short intervals. TTH supplies the near-
global length condition

                                         𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

    The TC1 testing family has only polylogarithmic modulus and AP complexity, so the Davenport
saving can be chosen to dominate the recorded losses.

5.1.3    X10: DFI Kloosterman-Fraction Estimate
X10 is the Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction estimate in the smooth weighted
form required by CKP. The CKP branch uses X10 only through CKPX10M, after G1a–G3a reduce
the nonzero-frequency terms to the DFI form and CKPD verifies the actual two-variable smooth
weight.
   No later Kloosterman-fraction strengthening is an active dependency.

5.1.4    X16: Shiu/AP Divisor-Average Input
X16 is the Shiu-type arithmetic-progression divisor-average input used in the BRS carrier-slice step
of the TC1 route. X16C proves the active normalized carrier estimate, including the divisor-function
second moment and the logarithmic floor needed by GEB.

5.1.5    Citation Boundary
Each external theorem is invoked only through its named stated interface: X1, X9L-GT, X10, and
X16. The manuscript does not appeal to informal variants of these results.



                                                 20
5.2     Full-Manuscript Integration Note
The citation-grade external inputs are stated and matched in Appendix B. The active external
inputs are X1, X9, X10, and X16.


6     Heath–Brown Decomposition and Typed Blocks
6.1     Heath–Brown Decomposition
6.1.1    B1 Blocks
Lemma B1 applies the fixed-depth Heath–Brown identity to both copies of Λ in 𝑅Λ (𝑁 ). It writes
𝑅Λ (𝑁 ) as a finite sum of B1 blocks. Each B1 block consists of two finite lists of product variables,
smooth dyadic cutoffs, and elementary coefficient types

                                      𝜇 · 1≤𝑦 ,    1,     log .

   For fixed 𝐽0 , the number of elementary variables in a B1 block is bounded, and the number of
dyadic cells is 𝐿𝑂(1) .

6.1.2    Exactness
The B1 decomposition is an identity before estimates are applied. Smooth dyadic partitions are
inserted as exact partitions of unity up to boundary terms already routed to Edge. Thus no main
term is lost at the decomposition stage.
Remark 6.1 (Structural Output). The output of B1 is a structural finite-convolution problem
together with the equation inherited from 𝑛1 + 𝑛2 = 𝑁 . The routing layer decides which grouped
variables, divisor relations, local congruences, and oscillatory features survive to terminal atoms.
    The subsequent proof uses only B1-origin atoms. This origin condition is important in the
GoodAWACK finite-grammar argument: formal affine systems that do not arise from the B1/B3/
F3/F4 grammar are not terminal descendants of the proof.

6.2     Full-Manuscript Integration Note
The proof-level B1/B3 derivation is in Appendix C. This section supplies the article-level bridge
from the weighted Goldbach sum to the finite typed dyadic block family.


7     Routing Grammar and Terminal Classes
7.1     Routing Exhaustion
7.1.1    Finite Grouping
Lemma B3 supplies a finite set of product-grouping candidates for each B1 block. Candidate labels
may overlap at this preliminary stage; they are not yet terminal classes.




                                                  21
7.1.2    Authoritative Routing Operations
Lemma F3 supplies the complete routing operations for actual F3 atoms. These include controlled
CRT absorption, ordinary divisor decisions through F4, square-divisor routing, grouping selection
or elimination, terminal Edge detection, terminal LocalDiag detection, and final terminal labelling.
    Lemma E5 is used only as content stability for transports generated by the routing grammar. It
is not an independent generator of terminal affine systems.

7.1.3    Exhaustion Theorem
Lemma F3T proves the finite routing exhaustion theorem:

        B1 + B3 + F3 + F4 + E5 =⇒ {Edge, CKP, GoodAWACK, LongAP/Local, LocalDiag}.

    More precisely, each B1-origin atom is partitioned into a finite disjoint sum of tagged terminal
atoms in exactly those five classes. No sixth terminal class remains.
    The proof uses a deterministic routing order and a well-founded routing measure M♯ . Nonterminal
operations strictly decrease this measure, while terminal rows assign one of the five tags. If a cell
visually satisfies more than one terminal predicate, the tag records the first applicable class in the
deterministic order. Therefore visual overlap does not imply double counting.

7.1.4    Terminal Interfaces
Once F3T has assigned terminal tags, the estimates are supplied by the corresponding branch
lemmas:


Edge → 𝐶1𝑃/𝐶1𝐴/𝐶1,    CKP → 𝐺8𝑎,   GoodAWACK → 𝐸10𝐿,      LongAP/Local → 𝐷1/𝐻4𝑀,     LocalDiag → 𝐻4𝑀.


7.2     Routing Tables
This appendix records the routing table used by Lemma F3T. The terminal predicates themselves
are the intrinsic predicates fixed in Lemma F3P; the branch estimates are invoked only after the
routing tag has been assigned.

7.2.1    Terminal Classes

   Class                           Entry condition                   Exit theorem
   Edge                            strict C1P saving predicate       C1P/C1A/C1
   LocalDiag                       forced local dependence or diag- H4M
                                   onal relation
   LongAP/Local                    F3P long AP/local fibre whose D1/H4M
                                   long-variable coefficients lie in
                                   the local coefficient algebra
   CKP                             balanced bilinear Kloosterman- CKPX10M/G8a
                                   fraction structure
   GoodAWACK                       Branch B affine/global-testing E10L
                                   residual with controlled origin




                                                 22
7.2.2   Canonical Routing Order
On each tagged B1-origin cell, F3T reads the following order.

  1. Empty or incompatible cells are zero.

  2. Strict C1P saving predicates go to Edge.

  3. Forced equality, proportionality, repeated factor, or local dependence goes to LocalDiag.

  4. Ordinary divisor and quotient predicates are decided by F4.

  5. Square-divisor obstructions are routed by F3.

  6. Controlled CRT restrictions are absorbed only when E5 verifies clean content stability.

  7. Remaining grouping alternatives are selected or eliminated from the finite B3 grouping set.

  8. The terminal predicate assigns one of the five terminal classes.

   Every nonterminal step strictly decreases M♯ . Therefore the process terminates.

7.2.3   Routing Table

  Row                     Source regime           Routing outcome       Reason no other termi-
                                                                        nal class receives the
                                                                        cell
  1                       empty support or in-      Edge-zero           no analytic mass re-
                          compatible congruences                        mains
  2                       boundary, short vol-      Edge                a strict C1P saving
                          ume, large content,                           predicate is present
                          square-divisor tail, high
                          Fourier tail, small con-
                          ductor, or Type I sav-
                          ing
  3                       forced equality, pro-     LocalDiag           independent CKP/
                          portionality, repeated                        GoodAWACK variables
                          factor, or local depen-                       are absent
                          dence
  4                       one long AP variable      LongAP/Local        the positive F3P pred-
                          and F3P local coeffi-                         icate already excludes
                          cient algebra                                 nonlocal long-variable
                                                                        coefficients; D1.2A ex-
                                                                        pands the resulting
                                                                        local algebra
  5                       central balanced bi-    CKP                   Edge/LocalDiag have
                          linear Kloosterman-                           already failed; the struc-
                          fraction form                                 ture is bilinear, not
                                                                        GoodAWACK




                                                 23
    6                     nonlocal non-CKP         GoodAWACK                E10YMX excludes un-
                          Branch B affine residual                          tagged rank-dropping
                          with controlled origin                            AFF residuals using the
                                                                            finite grammar inputs
                                                                            E10Y/E10X/E10M/
                                                                            E10K
    7                     ordinary divisor predi- Edge                      F4 supplies a C1P pred-
                          cate with short quotient                          icate
                          or saving
    8                     ordinary divisor predi- LocalDiag                 quotienting identifies
                          cate forcing local depen-                         active forms
                          dence
    9                     ordinary divisor pred- CKP                        F4 removes local and
                          icate preserving bal-                             short alternatives
                          anced bilinear structure
    10                    ordinary divisor pred- GoodAWACK                  quotient origin is
                          icate preserving con-                             tagged and E5-clean
                          trolled Branch B affine
                          residual
    11                    large square divisor      Edge                    square-divisor saving
                                                                            applies
    12                    small controlled square nonterminal decrease      F3 continues with
                          divisor or full-rank                              smaller M♯
                          CRT restriction
    13                    unresolved finite group- nonterminal decrease or B3 grouping set is finite
                          ing alternative          terminal row



    Rows 12–13 are not terminal rows. They are included to show that all nonterminal transitions
are among the allowed F3 operations.

7.2.4     Consequence
For fixed 𝐽0 , every B1-origin atom reaches exactly one tagged terminal class after finitely many
steps. This is the routing-exhaustion input used in the assembly theorem.

7.3      Full-Manuscript Integration Note
The exact routing grammar is proved in Appendix D. The routing table is used only as a finite
partition theorem: every B1/B3 descendant has one terminal destination, with no duplicate
contribution.


8        Edge and LongAP/Local Terms
8.1      Edge Estimates
8.1.1     Edge Predicates
Edge terms are terminal cells carrying a strict saving predicate. Lemma C1P defines these predicates
before any late branch estimate is invoked. The active predicates include boundary or short-volume

                                                 24
loss, Type I saving, large gcd/content, square-divisor tails, high Fourier tails, and small-conductor
layers.
    Lemma C1 proves that any cell satisfying one of the strict C1P Edge predicates contributes
𝑜(𝑁 ) after its allowed coefficient and polylogarithmic losses.

8.1.2    Admission
Lemma C1A proves the complementary admission statement needed by the manuscript: every
active terminal atom routed into Edge satisfies one of the strict C1P predicates. Thus Edge is not a
residual label. It is a verified saving class.
    The admission ledger treats the Edge sources from F3/F4, CKP excluded ranges, BRS carrier
slices, square-divisor routing, boundary terms, and high-frequency tails. Each source is paired with
a saving estimate and a summability check.

8.1.3    Summation
For fixed 𝐽0 , the number of Edge cells is 𝐿𝑂(1) . C1 supplies either a fixed power saving in 𝑁 or a
logarithmic saving chosen larger than the routing multiplicity. GEB therefore gives

                                         𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

8.2     LongAP/Local Terms
8.2.1    Terminal LongAP/Local Atoms
A LongAP/Local atom is a terminal B1-origin cell satisfying the intrinsic F3P LongAP/Local
predicate: its long-variable coefficients lie in the controlled local coefficient algebra on a long
arithmetic-progression fibre. It is not allowed to contain a surviving nonlocal 𝜇-, 𝜆-, Fourier-,
Kloosterman-, or nilsequence-type coefficient.

8.2.2    Exclusion of Nonlocal Coefficients
Lemma D1.2A proves that the F3P LongAP/Local predicate has the advertised consequence inside
the routed B1/F3 partition: no nonlocal arithmetic coefficient survives in a terminal LongAP/Local
atom. The proof combines the positive F3P predicate with the finite F3/F4 routing alternatives:

  1. a surviving oscillatory or Liouville/Mobius-type factor routes to GoodAWACK or CKP;

  2. a strict saving predicate routes to Edge;

  3. forced local dependence routes to LocalDiag;

  4. only local AP/congruence data remains in LongAP/Local.

   Thus D1 may replace F3P-LongAP/Local atoms by the explicit H4 tagged local projections
without discarding hidden arithmetic oscillation.
Remark 8.1 (Output). Lemma D1 gives, for each LongAP/Local terminal cell,

                          𝑅LongAP/Local (𝑁 ) = 𝑀LongAP/Local (𝑁 ) + 𝑜(𝑁 ),

   where the main term is the H4 Λ𝑄 -projection of the same tagged B1/F3 cell. Boundary and
smoothing errors are Edge-admitted.

                                                 25
8.3     Edge Admission
This appendix records the three-layer Edge interface: C1P defines the strict Edge predicates, C1A
verifies that every active Edge input satisfies one of them, and C1 estimates the admitted terms.

8.3.1    Edge Predicates
The active C1P predicates are:

  1. boundary or short-volume loss;

  2. Type I saving;

  3. large gcd or large content;

  4. large square-divisor tail;

  5. high Fourier tail;

  6. small conductor;

  7. incompatible or zero support.

   The exact numerical exponents are chosen inside the PAR/GEB hierarchy.

8.3.2    Admission Table

   Source                            Edge predicate                  Saving mechanism
   F3 empty or incompatible cell     zero support                    no mass
   F3 boundary cell                  boundary/short volume           volume loss
   F3 square-divisor tail            large square divisor            square-divisor summation
   F4 short quotient or divisor      Type I or short-volume saving   divisor/quotient loss
   CKP high-frequency tail           high Fourier tail               integration/Fourier decay
   CKP small-conductor layer         small conductor                 conductor saving
   CKP large 𝑔 or content            large gcd/content               divisor-bound loss
   BRS singular carrier slice resid- boundary or slice-floor Edge    X16BRS/X16C plus C1
   ual
   LongAP/Local boundary error boundary/short volume                 C1 boundary estimate




8.3.3    Summability
C1A pairs each active Edge source with one of the C1P predicates above. C1 then gives either a
power saving in 𝑁 or a logarithmic saving large enough to dominate the 𝐿𝑂(1) routing multiplicity.
Hence the total Edge contribution is 𝑜(𝑁 ).

8.4     Explicit Local Projection Algebra
This appendix records the local algebra evaluated by H4 and imported into the final assembly
through H4M. The term canonical local projection” is only shorthand for the concrete construction
below.

                                                  26
8.4.1     The Finite Local Model
Let

                           𝑤 = 𝑤(𝑁 ) → ∞,          𝑤 = 𝑜(log 𝑁 ),           𝑄=
                                                                                 ∏︁
                                                                                       𝑝.
                                                                                 𝑝≤𝑤

      The local model of the von Mangoldt weight modulo 𝑄 is

                                                     𝑄
                                       Λ𝑄 (𝑎) =         1        .
                                                    𝜙(𝑄) (𝑎,𝑄)=1

      It has average value one on Z/𝑄Z. Define

                                            1 ∑︁
                                𝜎𝑄 (𝑁 ) =             Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎).
                                            𝑄 𝑎 mod 𝑄

      The local Goldbach model at modulus 𝑄 is

                                       Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ).

      Endpoint and smooth-partition discrepancies are already Edge errors in C1 and are 𝑜(𝑁 ).

8.4.2     Tagged-Cell Projection
The exact B1 decomposition and the F3 tagged partition give a finite tagged identity

                                   𝑅Λ (𝑁 ) =                        𝑅ℬ,𝜏 (𝑁 ),
                                               ∑︁          ∑︁
                                                    𝑐ℬ
                                               ℬ         𝜏 ∈𝒯 (ℬ)

      where ℬ is the parent B1 block and 𝜏 is the complete routing tag of the terminal cell.
      For a tagged cell (ℬ, 𝜏 ), define

                                             Loc𝑄 𝑅ℬ,𝜏 (𝑁 )

   by replacing the arithmetic coefficients in that same tagged cell by their residue-class local
densities modulo 𝑄, while preserving:

  1. the parent B1 block ℬ;

  2. the routing tag 𝜏 ;

  3. the dyadic and smooth weights;

  4. the local congruence restrictions already present in the cell.

    This is a tag-preserving operation. It is not a branch-specific density and it does not identify
cells that merely have the same displayed algebraic shape.




                                                     27
8.4.3   H4M Admission Condition
A terminal local/main expression is admitted into the H4M local bridge only if it satisfies

                                    local
                                   𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).                  (E-adm)
   The active sources are exactly:

  1. LongAP/Local main terms from D1;

  2. CKP zero-frequency terms from G8a together with B1LD;

  3. LocalDiag cells produced by the routing table;

  4. explicitly admitted local boundary terms whose discrepancy is already covered by C1.

    D1 verifies (E-adm) for LongAP/Local cells after excluding nonlocal coefficients. G8a verifies it
for CKP ℎ = 0 cells by identifying the zero Fourier mode with the same tagged local replacement.
LocalDiag cells are admitted only when the diagonal specialization remains a tagged local projection;
otherwise the routing table sends the cell to Edge, CKP, GoodAWACK, impossible, or a continuing
routed case.

8.4.4   Linearity Over the B1/F3 Partition
The operator Loc𝑄 is linear on the finite tagged decomposition:
                                        ⎛         ⎞

                                   Loc𝑄 ⎝ 𝑐ℬ 𝑅ℬ,𝜏 ⎠ =    𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 .
                                         ∑︁           ∑︁

                                             ℬ,𝜏             ℬ,𝜏

   Since B1 is an exact finite identity and F3 is an exact tagged partition, the tagged local
projections reconstruct the local model of the original Goldbach convolution:

                                       Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).           (E-reconstruct)
                  ∑︁          ∑︁
                       𝑐ℬ
                   ℬ        𝜏 ∈𝒯 (ℬ)

    Equivalently, applying the local replacement after the tagged proof-tree partition gives the same
local main term as applying Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄) directly to the two original von Mangoldt factors
in 𝑅Λ (𝑁 ).

8.4.5   No Double Counting
Local terms are indexed by (ℬ, 𝜏 ). If two terms have different parent B1 blocks, they are distinct
summands of the exact B1 identity. If they have the same parent block but different tags, F3.15
gives disjoint tagged cells.
    Thus two local-looking formulas may be algebraically identical and still be different summands;
they are recombined only through the tagged linear sum in (E-reconstruct). Conversely, a local-
looking expression without a parent B1 block and routing tag is not an H4 input. Hence H4 neither
double-counts nor loses admitted local/main terms.




                                                        28
8.4.6     Local Factors and the Singular Series
By the definition of Λ𝑄 ,

                                       1 ∑︁       𝑄             𝑄
                        𝜎𝑄 (𝑁 ) =                     1(𝑎,𝑄)=1     1           .
                                       𝑄 𝑎 mod 𝑄 𝜙(𝑄)          𝜙(𝑄) (𝑁 −𝑎,𝑄)=1

   Since 𝑄 is squarefree, the CRT gives

                                              𝜎𝑄 (𝑁 ) =             𝜎𝑝 (𝑁 ),
                                                               ∏︁

                                                             𝑝≤𝑤

   where

                             1
                                            )︂2
                                       𝑝
                                 (︂
                   𝜎𝑝 (𝑁 ) =                      #{𝑎 mod 𝑝 : (𝑎, 𝑝) = 1, (𝑁 − 𝑎, 𝑝) = 1}.
                             𝑝        𝑝−1

   For 𝑝 = 2 and even 𝑁 , 𝜎2 (𝑁 ) = 2. For odd 𝑝, the two forbidden residues are 0 and 𝑁 . Hence
                                                    ⎧    𝑝
                                                    ⎪       ,              𝑝 | 𝑁,
                                                        𝑝−1
                                                    ⎪
                                                    ⎨
                                      𝜎𝑝 (𝑁 ) =                 1
                                                    ⎩1 −              ,    𝑝 ∤ 𝑁.
                                                    ⎪
                                                             (𝑝 − 1)2
                                                    ⎪


   Therefore
                                              ∏︁ (︂               1                   𝑝
                                                                          )︂ ∏︁
                            𝜎𝑄 (𝑁 ) = 2                   1−                             .
                                            3≤𝑝≤𝑤
                                                               (𝑝 − 1)2        3≤𝑝≤𝑤
                                                                                     𝑝−1
                                             𝑝∤𝑁                                𝑝|𝑁

   Letting 𝑤 → ∞,
                                                            ∏︁ 𝑝 − 1
                                      𝜎𝑄 (𝑁 ) → 2𝐶2                     = S(𝑁 ),
                                                           𝑝|𝑁
                                                                 𝑝−2
                                                           𝑝>2

   with
                                                    ∏︁ (︂         1
                                                                           )︂
                                         𝐶2 =               1−          .
                                                    𝑝>2
                                                               (𝑝 − 1)2

8.4.7     H4M Local Bridge Theorem
The admitted terminal local/main terms satisfy

                                       𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Indeed, by the admission condition (E-adm), every active local/main term is a tagged local
projection up to 𝑜(𝑁 ). By tagged linearity and no double counting, the sum of these projections is
𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ). By the CRT factor computation, 𝜎𝑄 (𝑁 ) → S(𝑁 ). Thus the local/main assembly
contributes exactly the Goldbach singular series main term.



                                                            29
8.5     Full-Manuscript Integration Note
The full Edge-admission ledger, Edge estimate, LongAP/Local local-coefficient expansion, local
projection algebra, and H4M local bridge are collected in Appendix E.


9     CKP Branch
9.1     The CKP Branch
9.1.1    CKP Claim
The CKP branch proves

                                    𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ),

   where 𝑀CKP (𝑁 ) is the explicit H4 tagged local projection obtained by the same Λ𝑄 -replacement
used in Appendix E.

9.1.2    Internal Reduction
The CKP proof follows the chain


𝐺1𝑎 + 𝐺2𝑎 + 𝐺3𝑎 + 𝐶𝐾𝑃 𝐷 + 𝐺4𝑎/𝑋10 + 𝑋10-𝐸𝑅 =⇒ 𝐶𝐾𝑃 𝑋10𝑀,                    𝐶𝐾𝑃 𝑋10𝑀 + 𝐵1𝐿𝐷 =⇒ 𝐺8𝑎.

    Here CKPX10M is only the nonzero-frequency analytic input; B1LD and G8a normalize the
zero-frequency local mode.
    Lemma G1a performs the gcd splitting 𝑢 = 𝑔𝑎, 𝑢′ = 𝑔𝑞, with (𝑎, 𝑞) = 1. Lemma G2a performs
the smooth AP/Fourier expansion. The zero-frequency term is local. Lemma G3a writes each
nonzero central frequency as a bilinear Kloosterman-fraction sum

                                                                ℎ𝑁𝑔 𝑎
                                                              (︂      )︂
                                       𝛼𝑔 (𝑎)𝛾𝑔 (𝑞)𝑊𝑔,ℎ (𝑎, 𝑞)𝑒
                                ∑︁
                                                                      .
                             𝑎∼𝐴 , 𝑞∼𝑄
                                                                 𝑞
                                𝑔        𝑔
                               (𝑎,𝑞)=1


9.1.3    X10 Match
X10 is invoked only for the central nonzero-frequency range, through CKPX10M. CKPD proves that
the actual two-variable Fourier weight 𝑊𝑔,ℎ (𝑎, 𝑞), including the dependence 𝑧(𝑎, 𝑞, 𝑦) = (𝑁𝑔 − 𝑎𝑦)/𝑞,
satisfies the smooth derivative hypotheses of the DFI Kloosterman-fraction theorem with only
polylogarithmic loss.
    Excluded nonzero-frequency ranges are not sent to X10. The X10ER routing statement sends
them to Edge through C1P/C1A/C1. The zero-frequency term is a separate local mode normalized
in G8a/LPI and later assembled by H4M.
Remark 9.1 (Output). The nonzero-frequency CKP contribution is 𝑜(𝑁 ). The zero-frequency term
is identified by B1LD with the local model imported by H4M. Therefore G8a supplies the CKP
input needed by I1.




                                                 30
9.2     CKP/X10 Smooth-Weight Matching
9.2.1    Target
The CKP branch proves

                                  𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ).

   The nonzero-frequency contribution is estimated by CKPX10M through X10. The zero-frequency
contribution is local and is passed to G8a/LPI and then H4.

9.2.2    DFI Theorem Used
The X10 input is the Duke–Friedlander–Iwaniec bilinear estimate for Kloosterman fractions. In the
smooth weighted form used here, one estimates
                                                                      𝑟𝑚
                                                                 (︂      )︂
                                               𝛼𝑚 𝛽𝑞 𝐹 (𝑚, 𝑞)𝑒
                                      ∑︁
                                                                         ,
                                  𝑚∼𝑀, 𝑞∼𝑄
                                                                       𝑞
                                   (𝑚,𝑞)=1

   where 𝐹 is supported on a dyadic box and has controlled derivatives up to the required fixed
order. In the CKP application the derivative-control parameter is a fixed power of 𝐿.

9.2.3    CKP Substitution
After G1a–G3a, the central CKP nonzero-frequency layer has the form

                                                                                ℎ𝑁𝑔 𝑎
                                                                           (︂       )︂
                        𝒪𝑔,ℎ =                 𝛼𝑔 (𝑎)𝛾𝑔 (𝑞)𝑊𝑔,ℎ (𝑎, 𝑞)𝑒
                                    ∑︁
                                                                                      .
                                 𝑎∼𝐴𝑔 , 𝑞∼𝑄𝑔
                                                                                 𝑞
                                   (𝑎,𝑞)=1

   The substitution into X10 is

                        𝑚 = 𝑎,       𝑀 = 𝐴𝑔 ,           𝑟 = |ℎ|𝑁𝑔 ,           𝑄 = 𝑄𝑔 .

   The coprimality condition is exactly (𝑎, 𝑞) = 1. For ℎ < 0, the same estimate is applied to the
conjugate phase, so the positive external integer parameter in X10 is 𝑟 = |ℎ|𝑁𝑔 . The case ℎ = 0 is
excluded from X10 and routed to the local term.

9.2.4    Smooth Weight
CKPD differentiates the actual CKP weight, not a separated surrogate. The weight contains the
Fourier-fibre dependence
                                                          𝑁𝑔 − 𝑎𝑦
                                         𝑧(𝑎, 𝑞, 𝑦) =             .
                                                             𝑞
   On the central support, CKPD proves
                                 ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ 𝐿𝐶 𝐴−𝑖 𝑄−𝑗
                         𝜕𝑎𝑖 𝜕𝑞𝑗 𝑊                                    (0 ≤ 𝑖, 𝑗 ≤ 2).
                                                    𝑔   𝑔

    Thus the X10 smooth-weight hypotheses hold with polylogarithmic loss, and CKPX10M packages
this check with the DFI matching and the final 𝑔, ℎ summation.

                                                     31
9.2.5    Excluded Ranges
The following ranges are not sent to X10:

         Excluded range                                  Routing
         ℎ=0                                             separate G8a/LPI local mode, then H4
         high Fourier frequency                          Edge
         small conductor                                 Edge
         large 𝑔 or large content                        X10ER, then Edge or empty/auxiliary exclu-
                                                         sion
         boundary or short-volume layer                  Edge



  After these exclusions, the remaining central nonzero-frequency contribution is 𝑜(𝑁 ) by
CKPX10M and GEB.

9.3     Full-Manuscript Integration Note
The full CKP branch proof is in Appendix F. The external DFI input itself is stated once in
Appendix B and is invoked through the CKPX10M nonzero-frequency interface after the CKPD
smooth-weight derivative check.


10       GoodAWACK Branch
10.1     The GoodAWACK Branch
10.1.1     Branch B Theorem
Lemma E10L proves that the total terminal GoodAWACK contribution is 𝑜(𝑁 ). The proof has two
components:

  1. the TC1 global testing route;
  2. the HighTC finite-grammar closure.

10.1.2     TC1 Global Testing
The TC1 route does not choose an arbitrary shifted short interval. It tests only structural B1-origin
coarea families whose cells have not already been routed away. Theorem TNG-A gives the single
near-global-or-routed interface:

               every unrouted TC1 test is near-global, or routed away before X9L-GT.

     In the near-global alternative the test satisfies

                                           𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

    Thus X9L-GT applies in its Davenport/AP form with polylogarithmic modulus and AP com-
plexity. The resulting orthogonality estimate is summable by GEB.
    In the routed alternative, TTD, ROC, BRS, and X16BRS/X16C send the test to Edge, LongAP/
Local, CKP, LocalDiag, or zero before X9L-GT is invoked.

                                                    32
10.1.3   Finite GoodAWACK Grammar
The HighTC branch is closed by the E10YMX finite combinatorial grammar theorem, which
consumes the E10Y/E10X/E10M/E10K source-grammar components. E10YMX proves that every
actual terminal GoodAWACK skeleton is generated from B1/B3 grouped cells by the listed B1/B3/
F3/F4/E5 grammar: fixing/projection, controlled CRT restriction, fixed divisor quotient, F4-tagged
variable quotient, local/diagonal/gcd dependence, CKP-balanced structure, Edge-type saving or
boundary routing, full-rank affine regrouping, post-terminal slicing after terminal vectors are fixed,
E5 auxiliary inheritance, and final terminal labelling. E5 is used only as content stability and is not
an independent terminal generator.
    E10X proves by induction on the E10Y-certified finite grammar that every rank-dropping affine
operation created along a derivation carries an allowed origin tag. E10M then proves that no
untagged rank-dropping AFF occurrence survives in an actual terminal GoodAWACK skeleton.
E10K converts this into AFF-origin completeness, and E10X eliminates the FreeAffineHighTC
residual.
    The formal 4𝐴𝑃 -like family 𝑌𝑖 = 𝑥 + 𝑖𝑟 remains useful as an interface test, but E10X proves
that it has no untagged actual terminal occurrence in the finite routing grammar.
Remark 10.1 (Output). TC1 contributes 𝑜(𝑁 ), singular tests are routed to already handled classes,
and HighTC finite-grammar residuals are eliminated structurally. Hence

                                      𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

10.2     TC1, BRS/TTH, and GoodAWACK Finite Grammar
10.2.1   TC1 Route
The TC1 branch tests structural B1-origin coarea families whose cells have not already been routed
away. It never invokes Liouville/AP orthogonality on an arbitrary shifted short interval.
   The route is packaged as Theorem TNG-A:

                   every unrouted TC1 test is either near-global or routed away.

    The closure barrier preventing rogue short-interval refinements is TTH-SC. It proves that every
short subtest of a released structural coarea test is either non-structural and reaggregated, or
structural and routed through TTD/ROC/BRS/X16BRS/X16C before X9L-GT is invoked.
    On the near-global alternative:

                                         𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

  At this length, X9L-GT supplies the needed Davenport/AP orthogonality with polylogarithmic
modulus and AP complexity.
  The bridge from a non-small TC1 macro-template to a measured family of Liouville tests is
TGT-MF. It proves the finite Fourier/coarea transfer

                                        GT-U2 =⇒ GT-Test,

   with a probability measure 𝜈𝜅 and fixed lower bound depending only on the macro-template.
Thus the route tests an averaged family, not a selected pointwise short interval.



                                                  33
10.2.2    Singular Tests
If a testing measure is singular, it is not sent to X9L-GT. The routed alternative of TNG-A, proved
by TTD/ROC/BRS with X16BRS/X16C, shows that an unrouted singular B1-origin coarea test
must enter one of the already controlled destinations:

                        Edge,   LongAP/Local,        CKP,   LocalDiag,   0.

    The carrier-slice estimates used in this step are supplied by X16BRS and X16C. X16C uses
the Shiu AP divisor-average input and the fixed divisor-function second moment recorded in the
bibliography.

10.2.3    Finite GoodAWACK Grammar
The HighTC part is closed structurally. E10Y proves completeness of the finite grammar for
actual terminal GoodAWACK skeletons. The start states are B1/B3 grouped cells, and the allowed
transitions are fixing/projection, controlled CRT restriction, fixed divisor quotient, F4-tagged
variable quotient, local/diagonal/gcd dependence, CKP-balanced structure, Edge routing, full-
rank affine regrouping, post-terminal slicing after terminal vectors are fixed, E5-clean auxiliary
inheritance, and terminal labelling. The proof of E5 content stability belongs to the routing/
transport appendix; the GoodAWACK argument imports only this clean interface.
    E10X proves a grammar invariant on the E10Y-certified grammar: every rank-dropping affine
operation generated by these transitions carries an allowed origin tag. E10M proves that every
rank-dropping affine occurrence in an actual terminal GoodAWACK skeleton is tagged by an
allowed origin. Therefore no untagged rank-dropping AFF source exists. E10K gives AFF-origin
completeness, and E10X removes the FreeAffineHighTC residual.

10.2.4    Formal Interface Examples
The formal 4𝐴𝑃 -like pattern 𝑌𝑖 = 𝑥 + 𝑖𝑟 is retained as a diagnostic interface example. It shows
why broad affine regrouping language is unsafe if read without origin data. It is not a terminal
obstruction because any actual terminal occurrence must either be full-rank safe, tagged and
rerouted, or excluded by the no-untagged-AFF theorem.
Remark 10.2 (GoodAWACK Output). Combining TC1 testing, singular rerouting, and finite-
grammar HighTC closure gives

                                    𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

10.3     Full-Manuscript Integration Note
The GoodAWACK proof splits into the TC1 near-global testing package in Appendix G and the
HighTC finite-grammar package in Appendix H. E10L is the branch theorem, E10YMX is the
reader-facing HighTC finite-grammar master theorem, E10Y proves completeness of the actual
GoodAWACK routing grammar, and E10X supplies the finite-grammar invariant.




                                                34
11       Local Main Term and Global Assembly
11.1     Local/Main Assembly
11.1.1    Explicit H4 Local Algebra
The local/main layer is assembled by a concrete finite local model, not by an undefined projection
convention. Let

                         𝑄=                 𝑤 = 𝑤(𝑁 ) → ∞,         𝑤 = 𝑜(log 𝑁 ),
                                ∏︁
                                     𝑝,
                               𝑝≤𝑤

     and define
                              𝑄                              1 ∑︁
                  Λ𝑄 (𝑎) =       1        ,      𝜎𝑄 (𝑁 ) =             Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎).
                             𝜙(𝑄) (𝑎,𝑄)=1                    𝑄 𝑎 mod 𝑄

     The local projection of the original Goldbach convolution is

                                          Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ).

    For a terminal tagged cell (ℬ, 𝜏 ), LPI defines Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) by replacing the arithmetic coefficients
inside that same tagged cell by their residue-class local densities modulo 𝑄, preserving the parent
B1 block, routing tag, dyadic weights, and local congruence data.

11.1.2    H4 Admission
H4 admits a terminal local/main expression only if it satisfies the tagged admission condition
                                  local
                                 𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).

   Thus a local-looking expression does not enter the main term merely because it resembles a local
density. It must be attached to a parent B1 block and a terminal routing tag, and its normalization
must be the single Λ𝑄 -replacement above.
   The admitted local sources are:

   1. LongAP/Local main terms whose F3P terminal predicate already forces the long-variable
      coefficients into the local coefficient algebra, with D1 expanding them into the tagged local
      projection;

   2. CKP zero-frequency terms from G8a/B1LD, after the zero mode is identified with the same
      tagged Λ𝑄 -replacement;

   3. LocalDiag terms, admitted only when the diagonal specialization is a tagged local projection;

   4. harmless local boundary contributions explicitly admitted by H4 and already bounded through
      C1.




                                                     35
11.1.3   Linearity and No Double Counting
The exact B1 decomposition and the exact F3 tagged partition give

                                              𝑅Λ (𝑁 ) =                            𝑅ℬ,𝜏 (𝑁 ).
                                                            ∑︁            ∑︁
                                                                   𝑐ℬ
                                                               ℬ        𝜏 ∈𝒯 (ℬ)

   The operator Loc𝑄 is linear on this finite tagged partition. Therefore

                                   Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).
              ∑︁          ∑︁
                   𝑐ℬ
               ℬ        𝜏 ∈𝒯 (ℬ)

   No double counting occurs because each local term is indexed by its parent B1 block ℬ and its
routing tag 𝜏 . Different B1 blocks are different summands of the exact B1 identity, and different
tags inside the same block are disjoint cells of the F3 partition. Algebraically identical LocalDiag-
looking formulas from different tags are therefore distinct partition summands, not duplicates.

11.1.4   Finite Local Factors
The finite density 𝜎𝑄 (𝑁 ) factors by the CRT. For 𝑝 ≤ 𝑤,

                                    1
                                                   )︂2
                                              𝑝
                                        (︂
                   𝜎𝑝 (𝑁 ) =                             #{𝑎 mod 𝑝 : (𝑎, 𝑝) = 1, (𝑁 − 𝑎, 𝑝) = 1}.
                                    𝑝        𝑝−1

   For even 𝑁 , 𝜎2 (𝑁 ) = 2. For odd 𝑝,
                                                           ⎧    𝑝
                                                           ⎪       ,                  𝑝 | 𝑁,
                                                               𝑝−1
                                                           ⎪
                                                           ⎨
                                             𝜎𝑝 (𝑁 ) =                 1
                                                           ⎩1 −              ,        𝑝 ∤ 𝑁.
                                                           ⎪
                                                                    (𝑝 − 1)2
                                                           ⎪


   Hence
                                                                   ∏︁ 𝑝 − 1
                                             𝜎𝑄 (𝑁 ) → 2𝐶2                         = S(𝑁 ).
                                                                   𝑝|𝑁
                                                                         𝑝−2
                                                                   𝑝>2


11.1.5   Weighted Assembly
Combining the LPI admission condition consumed by H4M, tagged linearity, no double counting,
and the H4 finite local factor computation gives

                               𝑀local (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Lemma I1 then combines the terminal estimates and H4M:

                               𝑅Λ (𝑁 ) = 𝑀local (𝑁 ) + 𝑜(𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

    GEB verifies that all terminal 𝑜(𝑁 ) terms remain 𝑜(𝑁 ) after the full polylogarithmic decomposi-
tion and routing summation.




                                                                    36
11.2     Full-Manuscript Integration Note
The H4M local bridge is proved in Appendix E and invoked here together with the global error
budget from Appendix A. The final weighted assembly is completed in Appendix I.


12       Prime-Power Removal and Final Handoff
12.1     Prime-Power Removal and Final Proof
12.1.1    Statement
Assume the weighted asymptotic I1:

                                      𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )

    for sufficiently large even 𝑁 . Then, after removing the negligible nontrivial prime-power
contribution by G2, the genuine prime-pair weighted sum is positive. Consequently every sufficiently
large even 𝑁 is a sum of two primes.

12.1.2    Setup
The von Mangoldt function is supported on prime powers:

                                                 log 𝑝,    𝑛 = 𝑝𝑘 , 𝑘 ≥ 1,
                                            {︃
                                   Λ(𝑛) =
                                                 0,        otherwise.

     Let 𝑅𝑝𝑝 (𝑁 ) denote the ordered weighted sum restricted to genuine prime pairs:

                                 𝑅𝑝𝑝 (𝑁 ) =                    (log 𝑝1 )(log 𝑝2 ).
                                                   ∑︁

                                               𝑝1 +𝑝2 =𝑁
                                              𝑝1 ,𝑝2 prime

     The ordered-pair convention is the same convention used in B1, I1, G2, G1, and G0H.

Proof. Lemma G2 removes the nontrivial prime powers 𝑝𝑘 , 𝑘 ≥ 2. There are 𝑂(𝑁 1/2 ) such prime
powers up to 𝑁 , and once one coordinate in 𝑛1 + 𝑛2 = 𝑁 is selected the other coordinate is
determined. Since Λ(𝑛) ≤ log 𝑁 , the total contribution from pairs in which at least one coordinate
is a nontrivial prime power is

                                      𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).

     Therefore I1 and G2 imply

                                      𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

     For even 𝑁 , the Goldbach singular series satisfies

                                          S(𝑁 ) ≥ 2𝐶2 > 0.

    Hence 𝑅𝑝𝑝 (𝑁 ) > 0 for all sufficiently large even 𝑁 . Since every summand in 𝑅𝑝𝑝 (𝑁 ) is
nonnegative and the summands on genuine prime pairs are positive, this positivity implies that at
least one ordered pair of primes 𝑝1 , 𝑝2 satisfies


                                                          37
                                             𝑝1 + 𝑝2 = 𝑁.



Remark 12.1 (Output). Lemma G0H records the ordered-pair normalization and the final logical
handoff

                                      𝐼1 + 𝐺2 =⇒ 𝐺1 =⇒ 𝐺0.

   The statement proved is the sufficiently-large binary Goldbach theorem. Finite verification below
the asymptotic threshold is outside the asymptotic claim of this manuscript.

12.2     Full-Manuscript Integration Note
The final handoff proofs are collected in Appendix I. This section records the logical endpoint of the
proof of Theorem 1.1.


A      Parameter Register and Global Error Budget
A.1     Parameter register
A.1.1     PAR. Global Parameter Register
PAR.0. Role Logical ID: PAR.
   This file fixes the structural constants used by the proof. Its purpose is to make all parameter
choices explicit and to prevent hidden dependencies between the Heath–Brown decomposition,
routing, Edge estimates, CKP/X10, BRS/TTH, and GoodAWACK.
   The register proves the following bookkeeping assertion: the displayed hierarchy of constants is
nonempty and is strong enough for all later uses of logarithmic losses, slice floors, near-global TC1
image lengths, and CKP/DFI smooth-weight thresholds.
   Used by: B1, C1, BRS, TTH, G3a, G8a, CKPD, X10, GEB, and I1.
   Uses: the constant outputs of X16C and CKPD.

PAR.1. Statement         There exist constants

                 0 < 𝜃 ≪ 𝜂 ≪ 1,       𝐽0 ,    𝐶0 , 𝐶1 , 𝐶16 , 𝜌16 , 𝐵16 , 𝐵𝜅 , 𝐵HF , 𝐶DFI , 𝐵

   which can be chosen in the order specified below and which satisfy all parameter inequalities
needed by the active proof tree.
   More precisely:

    1. the Heath–Brown depth 𝐽0 can be chosen above the structural lower bound 𝐽* (𝜂);

    2. all routing, Edge, CKP, BRS/TTH, X16, and X10 losses are bounded by fixed powers of log 𝑁
       once 𝐽0 is fixed;

    3. the later logarithmic exponents 𝐵16 , 𝐵𝜅 , 𝐵HF , 𝐶DFI , and 𝐵 can be enlarged without changing
       any earlier finite decomposition;

    4. the resulting global summability of terminal errors is available to Lemma GEB.



                                                  38
PAR.2. Order of choices       The parameters are chosen in the following order:

  1. choose small structural exponents 0 < 𝜃 ≪ 𝜂 ≪ 1;

  2. choose the Heath–Brown depth 𝐽0 ≥ 𝐽* (𝜂);

  3. fix the B1/B3/F3/F4 dyadic and routing complexity bounds 𝐶0 (𝐽0 );

  4. fix the C1 strict-Edge polylogarithmic loss 𝐶1 (𝐽0 );

  5. fix the X16-BRS carrier-slice constants 𝐶16 (𝐽0 ) and 𝜌16 (𝐽0 ) > 0 supplied by Lemma X16C;

  6. choose the X16 slice-floor exponent 𝐵16 large enough that the floor term 𝑋𝐶 (log 𝑁 )−𝐵16 is
     strict C1P Edge after X16 losses;

  7. choose the BRS/TTH loss 𝐵𝜅 larger than all preceding polylogarithmic losses and larger than
     𝐵16 ;

  8. choose the CKP high-frequency and DFI smooth-weight thresholds large enough to dominate
     the G2a/G3a/CKPX10M/X10 derivative losses, as quantified in Lemma CKPD;

  9. choose the auxiliary square-divisor exponent 𝐵 after 𝐶0 and 𝐶1 , enlarged whenever C1/F4
     square-divisor routing requires it.

PAR.3. Parameter Table

  Parameter               Meaning                     Source            Required condition
  𝐽0                      Heath–Brown identity        B1                Fixed, 𝐽0 ≥ 𝐽*
                          depth
  𝐽*                      lower bound ensuring        B1, PAR           𝐽* (𝜂) =
                          bounded routing/CS/                           max{10, ⌈(4𝜂)−1 ⌉ + 1}
                          CKP complexity                                is sufficient
  𝜃                       small-variable/range        B3, C1            0<𝜃≪𝜂
                          cutoff
  𝜂                       large gcd/content and       B3, G8a           fixed small positive
                          balanced-range cutoff                         number
  𝐶0                      number of typed/            B1, C1, F3        𝐶0 = 𝐶0 (𝐽0 )
                          routing cells
  𝐶1                      strict Edge coefficient/    C1                𝐶1 = 𝐶1 (𝐽0 )
                          polylogarithmic loss
  𝐷 = 𝐿𝐵                  large square-divisor        C1, F4            𝐵 > 𝐶1 + 𝐶0 + 10,
                          threshold                                     enlarged as needed
  𝐶16                     X16-BRS logarithmic         X16C              admissibly 𝐶16 =
                          loss                                          100𝐽02 + 100, after harm-
                                                                        less enlargement
  𝜌16                     X16-BRS power-saving X16C                     admissibly 𝜌16 =
                          remainder                                     1/(106 𝐽04 )
  𝐵16                     X16 slice-floor ex-  X16C, BRS                choose 𝐵16 > 𝐶0 + 𝐶1 +
                          ponent 𝑌16 =                                  𝐶16 + 20
                          max(𝑌 # , 𝑋𝐶 𝐿−𝐵16 )




                                                     39
  𝐵𝜅                     near-global TC1 image     BRS, TTH                choose 𝐵𝜅 > 𝐵16 + 𝐶0 +
                         loss in TTH                                       𝐶1 + 𝐶16 + 𝜌−1
                                                                                       16 + 20 after
                                                                           X16-BRS is fixed
  𝐵HF                    CKP high-frequency        G2a, G8a, X10           dominates G2a Fourier
                         cutoff |ℎ|𝑔 ≤ 𝐿𝐵HF                                decay and X10 smooth-
                                                                           weight loss
  𝐶DFI                   DFI smooth-weight         CKPD, X10               fixed by Lemma CKPD
                         derivative loss




   Here and below 𝐿 = log 𝑁 .

PAR.4. Minimal Consistency Checks             The following inequalities must hold simultaneously:

  1. C1/F4 square-divisor routing uses 𝐵 > 𝐶0 + 𝐶1 + 10.

  2. X16/BRS uses 𝐵16 > 𝐶0 + 𝐶1 + 𝐶16 + 20.

  3. BRS/TTH uses

                                 𝐵𝜅 > 𝐵16 + 𝐶0 + 𝐶1 + 𝐶16 + 𝜌−1
                                                             16 + 20.

  4. X9L-GT is invoked only after TTH supplies

                                            𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

  5. X10 is invoked only in the central CKP range and only after the two-variable smooth weight
     𝑊𝑔,ℎ (𝑎, 𝑞) satisfies the DFI derivative bounds with loss (log 𝑁 )𝐶DFI , as proved in CKPD.

  6. All sums over 𝑔-layers use constants independent of 𝑔. The only 𝑔-dependence allowed is
     through the explicit dyadic scales 𝐴𝑔 , 𝑄𝑔 , 𝑆𝑔 and through summable powers handled by G4a
     and G8a.

PAR.5. Proof of Nonemptiness            The lower bound for 𝐽* (𝜂) is conservative:

                                  𝐽* (𝜂) = max{10, ⌈(4𝜂)−1 ⌉ + 1}.

   The constant 10 covers the fixed finite Cauchy–Schwarz, generalized von Neumann, and routing-
complexity overheads. The term ⌈(4𝜂)−1 ⌉ + 1 ensures that the Heath–Brown cutoff is not coarser
than the large-gcd/content hierarchy used in the balanced CKP and TC1 ranges.
   One concrete witness is
                                     1             1
                                𝜂=      ,    𝜃=        ,       𝐽0 = 20.
                                     40           4000
   Then

                                𝐽* (𝜂) = max{10, ⌈(4𝜂)−1 ⌉ + 1} = 11,

   so 𝐽0 ≥ 𝐽* (𝜂). After 𝐽0 is fixed, Lemma X16C supplies admissible constants

                                                  40
                                                                  1
                               𝐶16 = 100𝐽02 + 100,      𝜌16 =             .
                                                                106 𝐽04

   Choose

                                𝐵16 = ⌈𝐶0 (𝐽0 ) + 𝐶1 (𝐽0 ) + 𝐶16 + 21⌉

   and then choose
                               ⌈︁                                             ⌉︁
                         𝐵𝜅 = 𝐵16 + 𝐶0 (𝐽0 ) + 𝐶1 (𝐽0 ) + 𝐶16 + 𝜌−1
                                                                 16 + 21 .

    The CKP high-frequency threshold 𝐵HF , the DFI derivative-loss constant 𝐶DFI , and the auxiliary
square-divisor exponent 𝐵 are then chosen after these quantities, large enough for the inequalities
in PAR.4. Enlarging any of these later constants is harmless because all affected families are finite
once 𝐽0 is fixed.
    This proves that the hierarchy in PAR.1 is nonempty.

PAR.6. Notational Conventions           The singular series is denoted throughout by

                                               S(𝑁 ).

   The proof does not use a separate Sing(N) symbol. Terminal class names are written in prose
as Edge, LongAP/Local, CKP, GoodAWACK, and LocalDiag, and in displayed formulae as

             Edge,      LongAP/Local,        CKP,       GoodAWACK,                 LocalDiag.

   No independent macros such as ∖𝐶𝐾𝑃 or ∖𝐺𝑜𝑜𝑑𝐴𝑊 𝐴𝐶𝐾 are part of the source convention.
Remark A.1 (PAR.7. Output). The proof tree should not use disconnected phrases such as "take all
parameters sufficiently large". All later parameter choices must be compatible with this register.
    If the Shiu/AP X16 proof, the CKP/X10 derivative appendix, or the MRT/PACK interface
changes, this parameter register must be checked before the proof tree can again be treated as
closed.

PAR.8. Logical Dependencies Internal inputs used: X16C, CKPD.
  Internal nodes served: B1, C1, BRS, TTH, G3a, G8a, X10, GEB, and I1.

A.2     Global error budget
A.2.1    GEB. Global Error Budget and Parameter Hierarchy
GEB.0. Role Logical ID: GEB.
    Lemma GEB is an internal bookkeeping lemma. It does not introduce a new analytic estimate.
Its role is to prove that, with the parameter hierarchy of PAR, all terminal error terms produced by
the proof tree are summable and contribute 𝑜(𝑁 ) after the finite B1/B3/F3/F4 decomposition.
    Used by: I1.
    Uses: PAR, B1, B3, F3, F3T, F4, C1A, C1, D1, G8a, CKPX10M, CKPD, E10L, TNG, X16BRS, X16C, TTH,
H4, and H4M.
    External inputs used through their stated forms: X9, X10, and X16.



                                                 41
GEB.1. Statement After fixing the parameters in the order prescribed by PAR, the following
assertions hold uniformly over all tagged terminal cells in the proof tree:

  1. every strict Edge contribution is 𝑜(𝑁 ) after summing over all cells;

  2. every nonzero CKP contribution is 𝑜(𝑁 );

  3. every GoodAWACK TC1 contribution surviving to X9L-GT is 𝑜(𝑁 );

  4. every singular BRS/X16 carrier-slice remainder is either strict Edge or 𝑜(𝑁 );

  5. all local/main terms admitted by D1, G8a, and LocalDiag recombine through H4M into
     S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Consequently the I1 assembly may write

                                      𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )

   without hiding any additional summation over terminal families.

GEB.2. Setup and Constant Order             Let 𝐿 = log 𝑁 . Constants are fixed in the following order:

  1. choose 0 < 𝜃 ≪ 𝜂 ≪ 1;

  2. choose 𝐽0 ≥ 𝐽* (𝜂);

  3. fix the finite routing and dyadic complexity constant 𝐶0 (𝐽0 );

  4. fix the strict Edge polylogarithmic loss 𝐶1 (𝐽0 );

  5. fix 𝐶16 (𝐽0 ) and 𝜌16 (𝐽0 ) > 0 from X16C;

  6. choose 𝐵16 > 𝐶0 + 𝐶1 + 𝐶16 + 20;

  7. choose 𝐵𝜅 > 𝐵16 + 𝐶0 + 𝐶1 + 𝐶16 + 𝜌−1
                                        16 + 20;

  8. choose the CKP high-frequency and DFI derivative thresholds after the preceding constants;

  9. enlarge the auxiliary square-divisor exponent 𝐵 after 𝐶0 and 𝐶1 .

   The hierarchy is nonempty by PAR. A concrete consistency witness is
                                     1              1
                                𝜂=      ,     𝜃=        ,      𝐽0 = 20.
                                     40            4000
   For this witness

                                𝐽* (𝜂) = max{10, ⌈(4𝜂)−1 ⌉ + 1} = 11,

   so 𝐽0 ≥ 𝐽* (𝜂). After 𝐽0 is fixed, Lemma X16C supplies
                                                                      1
                               𝐶16 = 100𝐽02 + 100,          𝜌16 =             .
                                                                    106 𝐽04

    Then 𝐵16 , 𝐵𝜅 , the CKP thresholds, and 𝐵 are chosen by the inequalities above. Enlarging any
later logarithmic exponent is harmless because 𝐽0 and the routing grammar are already fixed.

                                                   42
GEB.3. Polylogarithmic Multiplicity Principle For fixed 𝐽0 , the B1 Heath–Brown expansion,
the smooth dyadic partitions, and the B3/F3/F4 routing grammar create at most 𝐿𝐶0 terminal
cells after harmless enlargement of 𝐶0 .
    Therefore:

  1. a per-cell estimate 𝑂(𝑁 𝐿−𝐶0 −𝐴 ) with fixed 𝐴 > 0 sums to 𝑂(𝑁 𝐿−𝐴 ) = 𝑜(𝑁 );

  2. a per-cell estimate 𝑂(𝑁 1−𝜌 𝐿𝐶 ) with fixed 𝜌 > 0 sums to 𝑜(𝑁 );

  3. a normalized testing estimate 𝑜(1) multiplied by an 𝐿𝑂(1) -complexity family remains 𝑜(1) after
     the Davenport/AP saving exponent is chosen larger than the recorded polylogarithmic losses.

   This principle is applied only after the corresponding branch lemma has verified that its input is
one of the terminal classes.

GEB.4. Global Loss Table

  Source             Estimate before          Multiplicity or loss   Parameter condition Global conclusion
                     global summation
  B1 dyadic decompo- exact identity and       at most 𝐿𝐶0 cells      fixed 𝐽0             no error
  sition             smooth partition
  F3/F4 routing      exact tagged termi-      at most 𝐿𝐶0 cells      termination by M♯    no error
                     nal partition
  C1 Edge            𝑁 𝐿−𝐶0 −10 or            𝐿𝐶0 cells              C1P predicate cat- 𝑅Edge (𝑁 ) = 𝑜(𝑁 )
                     𝑁 1−𝜌 𝐿𝐶1 per ad-                               alogue, C1A admis-
                     mitted atom                                     sion, and fixed 𝜌 > 0
  D1 LongAP/Local    canonical LPI lo-        boundary terms         C1P/C1A/C1 and        local projection
                     cal projection plus      C1P/C1A-admitted       LPI/H4M compati- +𝑜(𝑁 )
                     boundary/error                                  bility
                     terms
  CKP excluded       strict Edge, high-       polylogarithmic 𝑔, ℎ- X10ER, C1P/C1A/       nonzero excluded
  ranges             frequency decay,         families              C1, G1a, G2a,         ranges are 𝑜(𝑁 );
                     small conductor,                               CKPX10M, G8a,         ℎ = 0 is an LPI
                     large 𝑔, or local zero                         B1LD, H4M             local term assembled
                     frequency                                                            by H4M
  CKP central        DFI-X10 bound            CKPD derivative        X10 hypotheses       𝑜(𝑁 )
  nonzero frequen-   in the central           loss and polyloga-     matched in CKPD
  cies               Kloosterman-             rithmic 𝑔, ℎ-sum       and packaged by
                     fraction range                                  CKPX10M; thresh-
                                                                     olds chosen after
                                                                     PAR
  GoodAWACK TC1        normalized X9L-        PACK family and        TNG/TTH give         𝑜(𝑁 )
  regular branch       GT Davenport/AP        polylogarithmic AP     𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅
                       estimate 𝑜(1)          complexity             and X9 is invoked
                                                                     only there
  GoodAWACK singu- singular tests route no independent               TTD/ROC/BRS/         handled by existing
  lar branch       to Edge, LongAP/        short-interval input      X16BRS/X16C          branches
                   Local, CKP, Local-
                   Diag, or zero
  X16/BRS carrier  𝑁 (log 𝑁 )𝐶16 𝑌16 /𝑋𝑃 + floor loss 𝑌16 =          𝐵16 > 𝐶0 + 𝐶1 +      strict Edge or 𝑜(𝑁 )
  slice            𝑁 1−𝜌16 (log 𝑁 )𝐶16     𝑋𝐶 𝐿−𝐵16 where            𝐶16 + 20
                   after normalization     needed




                                                          43
  GoodAWACK           untagged rank-          finite GoodAWACK E10YMX, consum-         no residual
  HighTC finite gram- dropping AFF is         grammar          ing E10Y/E10X/          FreeAffineHighTC
  mar                 impossible; tagged                       E10M/E10K
                      alternatives route to
                      existing classes
  H4M local bridge    𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ) for   finite CRT/local      𝑤(𝑁 ) → ∞, 𝑤(𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )
                      the admitted local      projection family     𝑜(log 𝑁 ) and no
                      model                                         residual local source
                                                                    by LPI/H4M




   Each row is used only through its named branch lemmas. Lemma GEB records the global
summation and compatibility of losses; it does not replace C1, G8a, E10L, X16C, X9, X10, H4M, or the
separate prime-power removal lemma G2.

GEB.5. Proof By B1, B3, F3P, F3, F3T, and F4, the weighted Goldbach sum is exactly decomposed
into a finite tagged family of terminal contributions. The number of terminal cells is bounded by
𝐿𝐶0 .
    For Edge terms, C1P defines the strict Edge predicates and C1A verifies that each active Edge input
satisfies one of them. Lemma C1 gives either 𝑁 𝐿−𝐶0 −10 or 𝑁 1−𝜌 𝐿𝐶1 per cell. The polylogarithmic
multiplicity principle therefore gives

                                              𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

   For LongAP/Local terms, F3P gives the positive local-coefficient predicate and D1 expands those
atoms into the LPI local projection. For LocalDiag terms, the intrinsic local-dependence tag is
admitted through LPI and assembled by H4M. Boundary and smooth-partition discrepancies are C1-
admitted. Hence these branches contribute the LPI local model assembled by H4M plus 𝑜(𝑁 ).
   For CKP terms, G8a separates zero and nonzero frequencies. The zero-frequency terms are
admitted into H4M through B1LD. Excluded nonzero ranges are routed through X10ER and C1P/
C1A/C1 inside CKPX10M. The central nonzero range is matched to the DFI theorem by CKPD and
X10, then summed over 𝑔, ℎ by CKPX10M; the remaining polylogarithmic losses are dominated by the
CKP thresholds chosen after PAR. Thus the total CKP nonlocal contribution is 𝑜(𝑁 ).
   For GoodAWACK terms, E10L applies the TC1/HighTC dichotomy. The TC1 regular branch is
packaged by TNG and reaches X9L-GT only after TTH supplies the near-global length

                                              𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

    The Davenport/AP saving exponent dominates the PACK and modulus losses recorded in PAR.
The TC1 contribution is therefore 𝑜(𝑁 ). Singular B1-origin tests are routed by TTD/ROC/BRS, with
X16BRS/X16Ccontrolling carrier-slice remainders; the slice-floor condition on 𝐵16 puts the residual
floor term inside strict Edge, and the power-saving term is summable. The HighTC/grammar branch
is closed by E10YMX, consuming the finite GoodAWACK grammar inputs E10Y/E10X/E10M/E10K,
so no untagged rank-dropping AFF residual remains.
    Lemma H4M then sums all admitted local projections into

                                               𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 )

   and proves 𝜎𝑄 (𝑁 ) → S(𝑁 ). Combining the rows of the global loss table gives

                                                       44
                                          𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

    This proves Lemma GEB.
Remark A.2 (GEB.6. Output). Lemma GEB supplies I1 with a single global error statement: after
all terminal branches are evaluated, the remaining nonlocal and boundary contributions are 𝑜(𝑁 ),
while the admitted local branches combine to S(𝑁 )𝑁 + 𝑜(𝑁 ).

GEB.7. Logical Dependencies Internal dependencies: PAR, B1, B3, F3, F3T, F4, C1A, C1, D1,
G8a, CKPX10M, CKPD, E10L, TNG, X16BRS, X16C, TTH, H4, and H4M.
   External dependencies: X9 and X10 only through their stated forms, and X16 only through the
X16C/X16BRSinterface.


B     External Inputs and Theorem Matching
B.1     Heath–Brown identity verification
B.1.1    X1. Heath–Brown Identity Input for B1
X1.0. Statement and Role This document verifies the external dependency X1 used by Lemma
B1. The required input is the exact Heath–Brown identity
                             𝐽0
                                               (︃        )︃
                                                    𝐽0
                    Λ(𝑛) =        (−1)                                               𝜇(𝑚1 ) · · · 𝜇(𝑚𝑗 ) log 𝑟1
                             ∑︁                                      ∑︁
                                         𝑗−1

                             𝑗=1
                                                    𝑗         𝑚1 ···𝑚𝑗 𝑟1 ···𝑟𝑗 =𝑛
                                                                    𝑚𝑖 ≤𝑦

    for 𝑛 ≤ 𝑦 𝐽0 . In B1 one takes 𝑦 = 𝑁 1/𝐽0 , so this must apply for every 𝑛 ≤ 𝑁 .
    The proof obligations for X1 are:

    1. Does the displayed formula match a valid Heath–Brown identity?
    2. Does the choice 𝑦 = 𝑁 1/𝐽0 put all B1 arguments in the exact range?
    3. Does the subsequent dyadic localization introduce any error or hidden coefficient type?

   External source:
   D. R. Heath-Brown, "Prime numbers in short intervals and a generalized Vaughan identity",
Canadian Journal of Mathematics 34 (1982), no. 6, 1365–1377, DOI 10.4153/CJM-1982-095-9.
   The proof below records the exact finite identity in the range used by B1; no analytic estimate
from the paper is invoked.

X1.1. Formal identity        Let 𝐽 ≥ 1, 𝑦 ≥ 1, and

                        𝜇𝑦 (𝑛) := 𝜇(𝑛)1𝑛≤𝑦 ,                      1(𝑛) := 1,             𝐿(𝑛) := log 𝑛.

    All convolutions below are Dirichlet convolutions. The identity needed by B1 is

                                      𝐽
                                                               (︃ )︃
                                                                  𝐽 (︁ *𝑗             )︁
                           Λ(𝑛) =          (−1)𝑗−1                    𝜇𝑦 * 𝐿 * 1*(𝑗−1) (𝑛)                        (X1.1)
                                     ∑︁

                                     𝑗=1
                                                                  𝑗

    for every 𝑛 ≤ 𝑦 𝐽 . Expanding the convolution gives exactly the B1 formula, with 𝑟1 carrying the
logarithm and 𝑟2 , . . . , 𝑟𝑗 carrying the 1-weights.

                                                                    45
X1.2. Proof of the identity Write formal Dirichlet series only as a coefficient bookkeeping
device:

                                         𝐷(𝑓 ; 𝑠) =              𝑓 (𝑛)𝑛−𝑠 .
                                                           ∑︁

                                                           𝑛≥1

   Then
                                                                                            𝜁 ′ (𝑠)
                    𝐷(1; 𝑠) = 𝜁(𝑠),       𝐷(𝐿; 𝑠) = −𝜁 ′ (𝑠),                 𝐷(Λ; 𝑠) = −           .
                                                                                            𝜁(𝑠)

   Let

                                             𝑀𝑦 (𝑠) = 𝐷(𝜇𝑦 ; 𝑠).

   The Dirichlet series of the right side of (X1.1) is
                                          𝐽
                                                            (︃ )︃
                                  ′                             𝐽
                              (−𝜁 (𝑠))        (−1)                𝑀𝑦 (𝑠)𝑗 𝜁(𝑠)𝑗−1 .
                                         ∑︁
                                                      𝑗−1

                                         𝑗=1
                                                                𝑗

   Factoring one 𝜁(𝑠)−1 , this becomes
                                          𝐽
                                 𝜁 ′ (𝑠) ∑︁
                                                            (︃ )︃
                                                    𝐽 (︀           )︀𝑗
                               −            (−1)𝑗−1      𝑀𝑦 (𝑠)𝜁(𝑠) .
                                 𝜁(𝑠) 𝑗=1           𝑗

   The binomial sum is
                                 𝐽
                                                  (︃ )︃
                                                    𝐽
                                      (−1)            𝐴𝑗 = 1 − (1 − 𝐴)𝐽 .
                                ∑︁
                                          𝑗−1

                                𝑗=1
                                                    𝑗

   Therefore the right side of (X1.1) has Dirichlet series
                                             (︁                               )︀𝐽 )︁
                                 𝐷(Λ; 𝑠) 1 − 1 − 𝑀𝑦 (𝑠)𝜁(𝑠)                                             (X1.2)
                                                      (︀
                                                                                       .
   Let

                                             𝐵 := 𝛿1 − 𝜇𝑦 * 1.

   For 1 < 𝑛 ≤ 𝑦,

                             (𝜇𝑦 * 1)(𝑛) =                   𝜇(𝑑) =           𝜇(𝑑) = 0,
                                                    ∑︁                 ∑︁

                                                  𝑑|𝑛, 𝑑≤𝑦             𝑑|𝑛

   and for 𝑛 = 1 the same convolution equals 1. Hence

                               𝐵(1) = 0,            𝐵(𝑛) = 0         (1 < 𝑛 ≤ 𝑦).

    Thus every nonzero coefficient of 𝐵 is supported on 𝑛 > 𝑦. Consequently every nonzero coefficient
of 𝐵 *𝐽 is supported on 𝑛 > 𝑦 𝐽 .
    From (X1.2), the difference between Λ and the right side of (X1.1) is Λ * 𝐵 *𝐽 . Since 𝐵 *𝐽 is
supported on 𝑛 > 𝑦 𝐽 , this difference has zero coefficient for every 𝑛 ≤ 𝑦 𝐽 . This proves (X1.1).

                                                           46
X1.3. Match with Lemma B1             Lemma B1 fixes a sufficiently large constant 𝐽0 and takes

                                                  𝑦 = 𝑁 1/𝐽0 .

   Therefore

                                                    𝑁 = 𝑦 𝐽0 .

   Every positive argument 𝑛 appearing in either copy of

                                    𝑅Λ (𝑁 ) =                   Λ(𝑛1 )Λ(𝑛2 )
                                                         ∑︁

                                                  𝑛1 +𝑛2 =𝑁


   satisfies 1 ≤ 𝑛 ≤ 𝑁 = 𝑦 𝐽0 . The exact identity above applies to both Λ(𝑛1 ) and Λ(𝑛2 ). The case
𝑛 = 1 is harmless because log 1 = 0, and 𝑛 = 0 is not an argument of Λ in the positive-integer
Goldbach convolution.
   The coefficient types in B1 are also exactly those generated by (X1.1):

                             𝜇(𝑚𝑖 )1𝑚𝑖 ≤𝑦 ,         log 𝑟1 ,        1 on 𝑟2 , . . . , 𝑟𝑗 .

    There is no missing factor of 𝑗. The logarithm is attached to one ordered 𝑟-variable; this
corresponds to the single factor −𝜁 ′ (𝑠)𝜁(𝑠)𝑗−1 , not to differentiating 𝜁(𝑠)𝑗 .

X1.4. Dyadic localization check After the identity, B1 inserts an exact smooth dyadic partition

                                                         𝜔𝑋 (𝑣) = 1
                                               ∑︁

                                                𝑋

    for each positive Heath–Brown variable 𝑣. On the support of a factorization of 𝑛 ≤ 𝑁 , every
variable is at most 𝑁 , so only 𝑂(log 𝑁 ) dyadic scales occur per variable. Since 𝐽0 is fixed, the total
number of dyadic blocks in the two-sided Goldbach decomposition is

                                              𝑂𝐽0 (log 𝑁 )4𝐽0 ,
                                                    (︀             )︀


   as stated in B1. This localization is exact and creates no analytic error term.
   The dyadic weights preserve the B1 coefficient classes:

                                 𝜇 · 1≤𝑦 · 𝜔𝑋 ,            (log) · 𝜔𝑋 ,        𝜔𝑋 .

   Their pointwise sizes are divisor/polylog bounded on each block, with the only unbounded
elementary factor being log 𝑟1 ≤ log 𝑁 .

X1.5. Output for B1        The X1 input supplies exactly what Lemma B1 uses:

   1. exact finite decomposition of every Λ(𝑛), 1 ≤ 𝑛 ≤ 𝑁 ;

   2. no truncation error from the Heath–Brown identity;

   3. no error from dyadic localization;

   4. no extra coefficient type beyond 𝜇1≤𝑦 , 1, and log;

                                                           47
  5. fixed-𝐽0 polylogarithmic block count.

   Thus the B1 output used by I1 is not conditional on an unverified analytic estimate. X1 is a
standard formal identity whose applicability is checked in the exact B1 range.

                Parameter check B.1 (X1.6. Parameter check and conclusion).
                            X1 is verified in the exact B1 range.
   The Heath–Brown identity used in Lemma B1 is valid for the chosen 𝐽0 and 𝑦 = 𝑁 1/𝐽0 . It
applies to every positive argument in the Goldbach convolution, and the subsequent smooth dyadic
decomposition remains exact. X1 is a closed external input for the proof tree.

X1.7. Logical dependencies           Internal dependency served: B1.

B.2     Near-global Davenport/AP verification
B.2.1    X9L-GT. Davenport/AP Input for TC1 Testing
X9L-GT.0. Statement and Role               Lemma X9L-GT states and verifies the external input

                                 X9L-GT      or   X9L-AVG-POLYLOG.

   It is the averaged Liouville/Fourier input used by TGT for MRT-admissible TC1 testing families.
   To avoid ambiguity, the statement has two logically separate layers.

  1. **General low-𝜃 target.** This is the broad low-𝜃 polylog-modulus AP-fibre estimate one
     might want for arbitrary 𝐻 ≥ 𝑋 𝜃 , 0 < 𝜃 < 1/3. This proof does not claim a published citation
     for that general target.

  2. Near-global X9L-GT theorem. This is the theorem invoked by the proof tree after TTH. In
     that route, every surviving B1-origin TC1 coarea test has near-global length 𝐻 ≥ 𝑋(log 𝑋)−𝐵 ,
     and Davenport/AP cancellation is sufficient.

    Only the second layer is used by this proof.
    The target is not pointwise shifted short-interval cancellation. The target is an averaged
statement stable under:

  1. arithmetic progression fibres 𝑛 = 𝑔𝑢 + 𝑏;

  2. 𝑔 ≤ (log 𝑋)𝐶 ;

  3. linear phases depending on the fibre;

  4. testing measures whose pushforward to starts is dominated by a polylogarithmic density;

  5. fibre lengths 𝑈 = 𝐻/𝑔, with the normalized sum divided by 𝑈 .

   The unused general low-𝜃 target is:


ordinary qualitative short-interval estimates do not by themselves prove the full low-𝜃 polylog-modulus form.

   This proof does not use that full low-𝜃 form. Its input is the following near-global theorem:

                                                     48
𝑋9𝐿-𝐴𝑉 𝐺-𝑃 𝑂𝐿𝑌 𝐿𝑂𝐺 is supplied for unrouted TC1 coarea tests by Davenport/AP whenever 𝐻 ≥ 𝑋(log 𝑋)−𝐵 .

    Thus the broader unused target is:

X9L-POLYLOG-MOD<1/3 : prove the same averaged normalized AP-fibre Fourier estimate for every fixed 0 < 𝜃 < 1/3, without the TTH near-global restriction.


   Logical dependencies are TGT, MRT, TTH, TNG, and the parameter register. X9L-GT is used
by TGT, TTD, TTH, TNG, and E10L.
   —

X9L-GT.1. External Source The external source is:
   H. Davenport, "On some infinite series involving arithmetical functions (II)", Quart. J. Math.
Oxford 8 (1937), 313–320, DOI 10.1093/qmath/os-8.1.313.
   We use the standard Davenport consequence: for every 𝐴 > 0,
                                                   ⃒              ⃒
                                                   ⃒              ⃒
                                              sup ⃒     𝜇(𝑛)𝑒(𝛼𝑛)⃒⃒ ≪𝐴 𝑌 (log 𝑌 )−𝐴 .                                                          (Dav)
                                                   ⃒ ∑︁           ⃒
                                                   ⃒
                                             𝛼∈R/Z ⃒𝑛≤𝑌           ⃒

   The AP/interval form for 𝜆 follows from 𝜆 = 𝜇 * 1□ , a square-divisor split, additive-character
expansion of the AP condition, and summation by parts for smooth weights.

X9L-GT.2. Statement: Required Normalized AP-Fibre Form The form needed by TGT
can be abstracted as follows.
   Fix 𝐶 > 0, 0 < 𝜃 < 1, and a testing measure 𝜈 whose start pushforward satisfies

                                                                                          𝑑𝑥
                                                        (start)# 𝜈 ≪ (log 𝑋)𝐶                .                                              (PACK)
                                                                                          𝑋
    For parameters 𝑝 in the test family, let

                             𝑔𝑝 ≤ (log 𝑋)𝐶 ,                  𝐻𝑝 ≍ 𝐻,           𝑈𝑝 = 𝐻𝑝 /𝑔𝑝 ,              𝐻𝑝 ≥ 𝑋 𝜃 .

    The needed Fourier test is of the shape
                                                       ⃒                              ⃒
                                                       ⃒ 1
                                                       ⃒                              ⃒
                                         ℒ𝑝 (𝜆) = sup ⃒⃒           𝜆(𝑔𝑝 𝑢 + 𝑏𝑝 )𝑒(𝛼𝑢)⃒⃒ ,
                                                              ∑︁                      ⃒
                                                 𝛼∈R/Z ⃒ 𝑈𝑝 1≤𝑢≤𝑈𝑝                    ⃒

    possibly with smooth weights of fixed/polylogarithmic complexity. The desired input is
                                                         ∫︁
                                                              |ℒ𝑝 (𝜆)|2 𝑑𝜈(𝑝) = 𝑜(1).                                                    (X9L-GT)

    The normalization by 𝑈𝑝 = 𝐻𝑝 /𝑔𝑝 is essential. Bounds normalized by the ambient length 𝐻𝑝
are not enough unless they save a factor 𝑔𝑝 .
    —




                                                                          49
X9L-GT.3. Scope Check: Limitation of the Unused Low-Theta Target                         The ordinary
Fourier theorem gives averaged cancellation for

                                        1   ∑︁
                                                  𝜆(𝑛)𝑒(𝛽𝑛).
                                        𝐻 𝑥<𝑛≤𝑥+𝐻

   For an AP fibre,

                              𝜆(𝑔𝑢 + 𝑏)𝑒(𝛼𝑢) =
                        ∑︁                            ∑︁
                                                                 𝜆(𝑛)𝑒(𝛼(𝑛 − 𝑏)/𝑔).
                        𝑢≤𝑈                         𝑏<𝑛≤𝑏+𝑔𝑈
                                                   𝑛≡𝑏 (mod 𝑔)

   Expanding the congruence by additive characters gives
                      ⃒                    ⃒         ⃒                    ⃒
                      ⃒ 1 ∑︁                         ⃒1
                      ⃒                    ⃒         ⃒                    ⃒
                             𝜆(𝑔𝑢 + 𝑏)𝑒(𝛼𝑢)⃒ ≤ 𝑔 sup ⃒                                      (AP-loss)
                                           ⃒              ∑︁              ⃒
                      ⃒
                      ⃒𝑈
                                           ⃒         ⃒          𝜆(𝑛)𝑒(𝛽𝑛)⃒⃒ .
                      ⃒ 𝑢≤𝑈                ⃒      𝛽 ⃒ 𝐻 𝑏<𝑛≤𝑏+𝐻           ⃒

    Thus a bare qualitative 𝑜(1) average for ordinary intervals does not imply the normalized AP-
fibre statement uniformly for 𝑔 ≤ (log 𝑋)𝐶 . One needs either:

  1. a logarithmic saving strong enough to absorb 𝑔;

  2. a theorem stated directly relative to AP length 𝑈 = 𝐻/𝑔;

  3. a proof that all TC1 moduli 𝑔 are bounded independently of 𝑁 .

   The TC1 route only gives 𝑔 ≤ (log 𝑋)𝐶 , not bounded 𝑔.
   —

X9L-GT.4. Proof: Near-Global Davenport/AP Transfer The proof applies X9L-GT only
after TTH, where every surviving B1-origin coarea test has

                                           𝐻 ≥ 𝑋(log 𝑋)−𝐵 .                                     (NG)
   For a fibre sum

                      𝑆=         𝜆(𝑔𝑢 + 𝑏)𝑒(𝛼𝑢),          𝐻 = 𝑔𝑈,       𝑔 ≤ (log 𝑋)𝐶 ,
                           ∑︁

                           𝑢≤𝑈

    expand the congruence 𝑛 ≡ 𝑏 (mod 𝑔) by additive characters and transfer the phase 𝑒(𝛼𝑢) to a
linear phase in 𝑛. This gives a loss ≤ 𝑔, and the normalization by 𝑈 = 𝐻/𝑔 gives a second factor 𝑔.
Davenport’s bound, applied to global prefixes and then differenced over the interval of length 𝐻,
gives
                                ⃒                     ⃒
                                ⃒1
                                ⃒ ∑︁                  ⃒
                                                              𝑋
                           sup ⃒⃒      𝜇(𝑔𝑢 + 𝑏)𝑒(𝛼𝑢)⃒⃒ ≪𝐴 𝑔 2 (log 𝑋)−𝐴 .                  (Dav-AP)
                                                      ⃒
                            𝛼 ⃒𝑈                              𝐻
                                   𝑢≤𝑈                ⃒

   Under (NG), the factor 𝑔 2 𝑋/𝐻 is at most a fixed power of log 𝑋. By choosing the Davenport
saving exponent larger than this polylogarithmic loss, we obtain arbitrary logarithmic saving for
the normalized AP fibre.
   For 𝜆, use

                                                     50
                                               𝜆(𝑛) =              𝜇(𝑛/𝑑2 ).                                       (Sq)
                                                           ∑︁

                                                           𝑑2 |𝑛

    The terms 𝑑 ≤ (log 𝑋)𝐷 are handled by the same Davenport/AP argument after changing
variables 𝑚 = 𝑛/𝑑2 ; the near-global condition is stable under this polylogarithmic square-divisor
division. The terms 𝑑 > (log 𝑋)𝐷 have PACK-averaged normalized contribution
                                                ≪ (log 𝑋)𝑂(𝐶) 𝐷−1 .
   Taking 𝐷 large gives the required 𝑜(1) bound.
   Thus the theorem supplied here is:


X9L-GT-NG : normalized AP-fibre Fourier cancellation holds for all unrouted TC1 coarea tests satisfying 𝐻 ≥ 𝑋(log 𝑋)−𝐵 .

   —

X9L-GT.5. Proof: Explicit AP/Congruence Transfer The Davenport step used above can
be isolated as follows.
    Let 𝑞 ≤ (log 𝑋)𝐶 , 𝐻 ≥ 𝑋(log 𝑋)−𝐵 , and 𝐼 = [𝑥, 𝑥 + 𝐻] ⊂ [𝑋, 2𝑋]. Then for every residue class
𝑏 (mod 𝑞) and every 𝛼 ∈ R/Z,
                                   ⃒                      ⃒
                                   ⃒                      ⃒
                                   ⃒                      ⃒
                                                𝜇(𝑛)𝑒(𝛼𝑛)⃒⃒ ≪𝐴 𝑋(log 𝑋)−𝐴
                                   ⃒    ∑︁                ⃒
                                   ⃒
                                   ⃒
                                   ⃒𝑛≡𝑏 𝑛∈𝐼
                                   ⃒                      ⃒
                                        (mod 𝑞)
                                                          ⃒

   with arbitrary 𝐴. Since the near-global range has 𝐻 ≥ 𝑋(log 𝑋)−𝐵 , this is 𝐻(log 𝑋)−𝐴+𝐵 , and
the exponent 𝐴 can be increased to absorb all fixed polylogarithmic losses.
   Indeed,
                                                      1 ∑︁
                                    1𝑛≡𝑏 (mod 𝑞) =               𝑒(𝑟(𝑛 − 𝑏)/𝑞),
                                                      𝑞 𝑟(mod 𝑞)

   so the left side is bounded by
                                                   ⃒             ⃒
                                                   ⃒ ∑︁          ⃒
                                               sup ⃒    𝜇(𝑛)𝑒(𝛽𝑛)⃒ .
                                                   ⃒             ⃒
                                                𝛽 ⃒  𝑛∈𝐼
                                                                 ⃒

    The interval sum is the difference of two Davenport prefix sums, hence has arbitrary logarithmic
saving relative to the ambient scale 𝑋. Passing from the 𝑛-sum to the normalized fibre sum with
𝑛 = 𝑞𝑢 + 𝑏 divides by 𝑈 = 𝐻/𝑞, so the normalization costs 𝑞𝑋/𝐻, a polylogarithmic factor in
the near-global range. Smooth weights are removed by a fixed partition and summation by parts,
costing only another polylogarithmic factor. The more conservative bound (Dav-AP) above records
an allowable 𝑞 2 𝑋/𝐻 loss; this is still polylogarithmic after TTH.
    For 𝜆, insert 𝜆(𝑛) = 𝑑2 |𝑛 𝜇(𝑛/𝑑2 ). The terms 𝑑 ≤ (log 𝑋)𝐷 are handled by the same congruence-
                         ∑︀

transfer lemma at scale (𝑋/𝑑2 , 𝐻/𝑑2 ); compatibility of 𝑑2 | 𝑛 with 𝑛 ≡ 𝑏 (mod 𝑞) only refines the
residue class by a polylogarithmic modulus. The tail 𝑑 > (log 𝑋)𝐷 is bounded on average by
  𝑑>𝐷 𝑑 , hence is 𝑂(𝐷 ) after the PACK normalization.
∑︀      −2                −1

    This is the precise route:
         TTH near-global length =⇒ Davenport AP/congruence transfer =⇒ X9L-GT-NG.
   —

                                                            51
X9L-GT.6. Scope Check: Unused Low-Theta Extension For a general range 𝐻 ≥ 𝑋 𝜃 ,
0 < 𝜃 < 1/3, the elementary Davenport/AP proof above loses the factor

                                                    𝑋
                                               𝑔2     .
                                                    𝐻
    This is no longer polylogarithmic. A qualitative 𝑜(1) short-interval Fourier theorem for ordinary
intervals also does not imply the normalized AP-fibre statement uniformly for 𝑔 ≤ (log 𝑋)𝐶 , because
the ordinary-to-AP reduction (AP-loss) costs 𝑔.
    Therefore the full low-𝜃 theorem

                                    X9L-POLYLOG-MOD<1/3

   is not asserted here. This is harmless for this proof, since TTH routes every surviving B1-origin
coarea test into the near-global range before X9L-GT is invoked.
   —

X9L-GT.7. Output for the Proof Tree            The proof tree records the

                                  X9L-GT/X9L-AVG-POLYLOG

   interface as the following sharper pair:

  1. Near-global part:

                             𝐻 ≥ 𝑋(log 𝑋)−𝐵         =⇒    𝑋9𝐿-𝐺𝑇 -𝑁 𝐺

   by Davenport/AP, the square-divisor transfer to 𝜆, and polylog tail summation.

  1. Unused general low-theta part:

                                    X9L-POLYLOG-MOD<1/3

   namely the same normalized AP-fibre averaged Fourier estimate for 𝐻 ≥ 𝑋 𝜃 , every fixed
0 < 𝜃 < 1/3, and 𝑔 ≤ (log 𝑋)𝐶 .
   There are two clean ways one could strengthen the unused general theorem:

  1. prove/cite X9L-POLYLOG-MOD<1/3 ;

  2. prove that the regular TC1 branch has bounded modulus 𝑔 = 𝑂𝜅 (1), so ordinary qualitative
     short-interval input loses only a fixed factor.

   Neither strengthening is needed here, because the proof uses the TTH near-global bypass.
   —




                                                 52
X9L-GT.8. Scope Separation          The general low-𝜃 target remains outside the proof:

X9L-POLYLOG-MOD<1/3 is not asserted as a consequence of the cited short-interval estimates.

   The proof tree invokes only the following narrower theorem:


B1-origin TC1 coarea tests satisfying TTH are controlled by the near-global/AP X9L-GT estimate.

   Thus the conclusion is:

                      X9L-GT is proved in the near-global form used here.

   What is proved for this route:

           unrouted B1-origin coarea tests satisfy the cited averaged AP-fibre input.

   No low-𝜃 external input is required, because TTH proves the stronger near-global range-origin
lower bound for every unrouted coarea test. The low-𝜃 theorem

                                    X9L-POLYLOG-MOD<1/3

   remains an unused general target only.
   —

X9L-GT.9. External Theorem and Proof

External sources    The external theorem package is:

  1. Davenport. H. Davenport, "On some infinite series involving

   arithmetical functions (II)", Quart. J. Math. Oxford 8 (1937), 313–320, DOI 10.1093/qmath/
os-8.1.313.
   We use the standard Davenport consequence: for every 𝐴 > 0,
                                   ⃒              ⃒
                                   ⃒              ⃒
                              sup ⃒     𝜇(𝑛)𝑒(𝛼𝑛)⃒⃒ ≪𝐴 𝑌 (log 𝑌 )−𝐴 .                     (Dav)
                                   ⃒ ∑︁           ⃒
                                   ⃒
                             𝛼∈R/Z ⃒𝑛≤𝑌           ⃒

    The same AP/interval form for 𝜆 follows from 𝜆 = 𝜇 * 1□ , with a square-divisor split.
    No other theorem is used here, because the proof uses the near-global Davenport/AP argument
after TTH.




                                                53
Exact input        For every fixed 𝐶, 𝐵, 𝐴 > 0, let a TC1 testing family satisfy:

  1. 𝑔𝑝 ≤ (log 𝑋𝑝 )𝐶 ;

  2. 𝑈𝑝 = 𝐻𝑝 /𝑔𝑝 ;

  3. 𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵 ;

  4. the start pushforward obeys

                                         (start)# 𝜈 ≪ (log 𝑋)𝐶 𝑑𝑥/𝑋;                      (PACK)

  5. all smooth weights have polylogarithmic 𝐶 𝐽 -complexity.

   Then
                        ⃒                                   ⃒2
                        ⃒ 1
              ∫︁        ⃒                                   ⃒
                   sup ⃒⃒          𝜆(𝑔𝑝 𝑢 + 𝑏𝑝 )𝑒(𝛼𝑢)𝑤𝑝 (𝑢)⃒⃒ 𝑑𝜈(𝑝) = 𝑜(1).         (X9L-GT-NG)
                              ∑︁                            ⃒
                    𝛼 ⃒ 𝑈𝑝
                            1≤𝑢≤𝑈𝑝                          ⃒

Proof of the input. First remove the smooth weight by a fixed finite smooth partition and summation
by parts. This only changes the logarithmic loss.
   For a fixed fibre, expand the congruence 𝑛 ≡ 𝑏𝑝 (mod 𝑔𝑝 ) by additive characters. This costs at
most 𝑔𝑝 . Apply Davenport’s bound (Dav) to global prefixes and take differences. The AP fibre is
normalized by 𝑈𝑝 = 𝐻𝑝 /𝑔𝑝 , so the total polylogarithmic loss is at most
                                               𝑋𝑝
                                         𝑔𝑝2      ≤ (log 𝑋𝑝 )2𝐶+𝐵 .
                                               𝐻𝑝

    Choosing the Davenport logarithmic saving exponent larger than 2𝐶 + 𝐵 + 𝐴 gives 𝑂((log 𝑋)−𝐴 )
for every near-global fibre, hence 𝑜(1) after PACK averaging.


Match to the proof tree          TTH proves in fact

                                          𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅

   for every unrouted B1-origin coarea test not already routed to C1, CKP, LocalDiag, LongAP/
Local or Impossible. Therefore the Davenport near-global part alone suffices for this proof.
   Thus X9L-GT is a proved external input for the proof tree.
   —

X9L-GT.10. Logical Dependencies External dependency: Davenport’s exponential-sum
estimate in AP/near-global form, as stated in X9L-GT.9.
    Internal dependencies served: TGT, TTD, TTH, TNG, TC1 global testing, E10L.




                                                      54
B.3     DFI/X10 Kloosterman-fraction verification
B.3.1     X10. DFI Kloosterman Fraction Input
X10.0. Role Logical ID: X10.
  Used by: G4a, CKPX10M, CKPD. I1 uses X10 only through the CKP branch.
  Interface data checked against: G1a, G2a, G3a, G4a, CKPD, X10ER, C1P, C1A, C1, and the
Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction theorem.
  This document states and verifies the external black-box X10 used in the CKP branch:

            𝐺3𝑎 + 𝐺4𝑎 + 𝐶𝐾𝑃 𝐷 + 𝑋10𝐸𝑅 + 𝐶1𝑃/𝐶1𝐴/𝐶1 =⇒ 𝐶𝐾𝑃 𝑋10𝑀 =⇒ 𝐺8𝑎.

    The external input is Duke–Friedlander–Iwaniec Theorem 2 for bilinear Kloosterman fractions,
together with the smooth-weight corollary stated below. Any alternate internal shorthand for the
bilinear Kloosterman-sum form is descriptive only, not a separate external source.
    The goal is not to reprove DFI. The goal is to prove that the CKP interface satisfies the
hypotheses of the cited theorem:

        Does the DFI theorem apply to the exact nonzero-frequency CKP sums used in 𝐺8𝑎?

   The statement includes the following compatibility check:

                    Are the restrictions of X10 already routed by the proof tree?

   The answer is:


Yes, provided all noncentral CKP ranges are routed through 𝑋10-𝐸𝑅 and Lemmas C1P/C1A/C1 as stated.

   —

X10.1. Required CKP form After Lemmas G1a, G2a, and G3a, a nonzero-frequency CKP
contribution has the form
                                                                    ℎ𝑁𝑔 𝑎
                                                                          (︂     )︂
                         𝒪𝑔,ℎ =          𝛼𝑔 (𝑎)𝛾𝑔,ℎ (𝑞)𝑊𝑔,ℎ (𝑎, 𝑞)𝑒
                                   ∑︁
                                                                          ,
                                𝑎∼𝐴 ,𝑞∼𝑄
                                                                     𝑞
                                    𝑔       𝑔
                                  (𝑎,𝑞)=1

   where
                                                𝑁
                                        𝑁𝑔 =      ,        𝑘 = ℎ𝑁𝑔 ,
                                                𝑔

   and
                                                     1             ℎ
                                                                  (︂ )︂
                                        𝑊𝑔,ℎ (𝑎, 𝑞) = 𝐹̂︀𝑎,𝑞
                                                     𝑞             𝑞

   is the smooth Fourier weight from Lemma G2a.
   In the balanced CKP range,

                                                                  𝑁 1/2+𝑂(𝜂)
                                𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,             𝑆𝑔 =              .
                                                                       𝑔

                                                      55
   The Fourier-weight bound is

                              |𝑊𝑔,ℎ (𝑎, 𝑞)| ≪𝐴 (log 𝑁 )𝐶 𝑔(1 + |ℎ|𝑔)−𝐴 .

   The central nonzero-frequency range is restricted to

                                           |ℎ|𝑔 ≤ (log 𝑁 )𝐵 .

   The complementary high-frequency range is already Edge by C1P/C1A/C1.
   It remains to prove:

                                                    𝒪𝑔,ℎ = 𝑜(𝑁 ),
                                         ∑︁ ∑︁

                                         𝑔|𝑁 ℎ̸=0

   after excluding C1-routed large-g, high-frequency, small-conductor, and boundary layers.
   —

X10.2. External theorem We use the following external DFI bilinear Kloosterman fraction
estimate. The identical theorem statement is repeated in Lemma CKPD, Section CKPD.1, so that
the CKP derivative appendix can be read independently of this file.
    The citation is
    W. Duke, J. B. Friedlander, H. Iwaniec, "Bilinear forms with Kloosterman fractions", Invent.
Math. 128 (1997), 23–43, DOI 10.1007/s002220050135.
    No later Kloosterman-fraction strengthening is used as an input. The only CKP external theorem
is DFI Theorem 2 together with the smooth-weight formulation in the same paper.
    Let
                                                                          𝑟𝑚
                                                                     (︂     )︂
                               𝐵𝑟 (𝑀, 𝑄) =
                                                    ∑︁
                                                           𝛼𝑚 𝛽𝑛 𝑒           ,
                                              𝑀 <𝑚≤2𝑀
                                                                           𝑛
                                              𝑄<𝑛≤2𝑄
                                               (𝑚,𝑛)=1

   where 𝑟 is a positive integer and 𝛼𝑚 , 𝛽𝑛 are arbitrary complex coefficients. DFI Theorem 2 gives

                       𝐵𝑟 (𝑀, 𝑄) ≪𝜀 ‖𝛼‖2 ‖𝛽‖2 (𝑟 + 𝑀 𝑄)3/8 (𝑀 + 𝑄)11/48+𝜀 .

   DFI also allows a smooth weight, supported on the same dyadic box and normalized by |𝐹 | ≤ 1,

                                                𝐹 (𝑚, 𝑛)

   provided its derivatives satisfy controlled bounds

                           𝐹 (𝑗,𝑘) (𝑚, 𝑛) ≪ 𝜂 𝑗+𝑘 𝑚−𝑗 𝑛−𝑘 ,          0 ≤ 𝑗, 𝑘 ≤ 2,

   at the cost of multiplying the right-hand side by a harmless factor

                                                     𝜂2.

   For our use, 𝜂 is at most a fixed power of log 𝑁 , so this is absorbed into the polylogarithmic loss.
   —


                                                     56
X10.3. Parameter and hypothesis matching

Parameter dictionary


   DFI object                       CKP object                       Source
   𝑚                                CKP inverse variable 𝑎           G1a/G3a
   𝑛                                CKP modulus variable 𝑞           G1a/G3a
   dyadic length 𝑀                  𝐴𝑔                               G8a central layer
   dyadic length 𝑄                  𝑄𝑔                               G8a central layer
   external integer 𝑟               𝑟 = |ℎ|𝑁𝑔                        G2a/G3a
   coprimality (𝑚, 𝑛) = 1           (𝑎, 𝑞) = 1                       G1a
   coefficient 𝛼𝑚                   𝛼𝑔 (𝑎)                           B1 finite-convolution inheri-
                                                                     tance
   coefficient 𝛽𝑛                   𝛾𝑔,ℎ (𝑞)                         G2a/G3a
   smooth weight 𝐹 (𝑚, 𝑛)           normalized Fourier fibre         CKPD
                                    ̃︁𝑔,ℎ (𝑎, 𝑞)
                                    𝑊
   phase 𝑒(𝑟𝑚/𝑛)                    𝑒(ℎ𝑁𝑔 𝑎/𝑞)                       G3a



    Thus the formal phase and coprimality conditions match exactly.
    Negative ℎ causes no problem: the corresponding phase is the complex conjugate/sign variant
of the same estimate. The case ℎ = 0 is not part of X10; it is the CKP zero-frequency local term
handled by Lemma G8a through the LPI projection and then assembled by H4M.

Hypothesis-by-hypothesis check


   DFI hypothesis                   CKP verification                  Routing if it fails
   Dyadic support 𝑚 ∼ 𝑀 , 𝑛 ∼ 𝑄     The tagged CKP layer has 𝑎 ∼ Boundary or short-volume fail-
                                    𝐴𝑔 , 𝑞 ∼ 𝑄𝑔 after G1a/G8a.        ures are C1P/C1A/C1 Edge
                                                                      inputs.
   Coprimality of inverted variable G1a imposes (𝑎, 𝑞) = 1.           Non-coprime pre-split layers
   and modulus                                                        are not sent to X10; they are
                                                                      resolved in the gcd split.
   Arbitrary complex coefficients B1 finite-convolution coeffi-       Coefficient-size failures are
   allowed with 𝐿2 -norms           cients satisfy divisor-type 𝐿2    Edge/large-content inputs
                                    bounds recorded in G3a/G4a.       through C1P/C1A/C1.
   Smooth two-variable weight       CKPD proves this for the ac-      Noncentral balance failures are
   with derivatives up to order two tual nonseparated 𝑊 ̃︁𝑔,ℎ (𝑎, 𝑞). routed by X10ER and C1P/
                                                                      C1A/C1 before X10.
   Positive external integer 𝑟      Use 𝑟 = |ℎ|𝑁𝑔 ; the sign of ℎ is ℎ = 0 is the local term and
                                    handled by conjugation.           is handled by G8a/LPI, then
                                                                      assembled by H4M.
   Uniformity in 𝑟 with loss (𝑟 + In the central frequency range High-frequency layers are C1P/
   𝑀 𝑄)3/8                          𝑟/(𝑀 𝑄) ≍ |ℎ|𝑔 ≤ (log 𝑁 )𝐵 .      C1A/C1 Edge inputs.
   Central balanced lengths         𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑁    1/2+𝑂(𝜂)
                                                            /𝑔.       Unbalanced and large-𝑔 layers
                                                                      are X10ER and C1P/C1A/C1
                                                                      inputs.



                                                  57
   —

X10.4. Coefficient admissibility DFI allows arbitrary complex coefficient sequences. Our
sequences satisfy the stronger bounds

                                 𝑔 (log 𝑁 ) ,                           𝑔 (log 𝑁 ) .
                                           𝐶                                      𝐶
                       ‖𝛼𝑔 ‖2 ≪ 𝐴1/2                        ‖𝛾𝑔,ℎ ‖2 ≪ 𝑄1/2
   These follow from the finite-convolution/divisor-bounded structure inherited from B1 and from
the CKP routing.
   Therefore the coefficient condition passes.
   —

X10.5. Smooth-weight admissibility The derivative check in this subsection is supplied in full
by Lemma CKPD. The display below is the proof-interface summary of that lemma.
   The Fourier weight is
                                                      1              ℎ
                                                                   (︂ )︂
                                         𝑊𝑔,ℎ (𝑎, 𝑞) = 𝐹̂︀𝑎,𝑞          .
                                                      𝑞              𝑞
   By Lemma G2a, in the central CKP range it satisfies
                                 𝑊𝑔,ℎ (𝑎, 𝑞) ≪𝐴 (log 𝑁 )𝐶 𝑔(1 + |ℎ|𝑔)−𝐴 .
                                                                  ̃︁𝑔,ℎ ≪ 1 and has smooth derivative
   Moreover, after normalizing by its supremum size, it satisfies 𝑊
bounds of the DFI weighted-corollary type:
                               ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ (log 𝑁 )𝐶 𝑎−𝑗 𝑞 −𝑘 ,
                       𝜕𝑎𝑗 𝜕𝑞𝑘 𝑊                                           1 ≤ 𝑗 + 𝑘 ≤ 2,
   provided
                                              |ℎ|𝑔 ≤ (log 𝑁 )𝐵 .
    The nonseparated dependence on both 𝑎 and 𝑞 is intentional. The weight is not absorbed into
𝛾𝑔,ℎ (𝑞) alone. The chain-rule terms are the ones proved in Lemma CKPD: on the central CKP
support (𝑁𝑔 − 𝑎𝑦)/𝑞 ≍ 𝑌 ′ , so differentiating 𝑊𝑌 ′ ((𝑁𝑔 − 𝑎𝑦)/𝑞) in 𝑞 gives a factor
                                         𝑁𝑔 − 𝑎𝑦
                                                 · (𝑌 ′ )−1 ≪ 𝑄−1
                                                               𝑔 ,
                                           𝑞2
and differentiating in 𝑎 gives
                                                  𝑦
                                                    · (𝑌 ′ )−1 ,
                                                  𝑞
which is admissible in the central balanced range 𝑌 ≍ 𝑌 ′ , 𝐴𝑔 ≍ 𝑄𝑔 . Mixed derivatives up to order
two are bounded in the same way, with only the finite B1 smoothness/polylogarithmic loss. Ranges
where these balance relations fail are not part of the X10 call; they are routed to X10ER and C1P/
C1A/C1 as excluded CKP boundary ranges.
   If |ℎ|𝑔 > (log 𝑁 )𝐵 , the term is not sent to X10; it is high-frequency Edge by C1P/C1A/C1.
   Therefore the smooth-weight condition passes with a polylogarithmic loss. It is no longer an
open internal obligation; it remains only a standard external-citation check that DFI’s weighted
formulation is invoked in the stated form.
   —

                                                       58
X10.6. Uniformity in 𝑘 = ℎ𝑁𝑔 DFI Theorem 2 is uniform in the positive integer external
parameter 𝑟, with right-hand side depending on 𝑟 ≡ 𝑘 through

                                                (𝑟 + 𝑀 𝑄)3/8 .

   In our central range,

                                                                    𝑁
                                         𝑀 𝑄 ≍ 𝐴𝑔 𝑄𝑔 ≍ 𝑆𝑔2 ≍           ,
                                                                    𝑔2

   while
                                                             |ℎ|𝑁
                                            |𝑘| = |ℎ|𝑁𝑔 =         .
                                                               𝑔

   Therefore
                                                 |𝑘|
                                                     ≍ |ℎ|𝑔.
                                                 𝑀𝑄

   On the central frequency range

                                              |ℎ|𝑔 ≤ (log 𝑁 )𝐵 ,

   we have

                                        |𝑘| + 𝑀 𝑄 ≪ 𝑀 𝑄(log 𝑁 )𝐵 .

   Thus the dependence on 𝑘 costs only a polylogarithmic factor. This is harmless.
   The case of small conductor 𝑞/(𝑞, 𝑘) ≤ (log 𝑁 )𝐵 is already routed through C1A to Lemma C1,
Edge predicate E5. DFI itself does not require (𝑘, 𝑞) = 1, since its theorem is stated for arbitrary
positive integer external parameter. Therefore the gcd (𝑘, 𝑞) creates no additional obstruction for
X10.
   —

X10.7. Loss accounting for one (𝑔, ℎ)-layer                Let

                                                                  𝑁 1/2+𝑂(𝜂)
                                  𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,           𝑆𝑔 =              .
                                                                       𝑔

    For simplicity write 𝑀 = 𝑄 = 𝑆𝑔 . DFI gives, with the normalized smooth Fourier weight
included,

               |𝒪𝑔,ℎ | ≪𝜀 (log 𝑁 )𝐶 𝑔(1 + |ℎ|𝑔)−𝐴 ‖𝛼𝑔 ‖2 ‖𝛾𝑔,ℎ ‖2 (|𝑘| + 𝑆𝑔2 )3/8 (2𝑆𝑔 )11/48+𝜀 .

   The prefactor 𝑔(1 + |ℎ|𝑔)−𝐴 in this display is precisely the unnormalized amplitude 𝒜𝑔,ℎ,𝑅
from CKPD.7, after absorbing fixed powers of log 𝑁 and choosing 𝐴 smaller than 𝑅 by a fixed
margin. Thus the displayed bound already includes the amplitude accounting for the normalization
              ̃︁𝑔,ℎ .
𝒲𝑔,ℎ = 𝒜𝑔,ℎ,𝑅 𝑊
   Using

                                       ‖𝛼𝑔 ‖2 ‖𝛾𝑔,ℎ ‖2 ≪ 𝑆𝑔 (log 𝑁 )𝐶 ,

                                                      59
   and

                                         |𝑘| + 𝑆𝑔2 ≪ 𝑆𝑔2 (log 𝑁 )𝐵 ,

   we get

                         |𝒪𝑔,ℎ | ≪𝜀 (log 𝑁 )𝐶 𝑔(1 + |ℎ|𝑔)−𝐴 𝑆𝑔 𝑆𝑔3/4 𝑆𝑔11/48+𝜀 .

   Since
                                                 3 11  95
                                            1+    +   = ,
                                                 4 48  48
   this becomes

                             |𝒪𝑔,ℎ | ≪𝜀 (log 𝑁 )𝐶 𝑔(1 + |ℎ|𝑔)−𝐴 𝑆𝑔95/48+𝜀 .

   Substituting 𝑆𝑔 = 𝑁 1/2+𝑂(𝜂) /𝑔 ≡ 𝑁 1/2 /𝑔 at the exponent level,

                     |𝒪𝑔,ℎ | ≪𝜀 𝑁 95/96+𝜀+𝑂(𝜂) (log 𝑁 )𝐶 𝑔 −47/48−𝜀 (1 + |ℎ|𝑔)−𝐴 .

   Thus one central CKP layer has a power saving over 𝑁 ≡ 𝑁 1 , namely approximately

                                             𝑁 −1/96+𝑂(𝜀+𝜂) .

   Choosing 𝜀, 𝜂 sufficiently small preserves a fixed power saving.
   —

X10.8. Summation over ℎ and 𝑔             The frequency sum is harmless because for large 𝐴,

                                               (1 + |ℎ|𝑔)−𝐴 ≪ 1.
                                          ∑︁

                                          ℎ̸=0

   More precise bounds give an additional 𝑔 −1 when useful, but this is not needed.
   The gcd parameter satisfies

                                                   𝑔|𝑁

   by G1a. Hence the number of possible 𝑔-layers is divisor-bounded:

                                          #{𝑔 : 𝑔 | 𝑁 } ≪𝜀 𝑁 𝜀 .

   Equivalently, this contributes only an 𝑁 𝑜(1) or polylogarithmic/divisor loss in the ledger-level
asymptotic accounting.
   Thus

                                      |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝑜(1)+𝑂(𝜂)+𝜀 = 𝑜(𝑁 ),
                           ∑︁ ∑︁

                           𝑔|𝑁 ℎ̸=0

   provided 𝜂 > 0 and the DFI 𝜀 > 0 are fixed so small that
                                                                 1
                                         𝑂(𝜂) + 𝜀 + 𝑜(1) <         .
                                                                96
   This leaves a fixed power saving over 𝑁 . All noncentral ranges are already routed to X10ER
and C1P/C1A/C1 before this summation is used.
   —

                                                     60
X10.9. Excluded-range routing
Lemma B.2 (Lemma X10ER). The X10 input applies only to the central CKP nonzero-frequency
range. Every CKP nonzero-frequency layer outside that central range is routed before the DFI
estimate is invoked:
  1. High Fourier frequency:
                                          |ℎ|𝑔 > (log 𝑁 )𝐵 .
   Routed to the Edge admission ledger C1A and then Lemma C1, Edge predicate E4.
  1. Small conductor:
                                        𝑞/(𝑞, 𝑘) ≤ (log 𝑁 )𝐵 .
   Routed to the Edge admission ledger C1A and then Lemma C1, Edge predicate E5.
  1. Large gcd/content:
                                              𝑔 > 𝑁𝜂
    or any large-g layer outside CKP balance.
    Routed by the gcd/content saving recorded in G1a and G8a to the Edge admission ledger C1A
before Lemma C1, Edge predicate E3.
  1. Short/boundary volume:
   Routed to the Edge admission ledger C1A and then Lemma C1, Edge predicates E1/E6/E7 as
appropriate.
   Therefore X10 is not responsible for all CKP-looking terms, only for the central nonzero-frequency
DFI range.
   —

X10.10. Compatibility of X10 restrictions with the proof tree The restrictions in X10
do not obstruct the Goldbach proof. They are not additional hypotheses; they are the interface
conditions separating the central DFI range from the noncentral ranges already handled elsewhere
in the proof tree.
    The correct CKP nonzero-frequency decomposition is:
       CKPℎ̸=0 = CentralDFI ⊔ HighFreq ⊔ SmallConductor ⊔ LargeG ⊔ Boundary/Short.
   Then the routing is:
                                        CentralDFI → 𝑋10,
                                    HighFreq → 𝐶1𝑃/𝐶1𝐴/𝐶1,
                                SmallConductor → 𝐶1𝑃/𝐶1𝐴/𝐶1,
                               LargeG → 𝑋10𝐸𝑅 → 𝐶1𝑃/𝐶1𝐴/𝐶1,
                                Boundary/Short → 𝐶1𝑃/𝐶1𝐴/𝐶1.
   Thus the fact that X10 is only used on the central range is correct and necessary.
   —

                                                 61
X10.11. Restriction-by-restriction routing check

   X10 restriction                   Required proof-tree support      Current source of support
   Only central balanced CKP is      B3/F3/F3T isolate CKP atoms, Lemmas B3, F3, F3T,
   sent to DFI                       CKPX10M sends only central CKPX10M, G8a
                                     nonzero layers to X10, and G8a
                                     keeps ℎ = 0 local
   High-frequency layers are ex-     Fourier decay must make |ℎ|𝑔 > Lemmas G2a, X10ER, C1A, C1
   cluded                            (log 𝑁 )𝐵 an Edge tail           E4
   Small-conductor layers are ex-    Small conductor must be Edge Lemmas C1A, C1 E5
   cluded                            only in CKP-normalized oscilla-
                                     tory scale
   Large-𝑔 layers are excluded       GCD splitting gives volume       Lemmas G1a, X10ER, C1A, C1
                                     saving 𝑁/𝑔 2                     E3
   Boundary/short-volume layers      Boundary and short-volume        Lemmas C1A, C1 E1/E6/E7
   are excluded                      atoms must satisfy strict Edge
                                     predicates
   Smooth weighted fibre expan-      AP expansion must use full       Lemmas G2a, CKPD,
   sion is required                  tagged fibre weight, not bare    CKPX10M
                                     𝑊𝑌 (𝑦) only
   GCD (𝑘, 𝑞) > 1 may occur          Small conductor cases are re-    Lemmas C1A, C1 E5, and the
                                     moved; DFI itself is uniform in X10 theorem statement
                                     external 𝑟 = 𝑘
   Summation over 𝑔 must be          G1a gives 𝑔 | 𝑁 , hence divisor- Lemmas G1a, CKPX10M
   harmless                          bounded number of 𝑔-layers



   Therefore the X10 restrictions are already accounted for in the proof tree. They do not create
an additional terminal class and do not leave an unhandled CKP residual.
   —

X10.12. No new residual class created by X10 restrictions The X10 restrictions would
leave a residual class only if one of the following failed:

  1. high-frequency terms were not actually Edge;

  2. small-conductor terms were not actually Edge in the CKP-normalized scale;

  3. large-𝑔 terms did not have volume saving;

  4. boundary/short-volume terms did not satisfy strict C1P predicates;

  5. the CKP AP expansion used the wrong bare weight rather than the full tagged fibre weight;

  6. B3/F3 failed to make the central/noncentral split exhaustive.

   The proof tree addresses exactly these risks:

                                    𝐶1𝑃/𝐶1𝐴/𝐶1     closes Edge tails,

                                 𝐺2𝑎   closes weighted AP expansion,

                                                   62
                           𝐺8𝑎    closes CKP normalization and routing,

                    𝐵3 + 𝐹 3 + 𝐹 4    close classification and routing exhaustion.

   Thus,

                    X10 restrictions do not interfere with the Goldbach proof.

   They are part of the correct division of labour:

        central CKP → 𝑋10,           noncentral CKP residuals → 𝑋10𝐸𝑅 → 𝐶1𝑃/𝐶1𝐴/𝐶1.

   —

X10.13. Conclusion

Conclusion
                              PASS with explicit routed restrictions.
   The DFI theorem applies to the exact CKP nonzero-frequency sums after the reductions in
Lemmas G1a, G2a, and G3a, provided the following restrictions are enforced:

  1. only central balanced CKP ranges are sent to X10;
  2. high-frequency, small-conductor, large-𝑔, and boundary ranges are routed through X10ER
     and C1P/C1A/C1;
  3. the smooth Fourier weight is normalized as a DFI-admissible smooth weight with at most
     polylogarithmic derivative parameter;
  4. finite-convolution coefficient losses remain polylogarithmic;
  5. the central frequency range satisfies

                                           |ℎ|𝑔 ≤ (log 𝑁 )𝐵 ;

  1. 𝑔-summation uses the fact that G1a gives 𝑔 | 𝑁 .

   Under these conditions,

                                                    𝒪𝑔,ℎ = 𝑜(𝑁 ).
                                         ∑︁ ∑︁

                                         𝑔|𝑁 ℎ̸=0


Output for the CKP Branch The external X10 input discharges the DFI applicability condition
in G4a, as packaged and summed by CKPX10M.
    Consequently, X10 is verified with restrictions. DFI Theorem 2 and its smooth-weight corollary
apply to the central CKP nonzero-frequency sums with 𝑀 = 𝐴𝑔 , 𝑄 = 𝑄𝑔 , and positive external
integer parameter 𝑟 = |ℎ|𝑁𝑔 . For ℎ < 0, the same estimate is applied to the conjugate phase. The
resulting saving is 𝑁 −1/96+𝑜(1) in the balanced range, sufficient after summation over ℎ and divisor-
bounded 𝑔-layers. Boundary, high-frequency, small-conductor and large-𝑔 ranges remain assigned to
X10ER and C1P/C1A/C1, and there is no residual CKP terminal class because all excluded ranges
are routed through Lemmas C1A, C1, G2a, G8a, and X10ER before X10 is invoked.
    —

                                                     63
X10.14. External theorem invocation

External source The external theorem used in X10 is:
    W. Duke, J. B. Friedlander, H. Iwaniec, "Bilinear forms with Kloosterman fractions", Invent.
Math. 128 (1997), 23–43, DOI 10.1007/s002220050135.
    The invoked result is Theorem 2 of that paper, together with the weighted variant obtained by
inserting a smooth function 𝐹 (𝑚, 𝑛) satisfying the derivative bounds stated after formula (1.8) in
the same paper. Thus the the citation is not "bare Theorem 2 only"; it is DFI Theorem 2 plus its
smooth-weight formulation, and the derivative hypotheses are part of the proof interface checked
above.

Statement used here Let 𝑀, 𝑄 ≥ 1, 𝑟 ≥ 1, and let 𝛼𝑚 , 𝛽𝑞 be arbitrary complex sequences
supported on 𝑚 ∼ 𝑀 , 𝑞 ∼ 𝑄. Let 𝐹 (𝑚, 𝑞) be a smooth weight supported in the same dyadic box,
with |𝐹 (𝑚, 𝑞)| ≤ 1, and satisfying, for 0 ≤ 𝑖, 𝑗 ≤ 2,

                                              𝜕𝑞 𝐹 (𝑚, 𝑞) ≪ 𝜂 𝑖+𝑗 𝑀 −𝑖 𝑄−𝑗 .
                                            𝑖 𝑗
                                           𝜕𝑚

   Then, for every 𝜀 > 0,


                               𝑟𝑚
                                 (︂         )︂
               𝛼𝑚 𝛽𝑞 𝐹 (𝑚, 𝑞)𝑒                   ≪𝜀 𝜂 2 ‖𝛼‖2 ‖𝛽‖2 (𝑟 + 𝑀 𝑄)3/8 (𝑀 + 𝑄)11/48+𝜀 .        (DFI-X10)
         ∑︁

      𝑚∼𝑀, 𝑞∼𝑄
                                𝑞
       (𝑚,𝑞)=1


    In the CKP application, 𝜂 ≤ (log 𝑁 )𝐶 , so the 𝜂 2 factor is absorbed into the existing polyloga-
rithmic loss.

Exact substitution      The central CKP nonzero-frequency sum has

                   𝑚 = 𝑎,        𝑞 = 𝑞,              𝑀 = 𝐴𝑔 ,        𝑄 = 𝑄𝑔 ,       𝑟 = |ℎ|𝑁𝑔 .

   The coprimality (𝑎, 𝑞) = 1 is supplied by Lemma G1a. The smooth weight is the normalized
Fourier fibre weight

                                                             ̃︁𝑔,ℎ (𝑎, 𝑞)
                                                  𝐹 (𝑚, 𝑞) = 𝑊

   from G8a.3/G3a.2. Explicitly, CKPD.4 defines

                                 1                                          𝑁𝑔 − 𝑎𝑦     ℎ𝑦
                                      ∫︁                               (︂         )︂ (︂      )︂
                 𝒲𝑔,ℎ (𝑎, 𝑞) =             𝜔𝐴 (𝑎)𝜔𝑄 (𝑞)𝑊𝑌 (𝑦)𝑊𝑌 ′                   𝑒 −           𝑑𝑦
                                 𝑞                                             𝑞         𝑞

    and 𝐹 = 𝑊̃︁𝑔,ℎ = 𝒜−1 𝒲𝑔,ℎ , with 𝒜𝑔,ℎ,𝑅 accounted for in X10.7. Its derivative bounds, including
                       𝑔,ℎ,𝑅
the chain-rule dependence of 𝐹𝑎,𝑞 on 𝑎 and 𝑞, are proved in Lemma CKPD. They use the central
frequency condition |ℎ|𝑔 ≤ (log 𝑁 )𝐵 and the central balance restrictions.
    The coefficient norms are

                                 𝑔 (log 𝑁 ) ,                                𝑔 (log 𝑁 ) .
                                           𝐶                                           𝐶
                       ‖𝛼𝑔 ‖2 ≪ 𝐴1/2                             ‖𝛾𝑔,ℎ ‖2 ≪ 𝑄1/2

   In the central balanced range


                                                            64
                             𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,         𝑆𝑔 = 𝑁 1/2+𝑂(𝜂0 ) /𝑔,
   and
                                     |ℎ|𝑁𝑔
                                           ≍ |ℎ|𝑔 ≤ (log 𝑁 )𝐵 .
                                     𝐴𝑔 𝑄𝑔
   Thus DFI-X10 gives
                      |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝑂(𝜂0 )+𝜀 (log 𝑁 )𝐶 𝑔 −47/48 (1 + |ℎ|𝑔)−𝐴 .
                        ̸ 0 and divisor-bounded 𝑔 | 𝑁 , this is 𝑜(𝑁 ) once 𝑂(𝜂0 ) + 𝜀 + 𝑜(1) < 1/96.
   After summing over ℎ =

Routing of excluded ranges The DFI theorem is invoked only for central balanced CKP nonzero
frequencies. All excluded ranges are already assigned before X10 is called:
  1. |ℎ|𝑔 > (log 𝑁 )𝐵 : high Fourier tail, routed by Lemmas G2a, X10ER, and C1;
  2. 𝑞/(𝑞, ℎ𝑁𝑔 ) ≤ (log 𝑁 )𝐵 : small conductor, routed by C1P/C1A/C1;
  3. large 𝑔: routed by Lemmas G1a, X10ER, C1P/C1A, and C1;
  4. boundary/short-volume ranges: routed by C1P/C1A/C1;
  5. ℎ = 0: local/main term handled by Lemma G8a through LPI and then assembled by H4M.
    Therefore X10 is complete as a proof unit: the cited theorem, parameter substitution, smooth-
weight condition, loss accounting and excluded-range routing are all explicit. The smooth-weight
derivative condition is supplied by the CKP/X10 derivative appendix.
    —

X10.15. Logical Dependencies This verification confirms the interface with the DFI theorem
as used in the proof tree. It does not independently reprove DFI.
    In a self-contained manuscript, the boxed X10.2/X10.14 invocation and the derivative proof from
Lemma CKPD should appear together in the CKP appendix before the DFI theorem is applied.
    The external-input verification has the following structure:
  1. state the external theorem;
  2. state the exact form used in the proof;
  3. match parameters;
  4. check losses and uniformity;
  5. check whether restrictions create new routing obligations;
  6. verify that those obligations are already handled by existing internal lemmas;
  7. state the conclusion.
   External dependency: Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction estimate in the
form stated in X10.2/X10.14.
   Internal interface data used for the verification: G1a, G2a, G3a, G4a, CKPD, X10ER, C1P,
C1A, and C1. The theorem CKPX10M consumes the verified X10 interface; X10 does not use
CKPX10M as an input.
   Children served: G4a, CKPX10M, CKPD, and I1 through the CKP branch.

                                                  65
B.4     Shiu/AP divisor-average verification
B.4.1     X16. Divisor-Sum Input for BRS
X16.0. Statement and Role Lemma X16 states and verifies the divisor-sum input X16 in the
form used by Lemma BRS. It should be read together with Lemma X16BRS, which separates the
carrier-type reductions from X16C, and with Lemma X16C, which proves the core carrier-slice
estimate.
    The BRS step is the critical TC1 structural step: BRS proves that a singular short-image B1-
origin coarea test is strict C1P Edge unless it already carries a routing tag. The only external/
standard input in that step is X16.
    The goal here is to make X16 precise:


X16 is the finite-convolution B1 carrier-slice divisor estimate used in BRS.1; it follows from X16C and Shiu AP divisor averages.


   Logical dependencies are X16BRS, X16C, BRS, TTH, F4, and Shiu’s arithmetic-progression
Brun–Titchmarsh theorem for multiplicative functions. X16 is used by BRS and by the TC1 near-
global-or-routed chain.
   External sources:

   1. P. Shiu, "A Brun–Titchmarsh theorem for multiplicative functions", Journal fuer die reine
      und angewandte Mathematik 313 (1980), 161–170, DOI 10.1515/crll.1980.313.161.

   2. G. Tenenbaum, "Introduction to Analytic and Probabilistic Number Theory", Graduate Studies
      in Mathematics 163, American Mathematical Society, 3rd ed., 2015, Ch. II.5, Theorem 5.

  Shiu supplies the AP divisor-average input; Tenenbaum supplies the fixed-depth divisor second
moment used inside X16C. No prime distribution theorem is used.
  —

X16.1. Statement Fix the Heath–Brown depth 𝐽0 . Let ℬ be a B1 typed dyadic block and let 𝐶
be a B1 carrier reaching BRS after C1 boundary removal and after all F4 tags have been applied.
The allowed carrier types are exactly those listed in BRS.1:

   1. grouped product carrier;

   2. Goldbach complementary carrier 𝑁 − 𝑃 ;

   3. quotient carrier 𝑠 from a recorded equation 𝐿 = 𝑑𝑠;

   4. controlled divisor quotient of one of the preceding carriers.

    Let 𝑋𝐶 be the dyadic height of 𝐶, and let 𝐼 be an additive interval. Put

                                        𝑌16 := max{|𝐼 ∩ Z|, 𝑋𝐶 (log 𝑁 )−𝐵16 }.

    Then

                                                                    𝑌16
                            Massℬ (𝐶 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶16               + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 ,                    (X16-BRS)
                                                                    𝑋𝐶


                                                              66
    where 𝐶16 , 𝜌16 > 0 depend only on 𝐽0 , the fixed dyadic partition, and the finite routing grammar.
    This is the exact X16 statement invoked by BRS.1. The reductions from the four carrier types to
the core product-carrier estimate are recorded in Lemma X16BRS; the core product-carrier estimate
is proved in Lemma X16C.
    —

X16.2. Setup: Proof Input The estimate is a fixed-order divisor-correlation bound for finite-
convolution carriers. A one-variable divisor average alone is insufficient. Lemma X16C reduces the
product carrier to the model correlation

                 𝜏𝐾1 (𝑝)         𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ≪ 𝑌16 𝑈 (log 𝑁 )𝐶 + 𝑁 1−𝜌 (log 𝑁 )𝐶 ,
           ∑︁              ∑︁

           𝑝∈𝐼             𝑢≍𝑈

    for fixed 𝐾1 , 𝐾2 , 𝐾3 , 𝑋𝑃 𝑈 ≍ 𝑁 , after dyadic localization and with the harmless polylogarithmic
losses coming from the B1 coefficient types 𝜇, 1, log.
    The floor 𝑋𝑃 (log 𝑁 )−𝐵16 is intentional. It avoids the false one-point claim that local divisor
factors at a highly composite carrier value are always polylogarithmic. BRS only needs the floor
version, because a marked image shorter than the floor is monotonically enlarged and still gives a
C1P-certified Edge saving once 𝐵16 is chosen large.
    The proof uses only classical divisor technology:

  1. finite-order divisor bounds for products of boundedly many B1 variables;

  2. dyadic grouping of the carrier 𝑃 , same-side complement 𝑈 , and opposite-side product 𝑄 =
     𝑁 − 𝑃𝑈;
                                                                𝐴 , applied to 𝑄 = 𝑁 − 𝑝𝑢 after
  3. Shiu’s arithmetic-progression Brun–Titchmarsh theorem for 𝜏𝐾
     fixing 𝑝 or 𝑢;

  4. the X16-LFA local-factor averaging lemma for non-coprime AP classes;

  5. partial summation for smooth dyadic weights;

  6. divisor-sum stability under fixed divisor quotients and polylogarithmic CRT restrictions.

   No prime distribution theorem is used in X16.
   —

X16.3. Proof Outline for X16-BRS The following is the reduction outline; the full proof of
the analytic correlation estimate, including the two Cauchy–Schwarz orientations, Shiu modulus
checks, and local-factor averaging, is Lemma X16C.
    First reduce every B1 carrier to a fixed-depth divisor majorant. Since the Heath–Brown depth
𝐽0 is fixed, every coefficient sequence produced by B1, B3, F3/F4 tags and E5-clean transports is
bounded by (log 𝑁 )𝑂𝐽0 (1) 𝜏𝐾 (·) for a fixed 𝐾 = 𝐾(𝐽0 ), after dyadic localization and after absorbing
smooth cutoffs by partial summation.
    For a grouped product carrier 𝐶 = 𝑃 , fixing 𝑃 = 𝑝 leaves a fixed-depth number of factorizations
of 𝑝, but it also leaves a genuine complementary correlation 𝑄 = 𝑁 − 𝑝𝑢. Thus

        Massℬ (𝐶 ∈ 𝐼) ≪ (log 𝑁 )𝐶                                𝜏𝐾1 (𝑝)         𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ,
                                                  ∑︁                       ∑︁

                                           𝑝∈𝐼16 ∩[𝑋𝑃 /2,3𝑋𝑃 ]             𝑢≍𝑈



                                                           67
    where 𝑋𝑃 𝑈 ≍ 𝑁 . This is not bounded by averaging only 𝜏𝐾1 (𝑝). Instead, for fixed 𝑝, the values
𝑁 − 𝑝𝑢 lie in one arithmetic progression modulo 𝑝; for fixed 𝑢, they lie in one arithmetic progression
modulo 𝑢.
    Lemma X16C applies Shiu’s AP theorem, combined with Cauchy–Schwarz and second moments
for fixed divisor functions, and proves

                  𝜏𝐾1 (𝑝)         𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ≪ 𝑌16 𝑈 (log 𝑁 )𝐶 + 𝑁 1−𝜌 (log 𝑁 )𝐶 .
           ∑︁               ∑︁

          𝑝∈𝐼16             𝑢≍𝑈

   Since 𝑈 ≍ 𝑁/𝑋𝑃 , this gives exactly
                                                       𝑌16
                                         𝑁 (log 𝑁 )𝐶       + 𝑁 1−𝜌 (log 𝑁 )𝐶 .
                                                       𝑋𝐶
    The complementary carrier 𝑁 − 𝑃 is identical after replacing 𝐼 by 𝑁 − 𝐼. A quotient carrier 𝑠
from 𝐿 = 𝑑𝑠 is reduced to the grouped product case for 𝑑𝑠; the factor 𝑑 changes both the interval
length and dyadic scale by the same controlled amount, so the ratio 𝑌16 /𝑋𝐶 is preserved up to
polylogarithmic losses. Controlled CRT restrictions split the interval into 𝑂((log 𝑁 )𝐶 ) residue
subintervals, and full-rank affine transports change lattice index and derivatives by 𝑂((log 𝑁 )𝐶 ).
These losses are absorbed in 𝐶16 .
    This reduction outline is deliberately standard rather than deep, but it is not the rejected one-
variable shortcut: the 𝑁 − 𝑝𝑢 correlation is retained and estimated by AP divisor averages. No
prime distribution theorem and no cancellation of Λ is used in X16.
    —

X16.4. Match to BRS.1

Grouped product carrier Fixing 𝐶 = 𝑛 leaves boundedly many factorizations of 𝑛 and
boundedly many remaining parent variables, all of fixed depth 𝑂(𝐽0 ). Summing over 𝑛 ∈ 𝐼 gives a
fixed-order divisor correlation. X16-BRS gives the relative factor 𝑌16 /𝑋𝐶 , with only polylogarithmic
loss.

Complementary carrier                If 𝐶 = 𝑁 − 𝑃 , then 𝐶 ∈ 𝐼 is equivalent to 𝑃 ∈ 𝑁 − 𝐼. The previous
case applies to 𝑃 .

Quotient carrier If 𝐿 = 𝑑𝑠 and 𝐶 = 𝑠, then 𝑠 ∈ 𝐼 restricts 𝑑𝑠 to total length 𝑂(𝐷𝑌 ) inside
dyadic scale 𝐷𝑋𝐶 . Applying the grouped-product carrier estimate to 𝑑𝑠 gives
                                                      𝐷𝑌
                                      𝑁 (log 𝑁 )𝐶16       + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 ,
                                                      𝐷𝑋𝐶
    which is X16-BRS.
    If the quotient relation instead forces local dependence, CKP-balanced structure, short residual
volume, or impossibility, F4 routes the atom away before BRS is invoked.

CRT and full-rank affine transports Controlled CRT restrictions and full-rank affine coordinate
changes alter indices and lengths by at most polylogarithmic factors in the BRS route. Those losses
are absorbed by 𝐶16 . Tagged rank drops do not enter BRS as untagged B1-origin carriers; they are
handled by ROC/BRS case 3 or by E10M.
    —

                                                           68
X16.5. Consequence for TTH          Combining X16-BRS with the singular image condition

                                    |𝐿𝑚 (Ω)| < 𝑋𝑚 (log 𝑋𝑚 )−𝐵

   and choosing 𝐵 larger than the fixed C1-estimate and X16 losses gives a C1P-certified Edge
bound:

                  Mass(𝐿𝑚 (Ω)) ≪ 𝑁 (log 𝑁 )−𝐶0 −10 + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 = 𝑜(𝑁 ).

   Therefore a TC1 coarea test that remains after ROC/BRS must satisfy the near-global lower
bound used by TTH:

                                        𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

    This is the precise reason the TC1 proof does not require a low-𝜃 X9L theorem.
    —

                 Parameter check B.3 (X16.6. Parameter Check and Output).
                  X16-BRS is isolated and proved via the X16C Shiu/AP proof.
   The Shiu invocation, the switch between the 𝑝- and 𝑢-directions, and the divisor-local-factor
averaging are proved in Lemma X16C. Thus the analytic proof obligation is supplied internally,
with Shiu as the only external theorem.
   The insufficient shortcut that bounds only 𝑛∈𝐼 𝜏𝑘 (𝑛) after fixing the carrier value is still not
                                              ∑︀

used, because the remaining variables impose a divisor correlation along 𝑁 − 𝑛𝑣. The new proof
controls that correlation directly.

X16.7. Logical Dependencies         Internal dependencies served: BRS, TTH, X16BRS, X16C.


C     Heath–Brown Decomposition Details
C.1     B1 typed Heath–Brown decomposition
C.1.1    B1. Typed Heath–Brown Decomposition
B1.0. Role Logical ID: B1.
  Lemma B1 is the first technical decomposition node. Its purpose is to replace the two von
Mangoldt factors in

                                  𝑅Λ (𝑁 ) =               Λ(𝑛1 )Λ(𝑛2 )
                                                ∑︁

                                              𝑛1 +𝑛2 =𝑁

   by a finite sum of typed smooth dyadic finite-convolution blocks. No estimate is made and no
contribution is discarded at this stage.
   Used by: B3, F3, F4, I1, H4, BGS, E10M, E10X, the CKP branch, the GoodAWACK branch, and
the X16 carrier-slice branch.
   Uses: PAR for the structural depth 𝐽0 , X1 for the Heath–Brown identity, and X2 for the smooth
dyadic partition of unity.




                                                  69
B1.1. Statement     For fixed sufficiently large 𝐽0 and 𝑦 = 𝑁 1/𝐽0 , the Goldbach sum has the exact
decomposition
                                       𝑅Λ (𝑁 ) =                     𝑐ℬ 𝑅ℬ (𝑁 ),
                                                            ∑︁

                                                          ℬ∈B𝐽0

   where each 𝑅ℬ (𝑁 ) is a typed smooth dyadic finite-convolution block of the form
                                                𝑟             𝑠
                                                     𝑎𝑖 +          𝑏𝑗 = 𝑁,
                                               ∏︁            ∏︁

                                               𝑖=1           𝑗=1

   with
                                                       𝑟, 𝑠 ≤ 2𝐽0 .
   The elementary coefficient types are
                                          𝜇 · 1≤𝑦 ,             1,        log .
   The number of blocks satisfies
                                          #B𝐽0 ≪𝐽0 (log 𝑁 )4𝐽0 .
   The decomposition is exact; no error term is created at the B1 stage.

B1.2. Parameter and Range Setup                   Fix the structural Heath–Brown depth
                                                         𝐽0 ≥ 𝐽* ,
   as allowed by PAR. The value 𝐽0 is a fixed constant of the proof, not a variable depending on 𝑁 .
   Set
                                                       𝑦 = 𝑁 1/𝐽0 .
   Then every 𝑛 ≤ 𝑁 satisfies
                                                         𝑛 ≤ 𝑦 𝐽0 .
   This is the range in which the exact Heath–Brown identity is applied.

B1.3. Exact Heath–Brown Identity For 𝑛 ≤ 𝑁 and 𝑦 = 𝑁 1/𝐽0 , X1 supplies the exact identity
                           𝐽0
                                          (︃      )︃
                                             𝐽0
                  Λ(𝑛) =        (−1)                                          𝜇(𝑚1 ) · · · 𝜇(𝑚𝑗 ) log 𝑟1 .
                           ∑︁                                 ∑︁
                                    𝑗−1

                           𝑗=1
                                             𝑗         𝑚1 ···𝑚𝑗 𝑟1 ···𝑟𝑗 =𝑛
                                                         𝑚1 ,...,𝑚𝑗 ≤𝑦

   Denote the inner 𝑗-th contribution by
                           Λ𝑗 (𝑛) =                            𝜇(𝑚1 ) · · · 𝜇(𝑚𝑗 ) log 𝑟1 .
                                                ∑︁

                                       𝑚1 ···𝑚𝑗 𝑟1 ···𝑟𝑗 =𝑛
                                         𝑚1 ,...,𝑚𝑗 ≤𝑦

   Then
                                       𝐽0
                                                                                       (︃   )︃
                                                                                         𝐽0
                           Λ(𝑛) =            𝑐𝑗 Λ𝑗 (𝑛),              𝑐𝑗 = (−1)𝑗−1
                                       ∑︁
                                                                                            .
                                       𝑗=1
                                                                                         𝑗

   This identity is exact on the range 𝑛 ≤ 𝑁 .

                                                             70
B1.4. Elementary Coefficient Types                 Each 𝑗-block contains 2𝑗 variables:

                                         𝑚1 , . . . , 𝑚𝑗 , 𝑟1 , . . . , 𝑟𝑗 .

   The elementary coefficient types are

                                        𝛼𝑚𝑖 (𝑚𝑖 ) = 𝜇(𝑚𝑖 )1𝑚𝑖 ≤𝑦 ,

                                              𝛼𝑟1 (𝑟1 ) = log 𝑟1 ,

   and

                                     𝛼𝑟𝑖 (𝑟𝑖 ) = 1                (2 ≤ 𝑖 ≤ 𝑗).

   Thus all elementary types belong to

                                              {𝜇 · 1≤𝑦 , 1, log}.

    After dyadic localization these coefficients become smooth dyadic coefficient sequences, but their
arithmetic type remains the same.

B1.5. Exact Smooth Dyadic Partition                     Let

                                              𝜔 ∈ 𝐶𝑐∞ ([1/2, 2])

   be non-negative and satisfy

                                                    𝑡
                                              (︂        )︂
                                                             =1        (𝑡 > 0).
                                    ∑︁
                                          𝜔
                                    𝑘∈Z
                                                   2𝑘

   For the dyadic scale 𝑋 = 2𝑘 , set

                                                         𝑡
                                                                  (︂   )︂
                                              𝜔𝑋 (𝑡) = 𝜔                    .
                                                         𝑋

   Then for every positive integer 𝑛,

                                               1=                𝜔𝑋 (𝑛),
                                                         ∑︁

                                                             𝑋

   where the sum ranges over dyadic scales. If 𝑛 ≤ 𝑁 , only 𝑂(log 𝑁 ) scales occur.
   For each variable 𝑣 in a Heath–Brown block, insert the exact partition

                                               1=                𝜔𝑉 (𝑣).
                                                         ∑︁

                                                             𝑉

   Since this is a partition of unity, the dyadic decomposition creates no error.




                                                             71
B1.6. Dyadically Localized Blocks for Λ                                         For a tuple of dyadic scales

                                                       X = (𝑀1 , . . . , 𝑀𝑗 , 𝑅1 , . . . , 𝑅𝑗 )

    define
                                                            ⎛                                  ⎞
                                                                 𝑗                                                           𝑗
                Λ𝑗,X (𝑛) =                                             𝜇(𝑚𝑖 )𝜔𝑀 (𝑚𝑖 )⎠ (log 𝑟1 )𝜔𝑅 (𝑟1 )                          𝜔𝑅𝑖 (𝑟𝑖 ).
                                             ∑︁                 ∏︁                                                          ∏︁
                                                            ⎝
                                                                                      𝑖                             1
                                    𝑚1 ···𝑚𝑗 𝑟1 ···𝑟𝑗 =𝑛       𝑖=1                                                          𝑖=2
                                      𝑚1 ,...,𝑚𝑗 ≤𝑦

    The exact decomposition becomes
                                                             𝐽0
                                                Λ(𝑛) =                         Λ𝑗,X (𝑛),
                                                             ∑︁          ∑︁
                                                                   𝑐𝑗                                𝑛 ≤ 𝑁.
                                                             𝑗=1           X

   The sum over X is finite. Every variable on the support is at most 𝑁 , so each variable has
𝑂(log 𝑁 ) possible dyadic choices. Let 𝐷𝐽0 (𝑁 ) denote the number of admissible dyadic choices.
Since the number of variables is at most 2𝐽0 ,

                                                           𝐷𝐽0 (𝑁 ) ≪𝐽0 (log 𝑁 )2𝐽0 .

B1.7. Goldbach Block Expansion                                   Insert the localized decomposition into

                                                      𝑅Λ (𝑁 ) =                           Λ(𝑛1 )Λ(𝑛2 ).
                                                                             ∑︁

                                                                         𝑛1 +𝑛2 =𝑁

    This gives the exact identity
                                                                   𝐽0
                                                𝑅Λ (𝑁 ) =                                   𝑅𝑗,𝑗 ′ ,X,Y (𝑁 ),
                                                                   ∑︁                 ∑︁
                                                                            𝑐𝑗 𝑐𝑗 ′
                                                                 𝑗,𝑗 ′ =1             X,Y

    where

                                           𝑅𝑗,𝑗 ′ ,X,Y (𝑁 ) =                          Λ𝑗,X (𝑛1 )Λ𝑗 ′ ,Y (𝑛2 ).
                                                                            ∑︁

                                                                        𝑛1 +𝑛2 =𝑁

    Expanding the convolutions, each such block has the form


𝑅𝑗,𝑗 ′ ,X,Y (𝑁 ) =                                                     𝐴𝑗,X (𝑚1 , . . . , 𝑚𝑗 , 𝑟1 , . . . , 𝑟𝑗 )𝐵𝑗 ′ ,Y (𝑚′1 , . . . , 𝑚′𝑗 ′ , 𝑟1′ , . . . , 𝑟𝑗′ ′ ),
                                          ∑︁

                     𝑚1 ···𝑚𝑗 𝑟1 ···𝑟𝑗 +𝑚′1 ···𝑚′𝑗 ′ 𝑟1′ ···𝑟𝑗′ ′ =𝑁
                                   𝑚1 ,...,𝑚𝑗 ≤𝑦
                                   𝑚′1 ,...,𝑚′𝑗 ′ ≤𝑦


    where
                                             ⎛                                   ⎞
                                                  𝑗                                                            𝑗
                                𝐴𝑗,X = ⎝               𝜇(𝑚𝑖 )𝜔𝑀𝑖 (𝑚𝑖 )⎠ (log 𝑟1 )𝜔𝑅1 (𝑟1 )                          𝜔𝑅𝑖 (𝑟𝑖 ),
                                                 ∏︁                                                           ∏︁

                                                 𝑖=1                                                          𝑖=2

    and 𝐵𝑗 ′ ,Y is defined in the same way on the second side.
    This is a typed dyadic finite-convolution block.

                                                                                72
B1.8. Block Count and Coefficient Bounds The number of choices of (𝑗, 𝑗 ′ ) is at most 𝐽02 .
For each side the number of dyadic choices is 𝑂𝐽0 ((log 𝑁 )2𝐽0 ). Hence

                                       #B𝐽0 ≪𝐽0 (log 𝑁 )4𝐽0 .

   Since 𝐽0 is fixed, this is a polylogarithmic number of blocks. Therefore any block-level estimate
with saving

                                          𝑂(𝑁 (log 𝑁 )−𝐴 )

   for sufficiently large 𝐴 = 𝐴(𝐽0 ) can be summed over all typed blocks. The general record needed
by the proof is the bound (log 𝑁 )4𝐽0 .
   On each dyadic block the elementary coefficients satisfy divisor-bounded estimates. For the
𝜇-variables,

                                       |𝜇(𝑚𝑖 )𝜔𝑀𝑖 (𝑚𝑖 )| ≤ 1.

   For unit variables,

                                          |𝜔𝑅𝑖 (𝑟𝑖 )| ≤ 𝐶𝜔 .

   For the logarithmic variable,

                                     | log 𝑟1 𝜔𝑅1 (𝑟1 )| ≪ log 𝑁.

   Thus each full coefficient in a typed block is polylogarithmically bounded:

                                         ≪𝐽0 (log 𝑁 )𝐶(𝐽0 ) .

   This is the coefficient loss allowed by the later branches.

B1.9. Proof By the Heath–Brown identity with fixed 𝐽0 and 𝑦 = 𝑁 1/𝐽0 , the function Λ(𝑛) has
an exact finite decomposition into the contributions Λ𝑗 (𝑛), 1 ≤ 𝑗 ≤ 𝐽0 , for all 𝑛 ≤ 𝑁 .
   For every variable in every Λ𝑗 , insert the exact smooth dyadic partition

                                           1=        𝜔𝑋 (𝑣).
                                                ∑︁

                                                𝑋

    The partition is exact, so this introduces no error and gives the localized pieces Λ𝑗,X .
    Insert the localized decomposition for both copies of Λ in 𝑅Λ (𝑁 ). Expanding the convolutions
gives a finite sum of typed dyadic finite-convolution blocks. The number of such blocks is

                                          𝑂𝐽0 ((log 𝑁 )4𝐽0 ).

   The coefficient types and bounds follow directly from the elementary coefficient list and the
smooth dyadic support. This proves Lemma B1.
Remark C.1 (B1.10. Output). Lemma B1 supplies the exact Heath–Brown typed dyadic decomposi-
tion with fixed sufficiently large 𝐽0 , polylogarithmically many blocks, and no error term.
    The exact smooth dyadic partition is fixed inside B1. Boundary and tail compatibility for
subsequent routing is checked later by B3, F3, F4, and C1; it is not an additional error in the B1
identity.

                                                  73
B1.11. Logical Dependencies Internal dependencies: PAR.
   External or standard dependencies: X1 and X2.
   Internal nodes served: B3, F3, F4, I1, H4, BGS, E10M, E10X, the CKP branch, the GoodAWACK
branch, and the X16 carrier-slice branch.

C.2     B3 block classification
C.2.1     B3. Block Classification Lemma
B3.0. Role Logical ID: B3.
    Lemma B3 sits between the exact B1 decomposition and the F3 routing theorem. It does not
estimate sums. Its purpose is to produce a finite and exhaustive preliminary classification of the
typed blocks produced by B1.
    Used by: F3, F4, I1, BGS, E10M, and E10X.
    Uses: B1 and the standard smooth dyadic partition input X2.

B3.1. Statement Let ℬ be any typed smooth dyadic finite-convolution block produced by Lemma
B1. Then B3 constructs a finite grouping set

                                                 𝒢(ℬ)

   with

                                          |𝒢(ℬ)| ≤ 24𝐽0 ,

   and assigns ℬ to at least one of the preliminary classes

                   TypeI/Edge,       LongAP/Local,           BranchB,     CKP,

   possibly with an additional LocalDiag flag.
   Moreover:

  1. every scale pattern after B1 is represented among the alternatives in 𝒢(ℬ);

  2. all admissible product groupings are finite and are passed to F3;

  3. TypeI/CKP and CKP/BranchB boundaries are handled as candidate overlaps, not as mutually
     exclusive hard cuts;

  4. every forced local dependence is flagged for LocalDiag;

  5. no residual classes

                      MultiBalancedResidual       or    MixedProductAffineResidual

      remain.




                                                 74
B3.2. Input from B1            By Lemma B1 there is an exact decomposition
                                             𝑅Λ (𝑁 ) =                𝑐ℬ 𝑅ℬ (𝑁 ),
                                                                 ∑︁

                                                             ℬ∈B𝐽0

      where each typed block has the form
                          𝑅ℬ (𝑁 ) =                              𝐴(𝑥1 , . . . , 𝑥𝑟 )𝐵(𝑦1 , . . . , 𝑦𝑠 ),
                                                  ∑︁

                                         𝑥1 ···𝑥𝑟 +𝑦1 ···𝑦𝑠 =𝑁

      with
                                                          𝑟, 𝑠 ≤ 2𝐽0 ,
      and all variables dyadically localized:
                                                𝑥𝑖 ∼ 𝑋𝑖 ,             𝑦𝑗 ∼ 𝑌𝑗 .
      The elementary coefficient types are inherited from B1:
                                            𝜇 · 1≤𝑁 1/𝐽0 ,            1,        log .
      No estimate is made in B3; only structural alternatives are recorded.

B3.3. Scale Vector and Finite Grouping Set For every dyadic block define the scale exponents
                                                  log 𝑋𝑖                      log 𝑌𝑗
                                           𝜉𝑖 =          ,            𝜐𝑗 =           .
                                                  log 𝑁                       log 𝑁
      Since the products satisfy
                                         𝑥1 · · · 𝑥𝑟 ≤ 𝑁,             𝑦1 · · · 𝑦𝑠 ≤ 𝑁,
      one has
                                    𝑟                                  𝑠
                                         𝜉𝑖 ≤ 1 + 𝑜(1),                     𝜐𝑗 ≤ 1 + 𝑜(1).
                                   ∑︁                                 ∑︁

                                   𝑖=1                                𝑗=1

      Define the finite grouping set
                            𝒢(ℬ) = {(𝐼, 𝐽) : 𝐼 ⊆ {1, . . . , 𝑟}, 𝐽 ⊆ {1, . . . , 𝑠}}.
      For (𝐼, 𝐽) ∈ 𝒢(ℬ), put
                                            𝑢𝐼 =                      𝑣𝐼 =
                                                    ∏︁                        ∏︁
                                                          𝑥𝑖 ,                      𝑥𝑖 ,
                                                    𝑖∈𝐼                       𝑖∈𝐼
                                                                               /

      and
                                           𝑢′𝐽 =                      𝑣𝐽′ =
                                                   ∏︁                         ∏︁
                                                          𝑦𝑗 ,                       𝑦𝑗 .
                                                   𝑗∈𝐽                        𝑗 ∈𝐽
                                                                                /

      Then every grouping gives
                                                   𝑢𝐼 𝑣𝐼 + 𝑢′𝐽 𝑣𝐽′ = 𝑁.
      The number of possible groupings is bounded by
                                                |𝒢(ℬ)| ≤ 2𝑟+𝑠 ≤ 24𝐽0 ,
      which is an absolute constant once 𝐽0 is fixed. This is the finite grouping set supplied to Lemma
F3.

                                                                 75
B3.4. Qualitative Scale Predicates           Fix small constants

                                             0<𝜃≪𝜂≪1

   as in PAR.
   A grouped factor 𝑢 is short if

                                                  𝑢 ≤ 𝑁 𝜃.

   A grouped factor 𝑢 is central if

                                        𝑁 1/2−𝜂 ≤ 𝑢 ≤ 𝑁 1/2+𝜂 .

   A factor is long if it is not short and has enough length for smooth AP/local or WACLE analysis:

                                                  𝑢 > 𝑁 𝜃.

   A grouping (𝐼, 𝐽) is CKP-balanced if

                                𝑢𝐼 ≍ 𝑁 1/2+𝑂(𝜂) ,         𝑢′𝐽 ≍ 𝑁 1/2+𝑂(𝜂) ,

    and both complementary factors 𝑣𝐼 , 𝑣𝐽′ are nontrivial long variables or controlled finite-convolution
factors.
    These predicates depend only on dyadic scales, so they are decidable from the scale vector of
the block.

B3.5. Preliminary Candidate Classes B3 assigns every typed block to a finite list of candidate
classes. A block may receive more than one candidate class. Lemma F3 later chooses the actual
route using 𝒢(ℬ). Thus B3 is responsible for exhaustive candidate generation, not for terminal
uniqueness.

TypeI/Edge Candidate A block is a TypeI/Edge candidate if, for some admissible grouping,
one side has a short grouped factor:

                                       min(𝑢𝐼 , 𝑣𝐼 , 𝑢′𝐽 , 𝑣𝐽′ ) ≤ 𝑁 𝜃 ,

   or if the scale vector gives explicit short residual volume after fixing all but one variable.
   Such atoms are not automatically terminal Edge. They are sent to F3/C1, where only the error
part with strict C1P saving is terminal Edge.

LongAP/Local Candidate A block is a LongAP/Local candidate if, after fixing all but one
long variable, the equation reduces to a controlled arithmetic progression or congruence count with
smooth weights and no remaining nonlocal oscillatory arithmetic coefficient.
   The schematic form is

                               𝑎𝑢 + 𝑏 ≡ 𝑁 (mod 𝑞),             𝑞 ≤ (log 𝑁 )𝐶 ,

   or an equivalent controlled local AP condition. The local main part is passed to D1/H4M.



                                                     76
CKP Candidate        A block is a CKP candidate if there exists a grouping such that

                                           𝑢𝑣 + 𝑢′ 𝑣 ′ = 𝑁,

   where 𝑢, 𝑢′ are central or balanced long finite-convolution factors and the coefficient sequences
remain divisor-bounded finite-convolution sequences of B1 type.
   This is the preliminary class later handled by the CKP package.

BranchB Candidate A block is a BranchB candidate if, after all short, local, CKP-balanced,
and collision candidates are removed, the residual structure is central-long affine/WACLE-type with
nonlocal oscillatory finite-convolution coefficients.
    Equivalently, BranchB is the residual non-short, nonlocal, non-CKP, non-diagonal preliminary
class with enough affine structure for GoodAWACK routing in F3/E10.

LocalDiag Flag If any grouping reveals forced equality, proportionality, repeated factor, fixed gcd-
local dependence, or affine dependence among active forms, B3 attaches a LocalDiag flag. Lemma
F3 then treats it as terminal LocalDiag or routes it through the corresponding local branch.

B3.6. Exhaustive Classification Algorithm                  For each typed block ℬ:

  1. build its finite grouping set 𝒢(ℬ);

  2. for each grouping (𝐼, 𝐽), compute the dyadic scales of

                                            𝑢𝐼 ,    𝑣𝐼 ,     𝑢′𝐽 ,   𝑣𝐽′ ;

  3. if a short factor or short residual volume is present, add a TypeI/Edge candidate;

  4. if a purely local AP configuration is exposed, add a LongAP/Local candidate;

  5. if a balanced bilinear finite-convolution structure is exposed, add a CKP candidate;

  6. if a forced local dependence or collision is exposed, add a LocalDiag flag;

  7. after all groupings are tested, if no previous candidate exhausts the nonlocal central-long part,
     add a BranchB candidate.

     Thus B3 never stops with an undefined residual. The default residual is not an unclassified class;
it is precisely a BranchB candidate, provided it is non-short, nonlocal, and non-CKP.

B3.7. Exclusion of MultiBalancedResidual                   There is no residual class

                                      MultiBalancedResidual.

   A multi-balanced product pattern means that more than one grouping produces central or near-
central factors. B3 does not require uniqueness of the grouping. All such groupings are placed in
the finite set

                                                   𝒢(ℬ).



                                                    77
    If any grouping produces a CKP-balanced bilinear form, a CKP candidate is added. If several
do, all are recorded as grouping alternatives and passed to the finite grouping-elimination protocol
in F3.
    Thus multi-balancedness is not a residual class. It is a finite multiplicity of CKP/BranchB
grouping alternatives:

                                      MultiBalanced ⊆ 𝒢(ℬ).

   Since

                                             |𝒢(ℬ)| ≤ 24𝐽0 ,

   this multiplicity is finite and is absorbed by F3.

B3.8. Exclusion of MixedProductAffineResidual                   There is no residual class

                                   MixedProductAffineResidual.

   A mixed product-affine residual would be a block that is:

  1. not short/TypeI;
  2. not purely LongAP/Local;
  3. not CKP-balanced under any grouping;
  4. not locally diagonal;
  5. but still contains nonlocal central-long finite-convolution structure.

    By definition B3 assigns exactly such residuals to the BranchB candidate class. This is not
circular: BranchB is the preliminary class designed for non-short, nonlocal, non-CKP central-long
affine/WACLE atoms. Lemmas F3 and F4 then decide whether the atom becomes GoodAWACK,
LocalDiag, CKP, Edge, or LongAP/Local.
    Thus the mixed residual is not unclassified:

                             MixedProductAffineResidual ⊆ BranchB.

B3.9. Boundary Conventions

TypeI and CKP        Suppose a grouping gives

                                             𝑢𝑣 + 𝑢′ 𝑣 ′ = 𝑁.

   If one of the grouped factors is short,

                                       min(𝑢, 𝑣, 𝑢′ , 𝑣 ′ ) ≤ 𝑁 𝜃 ,

   then B3 records a TypeI/Edge candidate. It may also record a CKP candidate for another
grouping if another grouping is balanced. There is no conflict: B3 generates candidates and F3 later
routes them.
   If all relevant grouped factors are long and two principal factors are central/balanced, B3 records
a CKP candidate.

                                                   78
CKP and BranchB If some grouping gives a balanced finite-convolution bilinear form compatible
with CKP, B3 records a CKP candidate.
   If no grouping gives CKP but the block remains non-short and nonlocal, the block is a BranchB
candidate. Thus

         BranchB = CentralLongNonlocal ∖ CKPCompatible ∖ LocalDiag ∖ TypeI/Edge.

  This definition is preliminary. Lemmas F3 and F4 may later reroute BranchB candidates into
CKP or LocalDiag if divisor/gcd structure becomes visible.

LongAP/Local and BranchB A block is LongAP/Local only if, after fixing auxiliary variables,
the remaining counting problem is purely local and has no nonlocal oscillatory coefficient.
    If an oscillatory coefficient remains, such as a Mobius/Liouville-type finite-convolution factor
in a long affine form, the block is not LongAP/Local. It is a BranchB candidate unless CKP or
LocalDiag applies.
    Thus D1 is not asked to hide nonlocal arithmetic estimates. The terminal LongAP/Local
coefficient-exclusion statement is proved later using this B3 boundary together with the F3/F4terminal
routing alternatives.

B3.10. Proof The number of variables in each B1 block is bounded by 2𝐽0 on each side. Therefore
the set of all product groupings is finite and has size at most 24𝐽0 . For each grouping, B3 tests the
dyadic scale predicates: short, central, long, local, balanced, and collision. Each test is a finite
condition on the scale vector and algebraic form of the block.
    If a short factor appears, a TypeI/Edge candidate is recorded. If the equation reduces to
controlled local AP counting without nonlocal oscillatory coefficients, a LongAP/Local candidate is
recorded. If a balanced finite-convolution bilinear structure is exposed, a CKP candidate is recorded.
If a forced local dependence is exposed, a LocalDiag flag is recorded. If, after all tests, a non-
short, nonlocal, non-CKP, non-diagonal central-long structure remains, it is by definition a BranchB
candidate.
    Thus every block receives at least one candidate class. Multi-balanced blocks produce several
grouping alternatives inside the finite set 𝒢(ℬ), and mixed product-affine residuals are precisely
BranchB candidates. Therefore no undefined residual class remains. This proves Lemma B3.
Remark C.2 (B3.11. Output). Lemma B3 supplies exhaustive preliminary block classification and a
finite grouping set for F3:

                             𝐵1 =⇒ 𝐵3 =⇒ 𝐹 3 =⇒ terminal routing.

   All B1 scale patterns are assigned to TypeI/Edge, LongAP/Local, CKP, or BranchB can-
didates, with LocalDiag flags when forced dependencies occur. No MultiBalancedResidual or
MixedProductAffineResidual remains.
   The output of B3 is not an analytic estimate. It is the finite candidate generation input used by
F3 to prove terminal routing.

B3.12. Logical Dependencies Internal dependencies: B1.
  External or standard dependencies: X2.
  Internal nodes served: F3, F4, I1, BGS, E10M, and E10X.



                                                 79
D      Routing Grammar and Complete Routing Exhaustion
D.1     F3F4M master routing theorem
D.1.1     F3F4M. Master Routing Exhaustion Theorem
F3F4M.0. Statement and Role Logical ID: F3F4M.
    This lemma is the reader-facing routing-exhaustion theorem for the B1/B3/F3/F4 layer. It
packages the intrinsic terminal predicate catalogue F3P, the F3 no-cycle routing theorem, the F3A
operation-completeness theorem, the F3T finite routing table, the F4 large-divisor decision theorem,
the E5-clean stability interface, and the C1P strict Edge predicate catalogue into one structural
theorem.
    The theorem proves only a finite exact partition statement. It does not use the analytic estimates
for Edge, CKP, GoodAWACK, LongAP/Local, or LocalDiag. Those estimates are supplied later by
the terminal branch theorems. The output of F3F4M is the assertion that every typed B1-origin
block is partitioned into tagged terminal atoms belonging to exactly one of five structural terminal
classes, with no sixth class and no unresolved mixed residual.
    —

F3F4M.1. State Space Let ℬ be a typed B1 block after smooth dyadic localization and B3
preliminary classification. A routing state over ℬ is a tuple

                                        𝒜 = (𝒱, ℒ, 𝒞, 𝒲, ℛ, 𝜏 ).                      (F3F4M-state)
    Here:

    1. 𝒱 is the finite variable list inherited from the fixed-depth Heath–Brown product variables and
       the B3 grouped variables;

    2. ℒ is the finite list of active affine, product, and grouped forms;

    3. 𝒞 is the finite list of congruence, divisibility, coprimality, quotient, local-dependence, and
       compatibility conditions;

    4. 𝒲 records dyadic smooth weights and coefficient type;

    5. ℛ records unresolved finite routing alternatives;

    6. 𝜏 is the accumulated routing tag.

   The state represents a subdomain Ω(𝒜) of the original summation domain and a corresponding
contribution 𝑅𝒜 (𝑁 ). The following invariant is maintained at every stage.

Invariant F3F4M-I         Every active state 𝒜 satisfies:

    1. Ω(𝒜) is described by a finite Boolean combination of the conditions in 𝒞, dyadic support
       constraints from 𝒲, and the fixed B1/B3 product data;

    2. 𝑅𝒜 (𝑁 ) is the exact weighted sum over Ω(𝒜);

    3. every unresolved item in ℛ belongs to one of the following structural types:



                                                   80
 grouping choice,    CRT or congruence restriction,    square-divisor condition,
 ordinary divisor or quotient predicate,   local-dependence predicate,
                                                                                   (F3F4M-alphabet)
 boundary/short-volume/high-content condition,        CKP balance candidate,
 LongAP/Local candidate,       GoodAWACK residual candidate.

   There are no other unresolved structural objects in a B1-origin state. This is where the fixed-
depth B1 identity, the finite B3 grouping list, F3A, and the F4 ordinary-divisor analysis enter the
theorem.
   —

F3F4M.2. Terminal Predicates          The terminal predicates are the intrinsic predicates of F3P:

             Edge,      CKP,       GoodAWACK,          LocalDiag,      LongAP/Local.
                                                                         (F3F4M-terminal-classes)
   They are structural predicates, not estimates.

  1. Edge means that the state is empty or carries one of the strict C1P saving certificates E1–E7.

  2. LocalDiag means that a forced equality, proportionality, repeated form, or fixed local depen-
     dence has become part of the structural data.

  3. LongAP/Localmeans that, after fixing auxiliary variables, one LPI-admissible long arithmetic-
     progression/local coefficient variable remains and no nonlocal oscillatory coefficient survives.

  4. CKP means that the state has the balanced two-sided bilinear Kloosterman-fraction shape
     required by the CKP interface, with the noncentral Edge and LocalDiag obstructions absent.

  5. GoodAWACK means that the state is a central-long affine Branch-B residual with controlled
     content and CRT data after Edge, LocalDiag, CKP, and LongAP/Local have all failed.

   The predicates are read in the deterministic F3T precedence order. If a state visually satisfies
more than one predicate, its terminal tag is the first applicable predicate in that order. The
remaining predicates are retained only as verification information and do not create another terminal
atom.
   —

F3F4M.3. Allowed Transitions Every nonterminal transition is one of the following exact
refinements.

T0. Empty or incompatible state If the conditions in 𝒞 are incompatible, the state contributes
zero and is terminal Edge-zero.

T1. Terminal labelling If one of the five terminal predicates applies, the state is labelled by
the first applicable predicate in the deterministic F3T order. This does not change the summation
domain and does not introduce an estimate.




                                                 81
T2. CRT or congruence restriction A congruence 𝐿(𝑧) ≡ 𝑎 (mod 𝑞), coprimality restriction,
or CRT compatibility condition splits the state into finitely many exact residue classes. Incompatible
classes go to T0; compatible full-rank restrictions remain in the routing process with the CRT tag
recorded.

T3. Square-divisor routing A large square-divisor condition is a strict Edge predicate. A small
square-divisor condition is absorbed as a controlled divisibility/CRT restriction and inherits the
CRT or divisibility tag.

T4. Grouping selection or elimination The finite B3 grouping list is reduced by selecting
one grouping candidate or eliminating an incompatible candidate. This is a finite exact split of the
candidate set, not an analytic estimate.

T5. Local-dependence decision An equality, proportionality, repeated form, fixed local
dependence, or one-form-determined-by-another condition is terminal LocalDiag if forced. If not
forced, the corresponding ambiguity is removed from ℛ.

T6. Strict Edge decision A boundary tail, short residual volume, large content/gcd layer, small
conductor, high-frequency tail, Type-I saving, or strict C1P predicate is terminal Edge.

T7. LongAP/Local decision A one-dimensional long AP/local cell satisfying the F3P local
coefficient predicate is terminal LongAP/Local. If a nonlocal 𝜇-, 𝜆-, Fourier-, Kloosterman-,
nilsequence-, or GoodAWACK-type oscillation remains, the state is not LongAP/Local and the
corresponding residual is routed by the CKP or GoodAWACK decisions.

T8. CKP decision A central balanced two-sided bilinear state satisfying the CKP structural
predicate is terminal CKP. The zero-frequency local mode is still structurally part of the CKP
branch, but is later assembled through the LPI/H4M local interface. Noncentral CKP obstructions
are not sent to CKP; they are strict Edge or routing-excluded states.

T9. Ordinary divisor or quotient decision An unresolved ordinary divisor predicate 𝑑 | 𝐿(𝑧),
quotient equation 𝐿(𝑧) = 𝑑𝑠, fixed-divisor quotient, or variable-quotient residual is handled by
F4. The F4 decision sends the state to Edge, LocalDiag, CKP, GoodAWACK, or to a controlled
nonterminal continuation with the quotient/divisor origin tag recorded.

T10. GoodAWACK residual labelling If the state is central-long, nonlocal, non-CKP, non-
Edge, and not LongAP/Local or LocalDiag, and all content/CRT data are controlled, it is terminal
GoodAWACK. This terminal label performs no coordinate operation.

T11. E5-clean transport E5-clean transport preserves controlled content and the terminal
tensor-test span. A full-rank transport does not change the terminal class. A rank-dropping
transport is allowed only with an inherited origin tag, and therefore is not an untagged new terminal
generator.
    These transitions are exhaustive by F3A for the F3-level operations and by F4 for ordinary
divisor and quotient predicates.
    —


                                                 82
F3F4M.4. Routing Measure              Define the strengthened routing measure

                M♯ (𝒜) = (𝐽free , 𝑅largeDiv , 𝐷unabsorbed , 𝐶coll , 𝐵amb )       (F3F4M-measure)
   with lexicographic order, where:

  1. 𝐽free counts unresolved free product/grouping degrees and candidate grouping choices;

  2. 𝑅largeDiv counts unresolved ordinary divisor and quotient predicates;

  3. 𝐷unabsorbed counts unresolved CRT, congruence, square-divisor, and controlled-divisibility
     restrictions;

  4. 𝐶coll counts unresolved local-dependence, equality, proportionality, and repeated-form deci-
     sions;

  5. 𝐵amb counts remaining terminal-class ambiguities.

   All components are finite nonnegative integers depending only on the typed B1/B3 data and
the fixed depth 𝐽0 .
   Each nonterminal transition T2–T9 or T11 either sends the state to a terminal class or strictly
decreases M♯ . The decrease is read as follows:

  1. grouping selection decreases 𝐽free ;

  2. F4 divisor/quotient decisions decrease 𝑅largeDiv , unless they terminate;

  3. CRT, congruence, square-divisor, and controlled divisibility absorption decrease 𝐷unabsorbed ,
     unless they terminate;

  4. local-dependence decisions decrease 𝐶coll , unless they terminate;

  5. deterministic terminal precedence removes terminal ambiguity and decreases 𝐵amb , unless the
     state is already terminal.

   Thus no infinite nonterminal path is possible.
   —

F3F4M.5. Master Theorem
Theorem D.1 (Theorem F3F4M). For every typed B1 block ℬ, the B3/F3/F4 routing process
produces a finite set of tagged terminal atoms

                                               {(ℬ, 𝜏 )}𝜏 ∈𝒯 (ℬ)

   such that:

  1. the partition identity


                                        𝑅ℬ (𝑁 ) =              𝑅ℬ,𝜏 (𝑁 )         (F3F4M-partition)
                                                      ∑︁

                                                    𝜏 ∈𝒯 (ℬ)

   holds before any terminal estimate is applied;

                                                      83
  1. each terminal atom satisfies exactly one terminal predicate in the F3T precedence order;

  1. the only possible terminal classes are

                  Edge,    CKP,     GoodAWACK,          LocalDiag,   LongAP/Local;

  1. every nonterminal state admits one of the transitions T2–T9 or T11, or is terminal by T0,
     T1, or T10;

  1. every nonterminal transition is an exact refinement of the current summation domain and
     strictly decreases M♯ ;

  1. no sixth terminal class and no unresolved mixed residual remains.

   —

F3F4M.6. Compressed Proof
Proof. Start from a typed B1 block ℬ. B1 has fixed depth 𝐽0 , and B3 creates only finitely many
grouped product candidates. Therefore the initial state space (F3F4M-state) is finite up to the
finite dyadic, grouping, congruence, divisor, quotient, local-dependence, and terminal-predicate data
listed in (F3F4M-alphabet).
    At each active state 𝒜, first test incompatibility. If the conditions are incompatible, T0 gives a
zero Edge atom. Otherwise test the five intrinsic terminal predicates in the deterministic F3T order.
If one applies, T1 gives a terminal tag. This step partitions no mass and uses no estimate.
    Assume now that 𝒜 is nonterminal. By Invariant F3F4M-I, some unresolved structural object
remains in the finite alphabet (F3F4M-alphabet). The exhaustive list T2–T11 covers the possibilities:
CRT and congruence restrictions are T2; square-divisor conditions are T3; grouping alternatives are
T4; local-dependence decisions are T5; strict Edge-saving certificates are T6; one-dimensional local
AP candidates are T7; balanced CKP candidates are T8; ordinary divisor and quotient predicates
are T9; central-long affine residuals are T10; and E5-clean content transports are T11.
    There is no additional operation type. The reason is structural. A B1-origin descendant is
built from fixed-depth product variables, finite B3 groupings, affine/product forms, congruence
and divisor data, dyadic weights, and finite coefficient labels. F3A proves that every F3-level
transformation of such data is one of the non-large-divisor transitions above, including square-divisor
routing and controlled CRT/divisibility absorption. F4 proves that every ordinary large-divisor
or quotient predicate is exhausted by the F4 decision tree. E5-clean transport is only a content-
stability operation on an already generated routing record; it is not an additional skeleton generator.
Thus any nonterminal state has one of the listed legal moves.
     Each legal nonterminal move is exact. CRT restrictions split into residue classes; square-divisor
and divisibility decisions split by exact divisibility identities; grouping selection splits a finite set
of B3 grouping candidates; F4 quotient decisions split by exact quotient scales and structural
alternatives; local-dependence and terminal-class decisions split by explicit Boolean predicates.
Hence the parent contribution is the sum of the child contributions at every refinement step.
     Each legal nonterminal move strictly decreases M♯ as recorded in F3F4M.4. Since M♯ takes
values in a finite product of N with lexicographic order, no infinite descending chain exists. Therefore
every routing path terminates after finitely many exact refinements.
     It remains to identify the terminal leaves. At a terminal leaf, no unresolved grouping choice, CRT/
divisibility condition, ordinary divisor/quotient predicate, local-dependence ambiguity, boundary/

                                                   84
short-volume/high-content obstruction, CKP balance candidate, LongAP/Local candidate, or
central-long GoodAWACK residual remains untested. Therefore one of the five F3P terminal
predicates must apply:

   1. incompatible, empty, boundary, short-volume, large square-divisor, large content/gcd, high-
      frequency, small-conductor, or strict-saving states are Edge;

   2. forced equalities, proportionalities, repeated forms, or fixed local dependences are LocalDiag;

   3. one-dimensional LPI-admissible local AP cells are LongAP/Local;

   4. balanced two-sided bilinear Kloosterman-fraction states are CKP;

   5. the remaining central-long nonlocal affine residuals are GoodAWACK.

    These five cases cover the terminal alternatives by the terminal adequacy statement of F3P and
the expanded F3T table. Visual overlap is resolved by the deterministic F3T precedence order, so
each terminal leaf has exactly one tag.
    The routing tree is finite, all refinements are exact, and terminal leaves are disjoint because
the accumulated tag 𝜏 records the complete path of exact decisions. Summing over all leaves gives
(F3F4M-partition). Since every leaf has one of the five terminal tags and no other terminal predicate
remains available, there is no sixth terminal class and no unresolved mixed residual. This proves
the theorem.
    —


F3F4M.7. Why the Verification Tables Are Not Extra Hypotheses The F3T and F4
tables are verification records for the finite state analysis above. They do not introduce downstream
analytic estimates into F3F4M.

   1. The F3T table expands the intrinsic F3P terminal predicates by B1 block type, B3 grouping
      type, dyadic regime, divisor/conductor state, coefficient type, terminal class, and exclusion
      reason.

   2. The F4 table expands T9 by listing the possible ordinary divisor and quotient cases and their
      structural exits.

   3. The C1P predicates enter only as structural Edge certificates. The C1 estimate is not used
      here.

   4. CKP, GoodAWACK, LongAP/Local, and LocalDiag branch estimates are not used here; only
      their structural entry predicates are used.

    Thus F3F4M is a finite partition theorem. Its verification tables are hand-checkable expansions
of the finite alphabet (F3F4M-alphabet), not additional proof assumptions.
    —




                                                 85
F3F4M.8. Interface Corollary
Corollary D.2 (Corollary F3F4M.1). After applying F3F4M to every typed B1 block, the weighted
sum decomposes into a finite disjoint terminal sum


    𝑅Λ (𝑁 ) = 𝑅Edge (𝑁 ) + 𝑅CKP (𝑁 ) + 𝑅GoodAWACK (𝑁 ) + 𝑅LocalDiag (𝑁 ) + 𝑅LongAP/Local (𝑁 ).
                                                                                 (F3F4M-output)
   The five terms are structural inputs to the terminal packages:

  1. Edge atoms satisfy a strict C1P Edge predicate and are admitted by the Edge branch before C1
     estimates them.

  2. CKP atoms have the balanced finite-convolution / Kloosterman-fraction structural form con-
     sumed by the CKP package.

  3. GoodAWACK atoms are actual B1/B3/F3/F4/E5-generated terminal skeletons consumed by the
     GoodAWACK finite-grammar package.

  4. LocalDiag atoms are LPI-admissible local projection atoms assembled by the local/main
     branch.

  5. Lo ng AP /L oc al atoms are local-coefficient long arithmetic-progression atoms normalized by
     D1 and then assembled by H4.

   No terminal estimate is part of this corollary. The corollary supplies only the exact structural
partition and the terminal interfaces.
   —

F3F4M.9. Logical Dependencies Internal dependencies: B1, B3, F3P, F3, F3A, F3T, F4, E5,
LPI, C1P, and PAR.
   Children served: H4M, I1, BGS, HGO2R, E10Y, E10M, E10X, E10K, E10L, TNGTTHM, and the full
manuscript routing appendix.

D.2     F3P intrinsic terminal predicates
D.2.1    F3P. Intrinsic Terminal Predicate Catalogue
F3P.0. Statement and Role Lemma F3P fixes the terminal predicates used by the F3/F4
routing layer. The predicates are intrinsic: they are stated in terms of the current tagged atom, its
coefficient algebra, its affine/product forms, and its unresolved structural conditions. They do not
use the estimates later proved by C1, D1, G8a, E10L, or H4.
   The output is a finite catalogue

              IsEdge,    IsCKP,    IsGoodAWACK,        IsLocalDiag,    IsLongAPLocal.

    The LongAP/Local predicate is positive. It is not defined as the residual class left after excluding
Edge, CKP, GoodAWACK, and LocalDiag. Instead it requires that all surviving long-variable
coefficients belong to the local coefficient algebra Cloc (𝑄𝜏 ).
    Logical dependencies are B1, B3, C1P predicate names, the F3/F4 atom interface, finite CRT
algebra, and the parameter register. The lemma is used by F3, D1, LPI, H4, and the proof ledger.
    —

                                                  86
F3P.1. Tagged Atom Data               A routed atom is a finite tagged object

                                           𝒜 = (𝒱, ℒ, 𝒞, 𝒲, ℛ, 𝜏 )

   where:

  1. 𝒱 is the finite variable list;

  2. ℒ is the finite list of affine/product forms;

  3. 𝒞 is the finite list of congruence, divisibility, coprimality, quotient, and local constraints;

  4. 𝒲 is the finite coefficient and smooth-weight data;

  5. ℛ is the finite unresolved routing set;

  6. 𝜏 is the complete routing tag.

    All complexity constants are bounded in terms of the fixed parameter hierarchy. The tag 𝜏
records the parent B1 block and every exact refinement already made by B3/F3/F4.
    —

F3P.2. Local Coefficient Algebra For a controlled modulus 𝑄𝜏 ≤ (log 𝑁 )𝐶𝜏 , define Cloc (𝑄𝜏 )
to be the algebra generated by:

  1. smooth dyadic weights of fixed differentiability complexity;

  2. constants depending only on the tag 𝜏 ;

  3. residue-class indicators 1𝐿(𝑧)≡𝑎 (mod 𝑞) with 𝑞 | 𝑄𝜏 ;

  4. coprimality indicators 1(𝐿(𝑧),𝑞)=1 with 𝑞 | 𝑄𝜏 ;

  5. fixed controlled-divisor factors whose divisor value is part of the tag;

  6. finite products and finite linear combinations of the preceding generators.

    This algebra is local: its values are determined by smooth position data and residue/coprimality
classes modulo 𝑄𝜏 . It contains no long-variable arithmetic oscillation.
    The following are explicitly not elements of Cloc (𝑄𝜏 ), unless the relevant expression has already
been fixed into tag data or reduced to residue/coprimality data:
                                                                     (︃         )︃
                                                                       𝑘𝐿(𝑧)
                        𝜆(𝐿(𝑧)),         𝜇(𝐿(𝑧)),     𝑒(𝛼𝐿(𝑧)),      𝑒       ,
                                                                         𝑞

    nonlocal finite-convolution descendants of these functions, nilsequence-type oscillations, unre-
solved ordinary divisor predicates, and unresolved quotient equations.
    —




                                                     87
F3P.3. Intrinsic Edge Predicate
                                             IsEdge(𝒜)
   holds if the tagged atom carries one of the strict saving predicates defined by C1P:
  1. smooth boundary or dyadic tail;
  2. large square-divisor tail;
  3. large-gcd or large-content volume saving;
  4. high Fourier tail;
  5. small-conductor layer with a C1P saving certificate;
  6. short residual volume;
  7. Type I short-variable error.
    An ordinary condition 𝑑 | 𝐿(𝑧) is not Edge by itself. It is Edge only if one of the displayed
saving predicates is present.
    —

F3P.4. Intrinsic CKP Predicate
                                             IsCKP(𝒜)
    holds if 𝒜 has a balanced finite-convolution bilinear form reducible, after controlled gcd/content
splitting and smooth dyadic localization, to
                                           𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁𝑔
   with non-short grouped variables on both sides, divisor-bounded coefficients, central/balanced
ranges, controlled content, and no forced local diagonal obstruction. The predicate is structural; the
DFI/X10 estimate is not part of the predicate.
   —

F3P.5. Intrinsic GoodAWACK Predicate
                                          IsGoodAWACK(𝒜)
   holds if the tagged atom is a central-long affine WACLE/GoodAWACK atom with:
  1. bounded affine complexity;
  2. smooth weight of polylogarithmic complexity;
  3. no forced local diagonal relation;
  4. no unresolved ordinary divisor or quotient predicate;
  5. at least one marked affine Liouville-type or finite-convolution affine form with controlled
     content;
  6. long fibre directions.
   This is the structural Branch B input class. The cancellation estimate is proved only later by
the GoodAWACK package.
   —

                                                 88
F3P.6. Intrinsic LocalDiag Predicate

                                           IsLocalDiag(𝒜)

    holds if the current tag contains a forced equality, proportionality, repeated form, gcd-local
dependence, or collision between relevant affine/product forms which makes the contribution a
tagged local projection source rather than an oscillatory error term.
    The predicate is positive: it requires an explicit forced relation in 𝒞 or in the recorded routing
tag. H4 later assembles the local projection; H4 is not used to define the predicate.
    —

F3P.7. Intrinsic LongAP/Local Predicate

                                         IsLongAPLocal(𝒜)

   holds if the tagged atom satisfies all of the following conditions.

  1. The remaining long variable is organized as a long arithmetic progression or a finite union of
     controlled AP fibres, with length at least a fixed power of 𝑁 in the current B1 scale.

  2. There is a controlled modulus 𝑄𝜏 ≤ (log 𝑁 )𝐶𝜏 .

  3. Every coefficient which still depends on a long AP variable belongs to Cloc (𝑄𝜏 ).

  4. The remaining constraints are only residue-class, coprimality, fixed controlled-divisor, smooth-
     weight, or endpoint constraints recorded by 𝜏 .

  5. There is no unresolved ordinary divisor predicate, quotient equation, balanced reciprocal-
     phase structure, marked Liouville/Mobius coefficient, nonlocal finite-convolution coefficient,
     Kloosterman phase, or nilsequence oscillation.

   Thus LongAP/Local is a positive local-coefficient condition:

                           IsLongAPLocal(𝒜) =⇒ 𝒲long (𝒜) ⊂ Cloc (𝑄𝜏 ).                       (F3P-L)
   D1 later evaluates such atoms by pure local AP counting and proves LPI-admissibility. D1 is
not used in the definition.
   —

F3P.8. Mutual Routing Adequacy At the terminal-labelling stage, an atom is tested against
the five predicates in the deterministic F3/F4 routing order. If none of the predicates holds, then
the atom still has a nonempty unresolved obstruction set:

                                             𝒪(𝒜) ̸= ∅.

    Indeed, failure of the LongAP/Local predicate means either the long-variable coefficient is not in
Cloc (𝑄𝜏 ), the modulus is uncontrolled, the AP length is not long, or an unresolved divisor/quotient
or oscillatory structure remains. These are precisely F3/F4 routing obstructions, not downstream
analytic questions.
    Therefore there is no sixth terminal predicate. A nonterminal atom is routed by the F3/F4
measure-decreasing procedure until one of the intrinsic terminal predicates holds.
    —

                                                 89
F3P.9. Output for D1        For every tagged terminal atom (ℬ, 𝜏 ),

                                         IsLongAPLocal(ℬ, 𝜏 )

   implies that every long-variable coefficient is local:

                                           𝑎𝜏 (𝑢) ∈ Cloc (𝑄𝜏 ).

   Equivalently, after expanding the finite local algebra, 𝑎𝜏 (𝑢) is a finite linear combination of
smooth dyadic weights multiplied by residue-class and coprimality indicators modulo 𝑄𝜏 , with tag
constants. This is the only structural input D1 needs before applying smooth AP counting.

D.3     F3 routing partition
D.3.1    F3. Routing Exhaustion / No-Cycle Theorem
F3.0. Role Logical ID: F3.
    Lemma F3 is the routing-exhaustion theorem for typed B1 blocks after the B3 preliminary
classification. It uses the strengthened measure

                         M♯ (𝒜) = (𝐽free , 𝑅largeDiv , 𝐷unabsorbed , 𝐶coll , 𝐵amb )

   which is designed so that ordinary divisor expansion or variable quotienting cannot create a
routing cycle. Generic Cauchy/cube operations and Fourier expansion are not F3 routing operations;
they are proof subroutines in the terminal E10, G8a, D1, and C1 packages.
   The theorem is

            RawBlock =⇒ Edge ⊔ CKP ⊔ GoodAWACK ⊔ LocalDiag ⊔ LongAP/Local,

   with a genuine no-cycle proof using the strengthened measure

                                                   M♯ .

    Used by: F4, F3A, F3T, BGS, HGO2R, E10L, I1, and the terminal branch assembly.
    Uses: B1, B3, F3P, F4, E5, LPI, and the proof parameter register. The terminal predicates are
structural predicates; the estimates for those terminal classes are proved later by C1, D1, G8a, E10L,
and H4.
    The standalone reader-facing form of the F3/F4 routing layer is F3F4M. It packages F3P, F3,
F3A, F3T, and F4 into one master routing theorem and interface corollary.
    —

F3.1. Scope of F3      F3 applies to atoms obtained after:

  1. exact B1 typed Heath–Brown decomposition;

  2. smooth dyadic localization;

  3. preliminary B3 block classification.

   Each atom has a finite description:


                                                    90
                                       𝒜 = 𝒜(𝒱, ℒ, 𝒞, 𝒲, ℛ),

   where:

  • 𝒱 is a finite list of variables;

  • ℒ is a finite list of affine/product forms;

  • 𝒞 is a finite list of congruence/divisibility/coprimality conditions;

  • 𝒲 is the smooth dyadic weights and coefficient types;

  • ℛ is a finite list of unresolved routing alternatives.

    All complexity constants are bounded in terms of fixed 𝐽0 .
    All routing steps in F3 are exact refinements of the current summation domain. No analytic
estimate and no 𝑜(𝑁 ) error is introduced by F3 itself. Error terms appear only after a terminal
atom is sent to a terminal estimate such as C1, D1, G8a, E10L, or H4.
    —

F3.2. Terminal predicates The intrinsic terminal predicate catalogue is Lemma F3P. It defines
five structural terminal predicates without using the downstream estimates for the corresponding
terminal classes. The present section records the predicates in the shorthand form used by the
routing algorithm.

F3.2.1. Edge terminal predicate

                                             IsEdge(𝒜)

   holds if the routing data of 𝒜 contains one of the strict Edge-saving predicates later estimated
by C1:

  1. smooth boundary / dyadic tail;

  2. large square-divisor tail;

  3. large-gcd / large-content volume saving;

  4. high Fourier tail;

  5. small-conductor layer with C1 saving;

  6. short residual volume;

  7. Type I short-variable error.

   Important restriction:

                                              𝑑 | 𝐿(𝑧)

   alone is not Edge unless it triggers one of the above saving predicates.



                                                  91
F3.2.2. CKP terminal predicate

                                             IsCKP(𝒜)

   holds if 𝒜 has balanced finite-convolution bilinear form reducible to

                                           𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁

    with two non-short grouped variables on each side, divisor-bounded coefficients, smooth dyadic
weights, and no forced local diagonal obstruction. This is a structural input class. The CKP package
later proves the estimate for this class.

F3.2.3. GoodAWACK terminal predicate

                                          IsGoodAWACK(𝒜)

   holds if 𝒜 is a central-long affine WACLE atom with:

  1. bounded affine complexity;

  2. smooth weight of polylogarithmic complexity;

  3. no forced local diagonal relation;

  4. no unresolved ordinary large divisor condition;

  5. at least one marked affine Liouville-type form with controlled content;

  6. long fibre directions.

    This is a structural input class. The Branch B / GoodAWACK package later proves the estimate
for this class.

F3.2.4. LocalDiag terminal predicate

                                           IsLocalDiag(𝒜)

    holds whenever the current atom contains a forced equality, proportionality, gcd-local dependence,
or collision between relevant affine/product forms that makes the contribution a canonical local
term rather than an oscillatory error.
    All such atoms are terminal and are passed to H4.

F3.2.5. LongAP/Local terminal predicate

                                            IsLongAP(𝒜)

    is the predicate IsLongAPLocal(𝒜) from F3P. It holds if the atom is purely local smooth
arithmetic-progression counting with:

  1. smooth weights;

  2. controlled local moduli 𝑄𝜏 ≤ (log 𝑁 )𝐶𝜏 ;


                                                 92
  3. every coefficient depending on the long AP variable lying in the local coefficient algebra
     Cloc (𝑄𝜏 );

  4. no unresolved ordinary divisor or quotient predicate;

  5. no unresolved 𝜇-, 𝜆-, Fourier-, Kloosterman-, reciprocal, finite-convolution, or nilsequence-
     type oscillation;

  6. long AP length.

   This is a structural input class. Lemma D1 later proves that the corresponding main term is
LPI-admissible; the final local bridge is supplied by H4M, using the H4 local algebra.
   —

F3.3. Residual obstruction set        For every nonterminal atom define a finite obstruction set

                                                    𝒪(𝒜)

   consisting of unresolved predicates of the following types:

  1. unresolved ordinary divisor condition;

  2. unresolved quotient equation;

  3. unresolved conductor decision;

  4. unresolved CRT/congruence restriction;

  5. unresolved grouping/balance alternative;

  6. unresolved local collision/dependence decision;

  7. unresolved choice between CKP and GoodAWACK normal form.

   Because the atom is produced from finite B1/B3 data, the set 𝒪(𝒜) is finite and

                                            |𝒪(𝒜)| ≪𝐽0 1.

   —

F3.4. Finite grouping set      Let

                                                        𝒢(𝒜)

   be the finite set of admissible unresolved product groupings inherited from the B1 typed block.
   A grouping is a choice of partition of product variables into grouped variables such as

                                      𝑢=                     𝑣=
                                           ∏︁                     ∏︁
                                                 𝑥𝑖 ,                   𝑥𝑖 ,
                                           𝑖∈𝐼                    𝑖∈𝐼
                                                                   /

   and similarly on the second side.
   Since the number of product variables is bounded by 2𝐽0 on each side,


                                                        93
                                             |𝒢(𝒜)| ≤ 𝐶(𝐽0 ).

   The regrouping protocol is:

  • if a grouping yields Edge, CKP, GoodAWACK, LongAP/Local, or LocalDiag, the atom
    becomes terminal;

  • if the grouping is checked and fails all terminal predicates, that grouping is removed from
    𝒢(𝒜) and is not revisited.

   Thus every unsuccessful regrouping strictly decreases

                                                  |𝒢(𝒜)|.

   —

F3.5. Complexity measure           Define the strengthened lexicographic measure

                  M♯ (𝒜) = (𝑂unresolved , 𝑅largeDiv , 𝐷unabsorbed , |𝒢(𝒜)|, 𝐶coll , 𝐽free ),

   where:

                                          𝑂unresolved = |𝒪(𝒜)|;

    𝑅largeDiv counts ordinary large-divisor predicates assigned to F4 processing;
    𝐷unabsorbed counts unresolved controlled CRT/congruence restrictions;
    |𝒢(𝒜)| counts unresolved grouping alternatives;
    𝐶coll counts unresolved collision/local-dependence decisions;
    𝐽free counts remaining free finite-convolution/product variables.
    The order is lexicographic. Therefore increasing 𝐽free is harmless if some earlier obstruction
component decreases.
    Since all entries are nonnegative integers bounded in terms of 𝐽0 and the dyadic data, there is
no infinite strictly decreasing sequence.
    —

F3.6. Allowed routing-level operations In Lemma F3, only the following are generic routing-
level operations:

  1. controlled CRT absorption;

  2. F4 large-divisor decision;

  3. square-divisor routing;

  4. finite grouping selection/elimination;

  5. terminal LocalDiag detection;

  6. terminal Edge detection by C1P predicates;

  7. terminal class labelling into CKP, GoodAWACK, LongAP/Local, Edge, LocalDiag.


                                                     94
   The following are not generic F3 routing operations:

                              Cauchy/cube,        Fourier expansion.

   They are post-terminal proof subroutines:

         GoodAWACK → 𝐸10,            CKP → 𝐺8𝑎,           LongAP → 𝐷1,     Edge → 𝐶1.

   —

F3.7. Controlled CRT Absorption          Suppose 𝒜 contains a controlled congruence

                              𝐿(𝑧) ≡ 𝑎 (mod 𝑞),          𝑞 ≤ (log 𝑁 )𝐶 .

   If the congruence is incompatible, the atom is empty and hence Edge-zero.
   If compatible, replace the lattice coset Λ by the subcoset

                                Λ′ = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 𝑎 (mod 𝑞)}.

   Then one unresolved congruence is removed:

                                𝐷unabsorbed (𝒜′ ) < 𝐷unabsorbed (𝒜),

    and no earlier component of M♯ increases. Content may increase by at most a polylogarithmic
factor, which is allowed by E5.
    Therefore

                                        M♯ (𝒜′ ) < M♯ (𝒜).

   If 𝑞 is not controlled, CRT absorption is not allowed as a generic F3 step. Such a case must be
routed through C1, F4, CKP, or LocalDiag depending on the source of the large modulus.
   —

F3.8. F4 Large-Divisor Decision       Suppose 𝒜 contains an unresolved ordinary divisor condition

                                             𝑑 | 𝐿(𝑧).

   F3 does not expand it blindly. It invokes F4 as a decision procedure.
   F4 has the following exhaustive output:

  1. if the divisor condition gives a strict C1P saving, route to Edge;

  2. if it creates balanced multiplicative structure, route to CKP;

  3. if it creates forced local dependence, route to LocalDiag;

  4. otherwise, after fixed quotienting/content stabilization, route to GoodAWACK.

  Thus either the atom is terminal, or the unresolved large-divisor predicate is removed from
𝒪(𝒜). In the nonterminal case:

                                 𝑂unresolved (𝒜′ ) < 𝑂unresolved (𝒜)

                                                 95
   or at least

                                    𝑅largeDiv (𝒜′ ) < 𝑅largeDiv (𝒜)

    with no earlier increase.
    The operation may introduce quotient variables and therefore may increase 𝐽free , but this is
irrelevant because 𝐽free is the last component of M♯ .
    Therefore every nonterminal F4 decision strictly decreases M♯ .
    —

F3.9. Square-Divisor Routing          If the obstruction is square-divisor type

                                              𝑑2 | 𝐿(𝑧),

   then either:

  1. 𝑑 > 𝐷, in which case C1 square-divisor Edge applies;

  2. 𝑑 ≤ 𝐷, in which case the condition is a controlled CRT/divisibility restriction and can be
     absorbed.

    In case 1 the atom is terminal Edge. In case 2 controlled absorption removes an unresolved
divisibility predicate, so M♯ decreases.
    Thus square-divisor routing is terminal or decreasing.
    —

F3.10. Finite Grouping Selection/Elimination               Suppose the atom is not terminal but has
unresolved grouping alternatives

                                             𝒢(𝒜) ̸= ∅.

   Choose one grouping 𝐺 ∈ 𝒢(𝒜).
   After applying this grouping, exactly one of the following happens:

  1. terminal Edge predicate holds;

  2. terminal CKP predicate holds;

  3. terminal GoodAWACK predicate holds;

  4. terminal LongAP/Local predicate holds;

  5. terminal LocalDiag predicate holds;

  6. no terminal predicate holds.

   In cases 1–5, routing terminates.
   In case 6, remove 𝐺 from the unresolved grouping set:

                                        𝒢(𝒜′ ) = 𝒢(𝒜) ∖ {𝐺}.

   Thus

                                                  96
                                       |𝒢(𝒜′ )| = |𝒢(𝒜)| − 1.

   No earlier obstruction component increases: the failed grouping is recorded as eliminated, not
converted into a new obstruction. Therefore

                                           M♯ (𝒜′ ) < M♯ (𝒜).

   —

F3.11. LocalDiag Detection If any forced equality, proportionality, gcd-local dependence, or
unavoidable collision is detected, then

                                            IsLocalDiag(𝒜)

   holds and 𝒜 is terminal.
   F3 does not perform indefinite partial diagonal extraction. LocalDiag detection is terminal.
   This avoids cycles of the form:

             partial collision extraction → new collision → partial extraction again.

   —

F3.12. Edge Detection       If any strict C1P Edge predicate holds, then

                                              IsEdge(𝒜)

   and the atom is terminal.
   F3 uses C1 only as a terminal detector. It does not label ordinary divisor conditions as Edge
unless a C1 saving predicate is explicitly satisfied.
   —

F3.13. Terminal Class Labelling If none of the unresolved obstruction operations applies and
no grouping alternative remains, then the atom has no unresolved divisor, congruence, grouping,
collision, conductor, or balance decision.
    Then B3 structural classification plus the terminal predicates imply exactly one of:

       IsEdge(𝒜),   IsCKP(𝒜),      IsGoodAWACK(𝒜),           IsLocalDiag(𝒜),   IsLongAP(𝒜).

  The only possible residual alternative would be a MixedResidual atom: not Edge, not CKP, not
GoodAWACK, not LocalDiag, not LongAP/Local.
  But such a MixedResidual atom would necessarily contain at least one unresolved item:

  • unresolved ordinary divisor;

  • unresolved quotient equation;

  • unresolved grouping alternative;

  • unresolved conductor decision;

  • unresolved local collision decision;

                                                  97
   • unresolved choice between multiplicative balanced and affine WACLE form.

   This contradicts

                                      𝒪(𝒜) = ∅,        𝒢(𝒜) = ∅.

   Therefore no MixedResidual terminal class exists.

Decidability at termination At the terminal-labelling stage there is no circular call back into
F4. Indeed, when

                                      𝒪(𝒜) = ∅,        𝒢(𝒜) = ∅,

    all conditions appearing in F3.2.2–F3.2.3 are decidable from the finite atom data.
    The phrase "no unresolved ordinary large divisor condition" in IsGoodAWACK means exactly
that the large-divisor component of 𝒪(𝒜) is zero. If such a condition were still present, F3.8 would
call F4 before terminal labelling.
    The phrase "no forced local diagonal obstruction" in IsCKP and IsGoodAWACK means that
the collision/local dependence component of 𝒪(𝒜) is zero. If a forced equality, proportionality, gcd-
local dependence, or conductor collapse were present, F3.11 would have already labelled the atom
LocalDiag or sent it to the appropriate F4/C1 branch.
    The remaining decisions are dyadic scale comparisons, coefficient type labels, and whether the
B3 grouping is multiplicative-balanced or affine-WACLE. These are read from the finite B1/B3 atom
description and require no further routing operation. Hence the terminal predicates are genuine
predicates at termination, not requests for another pass through F4.
    —

F3.14. Theorem F3’

Theorem D.3 (Theorem F3’). Let 𝒜 be any atom produced by B1 typed decomposition and B3
preliminary classification. Then after finitely many F3 routing steps, 𝒜 is written as a finite disjoint
sum of terminal atoms belonging to

             Edge,       CKP,       GoodAWACK,           LocalDiag,       LongAP/Local.

   No other terminal class occurs.

Proof. If 𝒜 is already terminal, there is nothing to prove.
   Otherwise 𝒜 has at least one unresolved obstruction or grouping alternative:

                                     𝒪(𝒜) ̸= ∅    or   𝒢(𝒜) ̸= ∅.

   Apply the appropriate routing operation:

   • controlled CRT absorption for controlled congruences;

   • F4 decision for ordinary large divisors;

   • square-divisor routing for square-divisor obstructions;

   • finite grouping selection/elimination for unresolved groupings;

                                                  98
  • terminal LocalDiag detection for forced dependence;

  • terminal Edge detection for strict C1P-saving predicates.

   By Sections F3.7–F3.12, every nonterminal operation strictly decreases

                                                  M♯ .

    Since M♯ takes values in N6 with lexicographic order, no infinite strictly decreasing sequence
exists. Hence the routing process terminates after finitely many steps.
    At termination, no unresolved obstruction and no unresolved grouping alternative remains. By
Section F3.13, the terminal atom must be one of

                       Edge, CKP, GoodAWACK, LocalDiag, LongAP/Local.

   Thus terminal exhaustion holds and no sixth terminal class exists. The theorem is proved.
   Lemma F3T expands this theorem into a finite row-by-row routing table indexed by B1 block
type, B3 grouping type, dyadic regime, divisor/conductor regime, coefficient type, terminal class,
and exclusion reason. Lemma F3T is a tabular restatement of the F3.6–F3.14 routing mechanism,
not an additional routing operation.
   —


F3.15. Partition Identity For every typed B1 block ℬ, B3/F3/F4 routing produces a finite
family of tagged terminal atoms

                                            {(ℬ, 𝜏 )}𝜏 ∈𝒯 (ℬ)

   such that the exact identity

                                      𝑅ℬ (𝑁 ) =              𝑅ℬ,𝜏 (𝑁 )                    (F3-partition)
                                                    ∑︁

                                                  𝜏 ∈𝒯 (ℬ)

   holds before any terminal estimates are applied.

Proof. The routing process is an iterated finite partition of the summation domain. B3 first
partitions a typed B1 block into finitely many grouping/candidate cells. Each subsequent F3/F4
step is one of the following exact operations:

  1. terminal labelling of the current cell;

  2. controlled CRT restriction, using the exact union over residue classes;

  3. divisibility or square-divisibility splitting, using identities such as 1 = 1𝑑|𝐿 + 1𝑑∤𝐿 ;

  4. F4 quotient/divisor decision, where every branch carries the inherited routing tag;

  5. finite grouping selection/elimination, which partitions the finite set of available grouping
     alternatives;

  6. LocalDiag, Edge, CKP, GoodAWACK, or LongAP terminal detection.



                                                   99
    No step discards mass. When a branch is later proved negligible, that estimate is made by the
corresponding terminal package, not by F3. Since F3.14 proves finite termination by the strictly
decreasing measure M♯ , the iterated finite partition reaches terminal cells after finitely many steps.
The tag 𝜏 records the complete splitting history, so distinct terminal tags correspond to disjoint
cells of the parent B1 block. Summing the terminal cell contributions gives (F3-partition). Lemma
proved.
    —


F3.16. Output to Terminal Packages Lemma F3 does not prove terminal estimates. It only
routes terminal atoms to the correct packages:

                                            Edge → 𝐶1,

                                            CKP → 𝐺8𝑎,

                                       GoodAWACK → 𝐸10,

                                    LongAP/Local → 𝐷1/𝐻4𝑀,

                                        LocalDiag → 𝐻4𝑀.

   The proof-subroutines are external to F3:

   • Fourier expansion is used inside G2a, D1, C1;

   • Cauchy/cube/Gowers machinery is used inside E10;

   • local projection algebra is evaluated inside H4 and imported into the final assembly through
     H4M.

   This separation is part of the no-cycle proof.
   —

F3.17. Consequence for the Proof Tree

                F3 proves routing exhaustion using the strengthened measure M♯ .

    Generic Cauchy/cube operations and Fourier expansion are not F3 routing operations. They are
treated inside the terminal packages. F4 remains the large-divisor decision subroutine used by F3.
    —

F3.18. Dependency Check The no-cycle routing proof uses the following supporting statements:

   1. B3 preliminary classification gives a finite set of admissible grouping alternatives.

   2. F4 large-divisor decision is exhaustive.

   3. the structural Edge predicates are strict terminal tags for genuine saving mechanisms;

   4. the structural predicates CKP, GoodAWACK, LongAP/Local, and LocalDiag are mutually
      adequate to classify atoms with no unresolved obstruction.

                                                 100
   Thus F3 is a routing theorem. It labels terminal branches but does not use the later estimates
that discharge those branches.
   Lemma F3T is the finite table associated with this theorem. It is a child of F3, not a new
hypothesis for F3.

F3.19. Logical Dependencies Internal dependencies: B1, B3, F4, E5, LPI, and the proof
parameter register.
    Internal nodes served: F4, F3A, F3T, F3F4M, BGS, HGO2R, E10L, I1, and the terminal branch
assembly.

D.4      F3 complete routing interface
D.4.1      F3A. Completeness of the F3.6 Routing Interface
F3A.0. Role Logical ID: F3A.
   This verification addresses the routing-interface condition needed in the Branch B / GoodAWACK
source check:

                E10M depends on F3.6 being the exhaustive list of F3 routing operations.

   The verification does not introduce a new routing procedure. It records the exact contract later
used by E10YMX/E10L:


every actual terminal GoodAWACK skeleton is generated by B1/B3 data, F3.6 routing operations, F4 decisions, and E5-clean stability only.
                                                                                  (F3-COMPLETE)
     Here "actual" has the same non-circular meaning as in E10Y: the object lies in the image of the
independently defined B1/B3/F3/F4 construction. The word does not mean "accepted by E10Y."
     The F3.6 list includes square-divisor routing, matching the square-divisor step used in the F3
decrease check and routing theorem.
     Lemma F3T gives the associated finite routing table. F3T does not add a new operation to the
list below; it expands the same operation list by B1 block type, B3 grouping type, dyadic regime,
divisor/conductor regime, coefficient type, terminal class, and exclusion reason.
     Used by: E10Y, E10M, E10K, E10L, and E10X.
     Uses: B1, B3, F3, F3T, F4, and E5. The E10 lemmas consume this interface; they are not
prerequisites for it.
     —

F3A.1. Complete F3.6 Operation List                            The generic F3 routing-level operations are exactly:

   1. controlled CRT absorption;

   2. F4 large-divisor decision;

   3. square-divisor routing;

   4. finite grouping selection/elimination;

   5. terminal LocalDiag detection;

   6. terminal Edge detection by C1P predicates;

                                                                 101
  7. terminal class labelling into CKP, GoodAWACK, LongAP/Local, Edge, or LocalDiag.

   The explicitly excluded operations are:

  1. Cauchy/cube operations;

  2. Fourier expansion;

  3. primitive analytic slicing after the terminal tensor-verification skeleton is fixed.

   Those excluded operations may occur inside post-terminal estimates, but they do not generate
new terminal GoodAWACK skeletons.
   —

F3A.2. Occurrence Map


   Interface location              Operation class                  Post-terminal interpretation
   Lemma F3, F3.7                  controlled CRT absorption        Full-rank finite-index restriction
                                                                    or impossible fibre; content
                                                                    controlled by E5.
   Lemma F3, F3.8                  F4 large-divisor decision        Delegates ordinary large-divisor
                                                                    and quotient cases to F4; out-
                                                                    put is Edge, CKP, LocalDiag,
                                                                    GoodAWACK with tags, or a
                                                                    decreasing continuation.
   Lemma F3, F3.9                  square-divisor routing           Large square divisors are C1
                                                                    Edge; small square divisors
                                                                    are controlled divisibility/CRT
                                                                    restrictions.
   Lemma F3, F3.10                 finite grouping selection/       Selects among B3 finite product
                                   elimination                      groupings; failed groupings are
                                                                    removed and not converted into
                                                                    new affine operations.
   Lemma F3, F3.11                 LocalDiag detection              Terminal detection; the atom
                                                                    leaves GoodAWACK.
   Lemma F3, F3.12                 Edge detection                   Terminal C1 detection; the
                                                                    atom leaves GoodAWACK.
   Lemma F3, F3.13                 terminal class labelling         Labelling only; no coordinate
                                                                    operation is performed.
   Lemma F3, F3.14                 routing theorem                  Uses exactly the operations
                                                                    above to prove termination and
                                                                    terminal exhaustion.



    This table is the interface used by E10M. In particular, any rank drop in an actual terminal
GoodAWACK record must come from F4 quotient/divisor data, LocalDiag/Edge/CKP tags, con-
trolled CRT/divisibility data, or post-terminal analytic slicing after terminality. None of these is an
untagged free affine regrouping.
    —


                                                 102
F3A.3. Exhaustiveness Theorem for the F3 Interface Let 𝒜 be a nonterminal atom
produced by B1 and B3. By F3.3 and F3.4, any reason why 𝒜 is not terminal belongs to:

  1. an unresolved ordinary divisor or quotient condition;

  2. an unresolved conductor/CRT/congruence restriction;

  3. an unresolved square-divisor obstruction;

  4. an unresolved product grouping or balance alternative;

  5. an unresolved local collision or dependence decision;

  6. an unresolved choice between CKP and GoodAWACK normal form.

   The F3.6 list covers these cases as follows.


       Nonterminal reason                           Covered by
       ordinary divisor or quotient condition       F4 large-divisor decision
       conductor/CRT/congruence restriction         controlled CRT absorption, or F4/C1/CKP/
                                                    LocalDiag if uncontrolled
       square-divisor obstruction                   square-divisor routing
       grouping or balance alternative              finite grouping selection/elimination
       local collision or dependence                terminal LocalDiag detection
       strict saving predicate                      terminal Edge detection
       no unresolved obstruction remains            terminal class labelling




   Therefore an atom with no applicable F3.6 operation has no unresolved obstruction and no
unresolved grouping alternative. F3.13 then forces one of the five terminal classes. This is exactly
the terminal exhaustion theorem F3.14.
   The same implication is tabulated in Lemma F3T.

Proof of interface completeness. Let 𝒜 be an actual nonterminal atom after B1/B3. By Lemma
F3.3, every unresolved reason preventing terminal classification belongs to the finite obstruction set
𝒪(𝒜), whose labels are:

  1. ordinary divisor;

  2. quotient;

  3. conductor;

  4. CRT/congruence;

  5. grouping/balance;

  6. local collision;

  7. CKP/GoodAWACK choice.


                                                  103
   By Lemma F3.4 every unresolved product regrouping belongs to the finite grouping set 𝒢(𝒜).
The F3.6 operations act on exactly these two finite sources of nonterminality:

  1. controlled CRT absorption acts on the CRT/congruence part of 𝒪(𝒜);

  2. F4 large-divisor decision acts on ordinary divisor, quotient and conductor entries of 𝒪(𝒜);

  3. square-divisor routing acts on square-divisor obstructions;

  4. finite grouping selection/elimination acts on 𝒢(𝒜);

  5. LocalDiag detection acts on forced local collision or dependence entries;

  6. Edge detection acts on strict C1P-saving predicates;

  7. terminal class labelling applies when both 𝒪(𝒜) and 𝒢(𝒜) are empty.

   No other nonterminal datum is present in the F3 atom description of Lemma F3.1. Thus any
proposed additional F3 routing operation would have to act on a datum outside 𝒪(𝒜) ∪ 𝒢(𝒜), or
on a datum already covered by one of the seven cases above. The first possibility is not an actual
B1/B3/F3 atom; the second is not a new operation. Hence the F3.6 list is complete for actual F3
routing.
   —


F3A.4. Consequence for E10YMX E10Y may cite F3.6 as the routing-level part of the skeleton-
generating grammar. E10M, E10X, and E10K may then use E10Y as the formal completeness input.
Under the operation list above:

  1. square-divisor routing is explicit and therefore no longer a hidden F3 operation;

  2. Cauchy/cube/Fourier/slicing operations are post-terminal estimates, not terminal-skeleton
     generators;

  3. arbitrary untagged rank-dropping affine regrouping is not an allowed routing operation.

   Thus:

       F3A + E10Y + E10M =⇒ the F3-complete routing premise used by E10K is explicit.

   —

F3A.5. Stability Rule Any change to F3 that adds, renames, or reinterprets a routing-level
operation must update this verification, E10Y, E10M.3, and E10K. Without that check, the F3-
complete routing premise is not valid for the changed routing interface.

F3A.6. Logical Dependencies Internal dependencies: B1, B3, F3, F3T, F4, and E5.
  Internal nodes served: E10Y, E10M, E10K, E10L, and E10X.




                                               104
D.5     F3 complete routing table
D.5.1    F3T. Finite Routing Table for B1-Origin Atoms
F3T.0. Role Logical ID: F3T.
    Lemma F3T is the tabular routing lemma associated with Lemmas B1, B3, F3P, F3, F4, E5, and
the LPI terminal-class interface. It does not introduce a new routing operation. It records the finite
case distinction which is implicit in F3.6–F3.15 and makes explicit where every B1-origin atom goes.
    Associated with F3; used by F3A, E10M, E10K, E10L, I1, and the manuscript routing appendix.
    Uses: B1, B3, the intrinsic predicate catalogue F3P, the F3 routing definitions F3.1–F3.15, F4,
E5, LPI, and the proof parameter register. The branch theorems C1, D1, G8a, E10L, and H4 estimate
or assemble the terminal classes after F3T has labelled them.
    The purpose is to prove the following interface statement.


For fixed 𝐽0 , every B1-origin atom is routed into exactly one of Edge, CKP, GoodAWACK, LocalDiag, or LongAP/Local.
                                                                                           (F3T)
    The word "exactly" refers to the tagged partition produced by Lemma F3. If a cell satisfies
more than one terminal predicate, the routing tag records the first applicable terminal class in
the deterministic order stated in F3T.2; the other predicates are retained only as supplementary
verification information and do not create additional terminal atoms.
    —

F3T.1. Finite input data Fix 𝐽0 . A B1 block consists of two Heath–Brown finite-convolution
sides. Each side has 2𝑗 variables with 1 ≤ 𝑗 ≤ 𝐽0 , hence the total number of elementary variables is
at most 4𝐽0 . After the exact dyadic partition, each elementary coefficient is of type

                                           𝜇 · 1≤𝑦 ,      1,      log .

   Thus a B1 block has only the following finite structural data:

  1. the two finite lists of elementary variables;

  2. their coefficient types;

  3. their dyadic scales;

  4. the Goldbach equation relating the two sides;

  5. the finite set of admissible product groupings generated by B3.

   B3 supplies the finite grouping set

                                           𝒢(ℬ),       |𝒢(ℬ)| ≤ 24𝐽0 ,

   and the preliminary candidate labels TypeI/Edge, LongAP/Local, CKP, BranchB, and LocalDiag.
Candidate labels may overlap at this stage; uniqueness is produced only after the F3 tagged routing
partition.
   —




                                                       105
F3T.2. Canonical tagged routing order The F3 routing table is read in the following
deterministic order on each current tagged cell.

  1. Empty or incompatible cells are discarded as zero Edge cells.

  2. Strict C1 saving predicates are routed to Edge.

  3. Forced equality, proportionality, repeated factors, or local dependence is routed to LocalDiag.

  4. Ordinary divisor and quotient predicates are processed by F4.

  5. Square-divisor obstructions are processed by the square-divisor routing of F3.

  6. Controlled CRT/congruence restrictions are absorbed if they are full-rank and controlled by
     E5; otherwise they are routed through F4, C1, CKP, or LocalDiag according to the source of
     the restriction.

  7. Remaining unresolved grouping alternatives are selected or eliminated from the finite B3
     grouping set.

  8. A cell with no unresolved obstruction and no unresolved grouping alternative is terminally
     labelled as CKP, GoodAWACK, LongAP/Local, Edge, or LocalDiag by the intrinsic F3P
     terminal predicates as implemented by F3.

   Each nonterminal application is one of the allowed F3.6 operations and strictly decreases the F3
measure M♯ . Therefore the table cannot be read indefinitely.
   —

F3T.3. Complete finite routing table         The table below uses the following abbreviations:

  • B1 type records the relevant finite-convolution source;

  • B3 grouping is the preliminary grouping/candidate pattern;

  • Regime is the dyadic or structural condition visible on the current tagged cell;

  • Divisor/conductor state records whether F4, square-divisor routing, or controlled CRT
    absorption is needed;

  • Coefficient type records the surviving arithmetic coefficient shape;

  • Terminal class is the class assigned by the canonical routing order;

  • Exclusion reason explains why the other terminal classes do not receive the same tagged
    cell.




                                               106
      Row   B1 block type         B3 grouping type      Dyadic/structural       Divisor/conductor      Remaining coeffi-       Routed terminal   Reason other ter-     Structural source
                                                        regime                  state                  cient type              class             minal classes are
                                                                                                                                                 excluded
      1     Any typed B1 block    Any grouping          Empty support           Inconsistent CRT/      None                    Edge-zero         The cell has zero     F3
                                                        or incompatible         divisibility con-                                                mass, so no analytic
                                                        congruences             straints                                                         terminal class is
                                                                                                                                                 created.
      2     Any typed B1 block    TypeI/Edge or any     Boundary tail, short    A strict C1P Edge      Divisor-bounded         Edge              CKP/GoodAWACK/ F3, C1P
                                  grouping              residual volume,        predicate E1–E7 is     finite-convolution                        LongAP require
                                                        large square-divisor    structurally present   coefficient                               a non-negligible
                                                        tail, large gcd/                                                                         central or long local
                                                        content layer, high                                                                      cell; LocalDiag
                                                        Fourier tail, small-                                                                     requires forced local
                                                        conductor layer, or                                                                      dependence.
                                                        Type I saving
      3     Any typed B1 block    LocalDiag flag from   Forced equality,        Any associated         Local congruence-       LocalDiag         Edge requires          F3, F4, LPI
                                  B3 or later F4 cell   proportionality, re-    divisor relation is    only data                                 a strict saving
                                                        peated factor, fixed    part of the local                                                predicate; CKP/
                                                        local dependence, or    dependence tag                                                   GoodAWACK re-
                                                        one active form de-                                                                      quire non-diagonal
                                                        termined by another                                                                      independent vari-
                                                                                                                                                 ables; LongAP/
                                                                                                                                                 Local requires a one-
                                                                                                                                                 dimensional AP/
                                                                                                                                                 local cell rather than
                                                                                                                                                 a diagonal relation.
      4     Any typed B1 block    LongAP/Local          After fixing auxil-     No unresolved          Local AP weight         LongAP/Local      A surviving non-       B3, F3P, F3, LPI
                                  candidate             iary variables, one     ordinary divisor       whose long-variable                       local oscillatory
                                                        long AP variable        or uncontrolled        coefficients lie in                       factor prevents
                                                        remains and the         conductor remains      Cloc (𝑄𝜏 ) and with                       F3P-LongAP/Local
                                                        F3P local coefficient                          no surviving nonlo-                       and routes to CKP
107




                                                        predicate holds                                cal 𝜇-, 𝜆-, Fourier-,                     or GoodAWACK;
                                                                                                       Kloosterman-, or                          short/boundary
                                                                                                       nilsequence-type                          loss routes to Edge;
                                                                                                       oscillation                               forced dependence
                                                                                                                                                 routes to LocalDiag.
      5     Any typed B1 block    CKP-balanced          Two long grouped        No small-conductor,    Arbitrary divisor-      CKP               Edge exclusions        B3, F3
                                  grouping              variables on each       large-𝑔, high-         bounded finite-                           have already been
                                                        side are central and    frequency, bound-      convolution coeffi-                       removed struc-
                                                        balanced                ary, or LocalDiag      cients allowed by                         turally; LocalDiag
                                                                                obstruction remains    the CKP structural                        was tested earlier;
                                                                                                       predicate                                 GoodAWACK is ex-
                                                                                                                                                 cluded by balanced
                                                                                                                                                 bilinear CKP struc-
                                                                                                                                                 ture; LongAP/Local
                                                                                                                                                 is excluded by two-
                                                                                                                                                 sided bilinear shape.
      6     BranchB candidate     Non-short, nonlocal, No ordinary large-   Controlled content         Bounded affine/         GoodAWACK         Edge, LocalDiag,       F3, E5
                                  non-CKP central-     divisor predicate is and controlled CRT         finite-convolution                        CKP, and Lon-
                                  long affine grouping unresolved; no local data                       coefficient with                          gAP/Local have
                                                       dependence; no CKP                              controlled content                        all failed by the
                                                       balance                                                                                   previous rows; the
                                                                                                                                                 cell is passed to
                                                                                                                                                 the GoodAWACK
                                                                                                                                                 finite-grammar pack-
                                                                                                                                                 age for the rank-
                                                                                                                                                 dropping AFF clo-
                                                                                                                                                 sure.
      7     Any typed B1 block    Any B3 grouping       Divisor or quotient     F4 Case I: short di- Divisor-bounded           Edge              F4 supplies the        F4
            with ordinary divi-                         condition 𝑑 | 𝐿 or      visor, short quotient, finite-convolution                        structural Edge tag;
            sor predicate                               𝐿 = 𝑑𝑠 remains          or explicit saving     coefficient                               no central terminal
                                                        unresolved              predicate                                                        class is entered.
      8    Any typed B1 block    Any B3 grouping        Divisor or quotient   F4 Case II: quoti-    Local congruence/    LocalDiag          The divisor re-         F4, LPI
           with ordinary divi-                          condition remains     enting forces local   diagonal data                           lation identifies
           sor predicate                                unresolved            dependence                                                    active forms, so
                                                                                                                                            independent CKP/
                                                                                                                                            GoodAWACK vari-
                                                                                                                                            ables are absent.
      9    Any typed B1 block    CKP-compatible         Divisor and quotient F4 Case III: bal-    Divisor-bounded        CKP                Short/local cases       F4
           with ordinary divi-   after quotienting      variables remain     anced multiplicative bilinear coefficient                      were removed
           sor predicate                                long and balanced    divisor structure                                              by F4 Cases I–II;
                                                                                                                                            GoodAWACK is ex-
                                                                                                                                            cluded by balanced
                                                                                                                                            bilinear structure.
      10   BranchB or affine     Non-short, nonlocal, Central-long affine     F4 Case IV: quotient Controlled affine     GoodAWACK          F4 has already          F4, E5
           residual after F4     non-CKP after        residual with con-      tag recorded; no     finite-convolution                       excluded Edge,
                                 quotienting          trolled quotient/       untagged variable    coefficient                              LocalDiag, and
                                                      content                 divisor survives                                              CKP; LongAP/Local
                                                                                                                                            is absent because
                                                                                                                                            the residual is not
                                                                                                                                            a one-dimensional
                                                                                                                                            local AP cell.
      11   Any typed B1 block    Any grouping           Large square divisor Square-divisor tail    Divisor-bounded      Edge               The structural          F3
                                                        𝑑2 | 𝐿 with 𝑑 > 𝐷                           finite-convolution                      square-divisor Edge
                                                                                                    coefficient                             predicate applies.
      12   Any typed B1 block    Any grouping           Small square divisor Controlled full-rank   Same coefficient     Nonterminal de-    No terminal class       F3, E5
                                                        or controlled divisi- divisibility/CRT      type with polylog    crease             is assigned yet;
                                                        bility condition      absorption            content loss                            the unresolved
                                                                                                                                            divisor component
                                                                                                                                            is removed and
                                                                                                                                            F3 continues with
                                                                                                                                            smaller M♯ .
108




      13   Any typed B1 block    Any grouping           Controlled CRT/       Full-rank finite-     Same coefficient     Nonterminal de-    The cell remains in     F3, E5
                                                        congruence restric-   index restriction     type with controlled crease             the routing process;
                                                        tion with full-rank                         content                                 if the restriction is
                                                        lattice image                                                                       incompatible Row 1
                                                                                                                                            applies.
      14   Any typed B1 block    Any unresolved         Candidate overlap,    No new divisor/       Same coefficient     Nonterminal de-    The selected group-     B3, F3
                                 grouping alternative   e.g. TypeI/CKP or     conductor operation   type                 crease or one of   ing either triggers a
                                                        CKP/BranchB                                                      Rows 2–6           terminal row or is
                                                                                                                                            eliminated from the
                                                                                                                                            finite B3 grouping
                                                                                                                                            set.
   Rows 12–14 are not terminal rows. They are included because they are the only nonterminal
operations that can occur before a terminal row is reached. In each case Lemma F3 proves strict
decrease of M♯ .
   —

F3T.4. Residual Exclusion Table The table F3T.3 is the formal routing table. The following
refinement records the same exhaustion in the order in which an external reader can check a putative
mixed residual. The last column is the reason why the row cannot produce a sixth terminal class.


  Surviving cell after   Active scale profile                     Terminal destination Why no mixed resid-
                                                Surviving coefficient
  earlier tests                                                                        ual remains
  Empty support or      none                none                  Edge-zero            The tagged cell has
  incompatible CRT                                                                     zero summation
                                                                                       domain.
  Boundary, short im- noncentral or too     divisor-bounded       Edge                 A C1P strict pred-
  age, large content/ small for a main      finite-convolution                         icate E1–E7 is
  gcd, square-divisor   term                coefficient                                present before any
  tail, high Fourier                                                                   central terminal la-
  tail, small conductor                                                                bel is allowed; C1A
                                                                                       later records the
                                                                                       source-specific ad-
                                                                                       mission.
  Forced equality, re- diagonal/local fibre congruence-only       LocalDiag            The independent
  peated form, propor-                      local coefficient                          variables needed for
  tional active forms,                                                                 CKP/GoodAWACK
  or local dependence                                                                  are absent; H4 ad-
                                                                                       mits the cell only as
                                                                                       a tagged canonical
                                                                                       local projection.
  One F3P-long AP/ one-dimensional          local residue-density LongAP/Local         The positive F3P
  local fibre with no   long local fibre    weight                                     predicate excludes
  nonlocal arithmetic                                                                  surviving 𝜇, 𝜆,
  coefficient                                                                          Fourier, Klooster-
                                                                                       man, or nilsequence
                                                                                       coefficients; D1.2A
                                                                                       then expands this
                                                                                       local algebra into
                                                                                       the tagged LPI pro-
                                                                                       jection.
  Balanced two-sided central balanced 𝑎, 𝑞 divisor-bounded        CKP                  The defining CKP
  multiplicative bilin- and dual variables  bilinear coefficients                      balance and gcd
  ear structure                                                                        split are present;
                                                                                       noncentral or ex-
                                                                                       cluded frequency
                                                                                       ranges have already
                                                                                       been sent to C1/
                                                                                       G8a local.




                                                        109
  Central-long affine    full central image,   bounded affine finite- GoodAWACK         The cell is passed to
  Branch B cell after    nonlocal, non-CKP     convolution coeffi-                      the GoodAWACK
  all strict savings,                          cient with controlled                    finite-grammar pack-
  local dependence,                            content                                  age. F3T itself only
  and CKP balance                                                                       records the struc-
  have failed                                                                           tural terminal label
                                                                                        and does not use
                                                                                        the downstream E10
                                                                                        estimate.
  Ordinary divisor/      depends on quotient divisor-bounded         Edge, LocalDiag,   F4 is a decision tree,
  quotient cell before   case                  coefficient           CKP, GoodAWACK, not a new terminal
  F4 finishes                                                        or nonterminal de- class; each outcome
                                                                     crease             is one of the existing
                                                                                        destinations.
  Controlled full-rank   unchanged after       same coefficient with nonterminal de-    E5/F3 only trans-
  CRT or grouping        finite-index restric- controlled content    crease             port the existing
  ambiguity              tion                                                           cell; the F3 measure
                                                                                        decreases and ter-
                                                                                        minal labelling is
                                                                                        postponed.




   Thus the apparently broad fallback phrase "central-long affine residual" has a precise meaning in
the active routing table. It is not "whatever remains". It is the row in which:

  1. no strict C1P Edge predicate is present;

  2. F3/F4 have not found a LocalDiag relation;

  3. the positive F3P LongAP/Local predicate has failed, so the cell is not a one-dimensional local-
     coefficient AP fibre;

  4. G8a/CKP balance is absent;

  5. all ordinary divisor, square-divisor, CRT, and grouping operations have either been absorbed
     with controlled content or have strictly decreased the routing measure; and

  6. the remaining affine data is an actual B1/B3/F3/F4-origin terminal GoodAWACK skeleton.

   If any one of these six checks fails, the cell is routed by an earlier row and does not enter
GoodAWACK. If all six checks pass, the cell is structurally a nonlocal central affine macro-template
with controlled content. The later GoodAWACK package proves that no untagged rank-dropping
AFF source survives in such actual terminal skeletons. This is why F3T does not create a hidden
MixedResidual class.
   —

F3T.5. Finiteness of the table             For fixed 𝐽0 , the table is finite for three reasons.

  1. B1 has at most 4𝐽0 elementary variables and only three elementary coefficient types.

  2. B3 has at most 24𝐽0 admissible product groupings.



                                                      110
    3. F3.6 has exactly seven allowed routing-level operation types, and F4 has exactly the four
       terminal outcomes Edge, LocalDiag, CKP, GoodAWACK, plus the controlled absorption/
       decrease case.

    All dyadic thresholds used in the rows are qualitative comparisons against fixed powers of 𝑁 or
log 𝑁 determined by the global parameter hierarchy. Therefore, after dyadic localization, each row
represents only finitely many tagged subcases, with polylogarithmic total multiplicity.
    —

F3T.6. Exhaustion theorem

Lemma D.4 (Lemma F3T). Let 𝒜 be any atom produced by Lemma B1 and preliminarily classified
by Lemma B3. Applying the canonical routing order of F3T.2 and the case table F3T.3 writes 𝒜 as
a finite disjoint sum of tagged terminal atoms in exactly the five terminal classes

                  Edge,           CKP,            GoodAWACK,                    LocalDiag,                  LongAP/Local.

     No sixth terminal class occurs.

Proof. Start with the finite B3 grouping set of 𝒜. If a terminal row 2–11 applies, the current cell is
labelled by the corresponding terminal tag. If a nonterminal row 12–14 applies, then the operation
is one of the allowed F3.6 operations and Lemma F3 proves that M♯ strictly decreases.
    Since M♯ is lexicographically well-founded and the grouping set is finite, the process terminates.
At termination no controlled CRT, ordinary divisor, square-divisor, grouping, saving, or local-
dependence question remains unresolved. The terminal-labelling rows 2–6 are then exhaustive by
the B3 preliminary classification, the F4 decision tree, and the residual exclusion argument in F3T.4.
    The deterministic order in F3T.2 assigns a unique terminal tag to each terminal cell. Candidate
overlaps do not create duplicate mass because F3.15 records the exact tagged partition identity.
Hence every B1-origin atom is partitioned into the five named terminal classes and no additional
MixedResidual class exists. Lemma proved.
   —


                                  Remark D.5 (F3T.7. Output).
       B1 + B3 + F3 + F4 + E5 =⇒ complete finite routing exhaustion for B1-origin atoms.
     The terminal estimates are not part of F3T. They are:


Edge → 𝐶1𝑃 → 𝐶1𝐴 → 𝐶1,   CKP → 𝐺8𝑎,   GoodAWACK → GoodAWACK package,   LongAP/Local → LPI-admissible local package,   LocalDiag → LPI-admissible local package.



   Thus F3T is the finite exhaustion statement that every B1-origin atom has a named terminal
destination.

F3T.8. Logical Dependencies Internal dependencies: B1, B3, the F3 routing definitions
F3.1–F3.15, F4, E5, LPI, and the proof parameter register.
   Internal nodes served: F3A, E10M, E10K, E10L, I1, and the manuscript routing appendix. F3T is
associated with F3 as its finite routing table; it is not a new hypothesis needed to prove F3.




                                                                       111
D.6     F4 large-divisor and quotient-tag routing
D.6.1    F4. Large Divisor Routing Lemma
F4.0. Role Logical ID: F4.
    Lemma F4 is the exhaustive large-divisor decision procedure used by Lemma F3. It proves the
following statement:

            ordinary large divisor predicates are never left as unresolved residual atoms.

   In other words, if an atom contains a condition

                                               𝑑 | 𝐿(𝑧)

   or a quotient equation

                                              𝐿(𝑧) = 𝑑𝑠,

   then it must be routed to one of the terminal classes

                        Edge,      CKP,       LocalDiag,           GoodAWACK,

    or it must strictly decrease the obstruction measure M♯ from Lemma F3.
    The purpose of Lemma F4 is to close precisely this decision step. LongAP/Local is not an
output of F4 itself; it is a separate terminal branch of the B3/F3 routing layer when the ordinary
large-divisor obstruction handled by F4 is absent or has already been removed.
    Used by: F3, F3T, BGS, HGO2R, E10L, and the GoodAWACK finite-grammar closure layer.
    Uses: the F3 atom interface and routing-measure definitions F3.1–F3.6, E5, LPI, X6, and
standard lattice/content algebra. The terminal outputs of F4 are structural tags; their estimates
are proved later by C1, D1, G8a, E10L, and H4.
    The standalone reader-facing form of the combined F3/F4 routing theorem is F3F4M. Lemma F4
supplies the large-divisor and quotient component of that master theorem.
    —

F4.1. What counts as an ordinary large-divisor predicate An ordinary large-divisor
predicate means a structural condition of one of the following types:

                                               𝑑 | 𝐿(𝑧),

                                              𝐿(𝑧) = 𝑑𝑠,

                                        𝑑 | gcd(𝐿1 (𝑧), 𝐿2 (𝑧)),

   where:

  • 𝐿, 𝐿1 , 𝐿2 are affine or product-grouped forms already produced by B1/B3;

  • 𝑑 is not a square-divisor variable already covered by C1;

  • the predicate does not by itself produce a summable tail of type



                                                  112
                                                      𝑑−2 ;
                                                ∑︁

                                                𝑑>𝐷

  • the density of the condition is ordinarily of size 1/𝑑, hence not automatically Edge.

   Important exclusion:

                                       𝑑 | 𝐿(𝑧)       ̸⇒      Edge.

   F4 exists precisely because the harmonic tail
                                                  ∑︁ 1

                                                  𝑑>𝐷
                                                        𝑑

   is not small.
   —

F4.2. Decision parameters        Let

                                            𝐿𝑁 = log 𝑁.

   We use three qualitative scale regimes for a divisor/quotient pair

                                            𝐿(𝑧) = 𝑑𝑠.

  1. Controlled/local divisor:

                                                𝑑 ≤ 𝐿𝐵
                                                     𝑁.

  1. Short-volume regime:

   one of the variables or resulting fibres has effective volume

                                   ≤ 𝑁 𝜀(𝑁 ),               𝑁 → 0.
                                                      𝜀(𝑁 )𝐿𝐶

  1. Central non-short regime:

    both the divisor variable and quotient variable are long enough that neither produces short-
volume Edge.
    The exact constants 𝐵, 𝐶 are fixed after B1/B3 and depend only on 𝐽0 . The word "long" always
means long enough to avoid C1 short-volume / Type-I saving.
    —




                                                  113
F4.3. Fixed divisor absorption           Suppose 𝑑 is fixed on the current atom and

                                                  𝑑 | 𝐿(𝑧).

   Define the restricted lattice coset

                                 Λ𝑑 = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 0 (mod 𝑑)},

   and the quotient form

                                               𝐿𝑑 (𝑧) = 𝐿(𝑧)/𝑑.

           𝑁 , this is controlled CRT absorption. It is handled by Lemma F3.7, and decreases
   If 𝑑 ≤ 𝐿𝐵

                                                 𝐷unabsorbed .

           𝑁 , fixed-divisor absorption is not automatically local. Then either:
   If 𝑑 > 𝐿𝐵

  1. the fibre volume is short and the atom is Edge by C1;
  2. the restriction forces local dependence and the atom is LocalDiag;
  3. the quotient produces a central-long affine atom with controlled content, hence GoodAWACK;
  4. or it produces balanced multiplicative bilinear structure, hence CKP.

   The exact alternative is decided by the scale and dependency tests below.
   —

F4.4. Content quotient lemma             Let

                                                𝑔 = contΛ (𝐿).

   On the restricted lattice Λ𝑑 , the quotient form satisfies
                                                       𝑔
                                      contΛ𝑑 (𝐿/𝑑) =        ≤ 𝑔.
                                                     (𝑔, 𝑑)
Proof. The image of the linear part on the original difference lattice is

                                                 ℓ(Λ0 ) = 𝑔Z.

   After imposing 𝑑 ̸= 0 and 𝑑 | 𝐿(𝑧), the difference lattice satisfies

                                   ℓ(Λ𝑑,0 ) = 𝑔Z ∩ 𝑑Z = lcm(𝑔, 𝑑)Z.

   Dividing by 𝑑, the image of the quotient linear part is
                                         1                𝑔
                                           lcm(𝑔, 𝑑)Z =        Z.
                                         𝑑              (𝑔, 𝑑)
   Hence the quotient content is
                                                    𝑔
                                                         ≤ 𝑔.
                                                  (𝑔, 𝑑)
   Lemma proved.
   —


                                                     114
F4.5. Variable divisor equation        Suppose the ordinary divisor predicate is represented as
                                             𝐿(𝑧) = 𝑑𝑠.
   This is the delicate case because it can introduce a new free variable.
   In Lemma F3, this is handled by placing 𝐽free last in M♯ . Therefore F4 only needs to prove that
the unresolved divisor predicate is either terminally classified or removed from the obstruction set.
   We split into cases.
   —

F4.6. Case I: short divisor or short quotient If either 𝑑 or 𝑠 lies in a short range such that
the resulting effective atom volume satisfies
                                       Voleff (𝒜𝑑,𝑠 ) ≪ 𝑁 𝐿−𝐶
                                                           𝑁 ,

   then the atom is Edge by C1 short-volume or Type-I criteria.
   More explicitly, if after fixing the long variables the remaining sum has at most
                                                𝑁 1−𝜌
   choices for some fixed 𝜌 > 0, then with divisor-bounded coefficients

                                                     𝑁 = 𝑜(𝑁 ).
                                    |𝒜𝑑,𝑠 | ≪ 𝑁 1−𝜌 𝐿𝐶
   Therefore:
                              ShortDivisor/ShortQuotient =⇒ Edge.
   —

F4.7. Case II: forced local dependence          If the equation
                                             𝐿(𝑧) = 𝑑𝑠
   or a gcd condition
                                        𝑑 | gcd(𝐿1 (𝑧), 𝐿2 (𝑧))
   forces two active forms to satisfy a relation on the current lattice,
                                            𝐿𝑖 = 𝑐𝐿𝑗 + 𝑏
   or forces a fixed local congruence class that determines one form from another, then the atom is
LocalDiag.
   This includes:
  1. proportional forms;
  2. repeated factors after quotienting;
  3. fixed gcd layers causing a local diagonal relation;
  4. quotient equations where 𝑠 is forced by another active affine form.
   Thus:
                             ForcedLocalDependence =⇒ LocalDiag.
   No further routing is performed inside F4.
   —

                                                 115
F4.8. Case III: balanced multiplicative divisor structure Suppose neither short-volume nor
LocalDiag applies, and the quotient equation produces two genuinely long multiplicative variables
or grouped products. Then after grouping, the atom has a balanced bilinear finite-convolution form
of the type

                                            𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁,

   or equivalently after gcd splitting,

                                   𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,      (𝑎, 𝑞) = 1.

   This is precisely the CKP terminal class, provided the ranges are central/balanced and the
coefficients remain finite-convolution/divisor-bounded.
   The coefficient condition is preserved because the divisor/quotient variables arise from B1 finite-
convolution factors and quotienting does not increase content by F4.4.
   Thus:

                        BalancedMultiplicativeDivisorStructure =⇒ CKP.

   The subsequent Fourier/Kloosterman-fraction analysis is not part of F4; it is handled by G8a.
   —

F4.9. Case IV: central-long affine residual           Suppose none of the previous cases applies:

  1. not Edge by short volume or C1-saving;

  2. not LocalDiag by forced dependence;

  3. not CKP by balanced multiplicative grouping.

    Then all multiplicative/divisor ambiguity has been absorbed or resolved, and the remaining
atom has central-long affine structure with Liouville-type factors. The quotient/content conditions
are controlled by F4.4 and E5. Therefore the atom satisfies the GoodAWACK terminal predicate of
Lemma F3:

                                          IsGoodAWACK(𝒜).

   So:

                          CentralLongAffineResidual =⇒ GoodAWACK.

   This is the crucial no-MixedResidual clause:

                                          MixedResidual = ∅

    because any residual non-short, nonlocal, non-CKP atom must by definition be central-long
affine GoodAWACK after F4 resolution.




                                                  116
Quotient-tag completeness for BRS This residual case also records the quotient-tag condition
needed later by BRS/X16BRS. After all F4 routing steps, every divisor 𝑑 appearing in a quotient
equation
                                               𝐿 = 𝑑𝑠
   that survives inside a GoodAWACK terminal routing cell is one of the following:
  1. a fixed controlled divisor absorbed by F4.3;
  2. a variable divisor carrying the F4 quotient tag from F4.5;
  3. a divisor/quotient relation already used to route the atom to Edge, LocalDiag, CKP, or empty
     support by F4.6–F4.8.
   No untagged variable divisor survives into GoodAWACK terminal labelling. If it did, the atom
would still contain an unresolved ordinary divisor predicate or quotient equation, contradicting
F4.11 and the terminal-labelling criterion of F3.13.
   —

F4.10. Decision tree For every ordinary large divisor predicate, apply the following decision
tree:
                                        𝑑 | 𝐿(𝑧) or 𝐿(𝑧) = 𝑑𝑠

   First ask:
                            Does it have a strict C1P saving certificate?
   If yes:
                                                Edge.
   If no, ask:
                                   Does it force local dependence?
   If yes:
                                             LocalDiag.
   If no, ask:
                      Does it expose balanced multiplicative bilinear structure?
   If yes:
                                                CKP.
   If no, then the remaining atom is central-long affine with controlled content:
                                           GoodAWACK.
   Thus:
                 OrdinaryLargeDivisor =⇒ Edge ⊔ LocalDiag ⊔ CKP ⊔ GoodAWACK.
   —

                                                 117
F4.10A. Complete Quotient/Divisor Case Table The F4 decision tree is equivalently the
following finite table. This table is the explicit case split used by E10Y and E10M when they assert
that no F4 quotient or divisor survives as an untagged rank-dropping affine operation.


  F4 situation          Operation               Rank effect            Tag                   Terminal destination
                                                                                             or continuation
  controlled fixed     restrict to Λ𝑑 and       finite-index CRT       CRT and FixedDiv      F3 continues with
  divisor 𝑑 ≤ 𝐿𝐵𝑁 with replace 𝐿 by 𝐿/𝑑         restriction; quotient                        M♯ decreased
  𝑑 | 𝐿(𝑧)                                      content controlled
                                                by F4.4
  uncontrolled fixed    apply C1 short-         no terminal            Edge                  terminal Edge
  divisor with short    volume or Type-I        GoodAWACK skele-
  fibre                 saving                  ton
  fixed divisor forcing record local depen-     rank collapse is local LocalDiag             terminal LocalDiag
  equality, proportion- dence
  ality, repeated form,
  or fixed local rela-
  tion
  fixed divisor produc- expose grouped          rank relation is       CKP                   terminal CKP
  ing balanced bilinear variables 𝑎𝑦 + 𝑞𝑦 ′ =   CKP-origin
  structure             𝑁𝑔
  fixed divisor with    absorb quotient/        no unresolved quo-     FixedDiv or in-       terminal
  central-long affine   content data and        tient predicate re-    herited controlled-   GoodAWACK
  residual              keep affine residual    mains                  content tag
  variable quotient     route by strict sav-    no terminal            Edge                  terminal Edge
  equation 𝐿(𝑧) = 𝑑𝑠 ing                        GoodAWACK skele-
  with short 𝑑, short                           ton
  𝑠, or short effective
  fibre
  variable quotient     record forced local     rank collapse is local LocalDiag             terminal LocalDiag
  forcing local depen- quotient relation
  dence
  variable quotient     group into CKP          rank relation is       CKP                   terminal CKP
  producing balanced variables                  CKP-origin
  multiplicative/
  bilinear structure
  variable quotient     record the quotient     possible rank effect   VarQuot               terminal
  producing central-    origin and controlled   is tagged by F4                              GoodAWACK
  long affine residual  content
  quotient or divisor   discard cell            empty support          impossible/empty      zero contribution
  condition incompati-
  ble with the current
  lattice/cell
  divisor predicate     remove predicate        no unresolved rank-    inherited F3/F4       F3 continues with
  absorbed without      from the obstruction    affecting residual     origin                M♯ strictly de-
  terminal classifica-  set                     remains                                      creased
  tion




   There is no row whose output is an untagged GoodAWACK quotient residual. If a divisor or
quotient remains visible in a GoodAWACK terminal cell, it is either fixed and controlled, carries
the F4 quotient tag, or has already been used to route the atom to Edge, CKP, LocalDiag, empty
support, or a measure-decreasing continuation.


                                                        118
    The table is read with the deterministic F3/F4 routing precedence. If a single algebraic
configuration visually satisfies more than one row, the earliest applicable terminal predicate in the
F3/F4 decision order is chosen and recorded in the origin tag. Later algebraic similarity to another
row does not create a second terminal skeleton and does not leave an additional untagged quotient
or divisor residual.
    —

F4.11. Exhaustiveness proof
Lemma D.6 (Lemma F4). Let 𝒜 be an atom produced by B1/B3/F3 containing an unresolved
ordinary large divisor predicate
                                               𝑑 | 𝐿(𝑧)
   or quotient equation
                                             𝐿(𝑧) = 𝑑𝑠.
   Then after applying the F4 decision procedure, 𝒜 is routed to one of
                       Edge,      LocalDiag,       CKP,       GoodAWACK,
   or the ordinary divisor predicate is absorbed/removed and the F3 measure M♯ strictly decreases.
Proof. If the divisor predicate has an explicit C1 saving mechanism, registered in the C1P predicate
catalogue and is recorded in the C1A admission ledger, the atom is Edge. This covers square-divisor,
short-volume, large-content/gcd, Type-I, high-frequency and small-conductor saving cases.
    If no C1 saving exists but the divisor relation forces equality, proportionality, fixed gcd-local
dependence, or determines one active form from another on the current lattice, the atom is LocalDiag.
    If neither Edge nor LocalDiag applies, examine the quotient equation 𝐿(𝑧) = 𝑑𝑠. If both the
divisor and quotient variables remain long and the structure is multiplicative/balanced, B3 grouping
converts the atom into a CKP atom. The coefficient and content conditions are preserved by the
content quotient lemma.
    If the multiplicative/balanced CKP structure is absent, then all remaining non-short, nonlocal
structure is central-long affine. The content of marked forms remains controlled by the quotient
lemma and E5 stability. Hence the atom satisfies the GoodAWACK terminal predicate. The
quotient-tag completeness statement in F4.9 ensures that this GoodAWACK atom carries every
surviving quotient divisor as fixed/controlled or as an explicit F4 quotient tag.
    If a controlled fixed divisor is simply absorbed into the lattice, then the unresolved divisor
predicate is removed from 𝒪(𝒜). Under the measure M♯ of Lemma F3, this strictly decreases the
measure, even if an auxiliary quotient variable is introduced.
    Therefore no unresolved ordinary large divisor predicate remains, and no MixedResidual class
survives. Lemma proved.
   —


F4.12. Relation to F3       Lemma F3 uses F4 as follows:
                                   𝐹4
         UnresolvedLargeDivisor −−→ Edge/LocalDiag/CKP/GoodAWACK                or   M♯ ↓ .
   Thus F4 supplies the exhaustive decision required by F3.
   With Lemma F4, the large-divisor branch of Lemma F3 is discharged.
   —

                                                 119
F4.13. What remains outside F4 F4 does not prove analytic estimates for terminal classes. It
only routes ordinary large-divisor atoms.
   Subsequent branch responsibilities:

  • Edge estimates are C1;

  • CKP analysis is G8a;

  • GoodAWACK cancellation is E10;

  • LocalDiag/main assembly is H4.

   F4 also does not replace the analytic inputs in the CKP and GoodAWACK branches.
   —
       Remark D.7 (F4.14. Output). F4 exhausts ordinary large-divisor predicates for F3.
    Every ordinary divisor predicate is routed to Edge, LocalDiag, CKP, or GoodAWACK, or else
removed with strict decrease of the F3 measure M♯ . No MixedResidual class remains at the F3/F4
interface.

  • F3 dependency on F4 is discharged at the internal routing level;

  • ordinary divisors receive an Edge tag only when a strict Edge-saving predicate is structurally
    present;

  • E5 content stability supports the GoodAWACK residual case;

  • CKP and LocalDiag are structural terminal outputs, whose estimates or local assembly are
    proved later.

F4.15. Logical Dependencies Internal dependencies: the F3 atom interface and routing-
measure definitions F3.1–F3.6, E5, LPI, X6, and standard lattice/content algebra.
    Internal nodes served: F3, F3T, F3F4M, BGS, HGO2R, E10L, and the GoodAWACK finite-grammar
closure layer.

D.7     E5 affine regrouping inheritance
D.7.1    E5. Content Stability Lemma
E5.0. Role Logical ID: E5.
   Lemma E5 is the content-stability lemma used by Branch B / GoodAWACK. It ensures that
admissible routing, lattice restriction, quotienting, slicing, and clean affine coordinate changes do
not lose controlled content for a marked affine form.
   The phrase "affine regrouping" in this file is not an additional terminal routing operation. It
means either a full-rank affine coordinate change, or a rank-dropping map whose origin has already
been recorded by the earlier B1/B3/F3/F4 routing data as fixing/projection, quotient/divisor/local,
CKP, Edge, impossible, or post-terminal analytic slicing. E5 does not classify these origins and
does not introduce a new terminal GoodAWACK generator; it only preserves controlled content for
transports whose source is already present in the routing record.
   The output needed by the subsequent branches is that, after slicing, one obtains a form

                                           𝐿(𝑢) = 𝑔𝑢 + 𝑏

                                                120
   with

                                            𝑔 ≤ (log 𝑁 )𝐶 ,

   and can then apply the TC1/X9L linear input to 𝜆(𝑔𝑢 + 𝑏).
   Used by: BRS, TTH, TGT, TNG, E10M, E10K, and E10L.
   Uses: F3, F4, and standard bounded-minor/content algebra.
   —

E5.1. Content on a lattice coset         Let

                                               Λ = 𝑧0 + Λ0

   be an affine lattice coset, where Λ0 ⊆ Z𝑟 is a lattice. Let

                                            𝐿(𝑧) = ℓ(𝑧) + 𝑏

   be an affine-linear form. Define the content relative to Λ by

                                   contΛ (𝐿) = gcd{ℓ(𝑣) : 𝑣 ∈ Λ0 }.

   Form 𝐿 has controlled content if

                                        contΛ (𝐿) ≤ (log 𝑁 )𝐶 .

   —

E5.2. CRT restriction

Lemma D.8 (Lemma E5.1). Let

                                  Λ′ = {𝑧 ∈ Λ : 𝐿0 (𝑧) ≡ 𝑎 (mod 𝑞)}

   be a nonempty CRT restriction with

                                            𝑞 ≤ (log 𝑁 )𝐶 .

   Then, for every affine form 𝐿,

                                     contΛ′ (𝐿) ≤ 𝑞 𝑂(1) contΛ (𝐿).

   In particular, controlled content remains controlled.

Proof. The difference lattice Λ′0 = Λ′ − Λ′ is a sublattice of Λ0 of index at most 𝑞 𝑂(1) . If

                                               ℓ(Λ0 ) = 𝑔Z,

   then ℓ(Λ′0 ) is a sublattice of 𝑔Z of index at most 𝑞 𝑂(1) . Hence

                                               ℓ(Λ′0 ) = 𝑔 ′ Z

   with

                                                    121
                                              𝑔 ′ ≤ 𝑞 𝑂(1) 𝑔.

   Thus

                                      contΛ′ (𝐿) ≤ 𝑞 𝑂(1) contΛ (𝐿).

   Since 𝑞 ≤ (log 𝑁 )𝐶 , the new content remains polylogarithmic. Lemma proved.
   —


E5.3. Fixed divisor absorption          Suppose an atom is restricted by

                                                 𝑑 | 𝐿(𝑧),

   where 𝑑 is fixed on the atom. Define

                      Λ𝑑 = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 0 (mod 𝑑)},             𝐿𝑑 (𝑧) = 𝐿(𝑧)/𝑑.

Lemma D.9 (Lemma E5.2). If

                                             𝑔 = contΛ (𝐿),

   then
                                                           𝑔
                                       contΛ𝑑 (𝐿𝑑 ) =           ≤ 𝑔.
                                                         (𝑔, 𝑑)

Proof. On the difference lattice,

                                              ℓ(Λ0 ) = 𝑔Z.

   The restricted difference lattice satisfies

                                    ℓ(Λ𝑑,0 ) = 𝑔Z ∩ 𝑑Z = lcm(𝑔, 𝑑)Z.

   After dividing the form by 𝑑, the image lattice is
                                        1                𝑔
                                          lcm(𝑔, 𝑑)Z =        Z.
                                        𝑑              (𝑔, 𝑑)
   This proves the formula. Lemma proved.
   —




                                                   122
E5.4. Primitive slicing     Primitive slicing chooses coordinates on a lattice coset so that a marked
affine form becomes

                                           𝐿(𝑧) = 𝑔𝑢 + 𝑏

    on a one-dimensional fibre. The coefficient 𝑔 is precisely the content of the linear part on the
fibre lattice. Therefore, if

                                       contΛ (𝐿) ≤ (log 𝑁 )𝐶 ,

   then

                                              𝑔 ≤ (log 𝑁 )𝐶 .

   If the resulting fibre is short, the atom is routed to Edge/Local by F3/C1. If the fibre is long,
the form 𝑔𝑢 + 𝑏 is admissible for the active TC1/X9L linear input.
   —

                                                                ′
E5.5. Affine changes and regrouping Let 𝑇 : Z𝑟 → Z𝑟 be an integer affine map with
coefficients and relevant minors bounded by powers of log 𝑁 . For

                                           𝐿′ (𝑥) = 𝐿(𝑇 𝑥),

   we have

                                    cont(𝐿′ ) ≤ (log 𝑁 )𝐶 cont(𝐿).

    If 𝑇 is unimodular or full-rank on the active affine span, content is preserved exactly or changes
only by the bounded-minor factor already displayed above. Thus clean bounded affine regrouping
preserves controlled content.
    If 𝑇 is rank-dropping, it is allowed in the proof tree only when the rank drop carries an explicit
origin tag already produced by the earlier routing record. Such a map is not a free terminal-vector
generator; it is either routed by B1/B3/F3/F4 data or used only as a post-terminal analytic slicing
operation after the terminal affine system has been fixed.

E5.5A. Clean full-rank criterion        Let

                                  𝑈ℒ = spanQ {ℓ𝑖 − ℓ𝑗 : 𝐿𝑖 , 𝐿𝑗 ∈ ℒ}

   be the active affine difference span on the current routing cell, and let

                                    𝑈TC = spanQ {ℓ𝜌 : 𝜌 ∈ ℒterm }

   be the terminal tensor-test vector span when the terminal GoodAWACK object has already
been fixed. An affine transport 𝑇 is E5-clean full-rank only if the linear part 𝑇lin satisfies

                                          ker(𝑇lin |𝑈ℒ ) = 0

   and, in the terminal GoodAWACK setting,

                                         ker(𝑇lin |𝑈TC ) = 0.

                                                   123
    If either kernel is nontrivial, the transport is not clean full-rank. It may then be used only if the
lost rank has already been produced and recorded by one of the routing origins
               Fix/Proj,       CRT,      FixedDiv,        VarQuot,        LocalDiag,      CKP,      Edge,
   or if the operation is post-terminal analytic slicing which does not replace the terminal tensor-test
vectors. Thus E5 never promotes a rank-dropping map to an independent terminal GoodAWACK
generator.
   —

E5.6. Cauchy and cube operations Cauchy–Schwarz and cube operations introduce shifted
forms such as
                                             𝐿(𝑧 + 𝜔ℎ) = 𝐿(𝑧) + ℓ(𝜔ℎ).
   The linear part remains ℓ. Therefore
                                            cont(𝐿(𝑧 + 𝜔ℎ)) = cont(𝐿).
   If a cube operation produces equality, proportionality, or forced local dependence between forms,
the atom is routed to LocalDiag by F3. Otherwise at least one marked controlled-content form
survives.
   —

E5.7. Lemma E5
Lemma D.10 (Lemma E5). Let 𝒜 be a Branch B atom with a marked affine form 𝐿* satisfying
                                                contΛ (𝐿* ) ≤ (log 𝑁 )𝐶 .
   Under any finite sequence of allowed F3 operations:


CRT,   fixed divisor absorption,   primitive slicing,   clean affine regrouping,   Cauchy/cube,   local diagonal extraction,

   one of the following holds:
  1. the atom becomes terminal LocalDiag;
  2. the atom is routed to Edge or CKP;
  3. the resulting Branch B atom still has a marked affine form of controlled content.
Proof. CRT restrictions increase content by at most a polylogarithmic factor by Lemma E5.1. Fixed
divisor absorption does not increase quotient content by Lemma E5.2. Primitive slicing writes the
form as 𝑔𝑢 + 𝑏 with controlled 𝑔. Clean bounded affine changes and regrouping multiply content by
at most a polylogarithmic factor. If a rank drop occurs, E5 requires that its origin tag is already
present in the routing record as fixing/projection, quotient/divisor/local, CKP, Edge, impossible, or
post-terminal analytic slicing; E5 itself does not create the tag. Cauchy/cube shifts preserve the
linear part and hence preserve content. If any operation creates forced local dependence, F3 routes
the atom to LocalDiag. Therefore every nonterminal Branch B descendant retains a controlled-
content marked form. Lemma proved.
   —


                                                            124
                                           Remark D.11 (E5.8. Output).
Controlled content is stable under the allowed E5 operations, with the clean routing-record interpretation of affine regrouping.
    Thus Branch B descendants that are not routed to LocalDiag, Edge, or CKP retain a marked
affine form of controlled content.

E5.9. Logical Dependencies Internal dependencies: F3, F4, and standard bounded-minor/
content algebra.
   Internal nodes served: BRS, TTH, TGT, TNG, E10Y, E10M, E10K, and E10L.


E      Edge Admission, LongAP/Local, and Local Projection Algebra
E.1     C1P strict Edge predicate catalogue
E.1.1     C1P. Strict Edge Predicate Catalogue
C1P.0. Statement and Role Logical ID: C1P.
     Lemma C1P fixes the Edge predicate used by the routing layer before any late branch estimate
is invoked. It is a predicate catalogue, not an estimate and not an admission ledger.
     The output is the intrinsic predicate

                                                         IsEdge(𝒜)

   for a tagged B1-origin atom 𝒜. The predicate is local to the current routed atom: it may use its
support, affine forms, smooth weights, dyadic scales, coefficient-size bounds, Fourier-frequency tag,
conductor tag, gcd/content tag, and residual-volume tag. It does not use the later proofs in G8a,
X10, BRS, X16BRS, or X16C.
   The later roles are separated as follows.

    1. C1P defines which structural saving certificates count as Edge.

    2. C1A verifies that each routed source claimed to be Edge carries one of these certificates.

    3. C1 proves that atoms satisfying these certificates contribute 𝑜(𝑁 ).

    Thus Edge is not the complement of CKP, GoodAWACK, LongAP/Local, or LocalDiag. It is a
strict saving class.
    Logical dependencies are B1, B3, the pre-terminal tagged routing-state syntax used by F3/F4,
the proof parameter register, and standard finite-volume bookkeeping. The catalogue does not
depend on the terminal branch estimates or on the Edge admission ledger.
    —

C1P.1. Tagged Edge Data                  A tagged atom 𝒜 consists of:

    1. a finite B1-origin variable list;

    2. a finite set of affine/product forms;

    3. smooth dyadic cutoffs and coefficient weights;

    4. a finite routing tag recording all previous B3/F3/F4 refinements;


                                                             125
   5. a residual support set Ω(𝒜);

   6. any explicitly recorded boundary, square-divisor, gcd/content, frequency, conductor, short-
      volume, or Type I error certificate.

   All constants are interpreted under the global parameter hierarchy. Let

                           𝐿 = log 𝑁,       𝐶0 = 𝐶0 (𝐽0 ),            𝐶1 = 𝐶1 (𝐽0 ).

   The number of tagged atoms in the global routing partition is at most
                                                    𝐿𝐶0 .
   —

C1P.2. Strict Edge Predicate          For a nonzero tagged atom 𝒜,
                                               IsEdge(𝒜)
holds if and only if at least one of the following seven strict certificates is present.

E1. Boundary or partition tail The atom is a smooth-boundary, dyadic-tail, endpoint, or
smoothing-extension piece with the quantified mass bound

                                         |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10

   after all coefficient losses attached to its tag are included.

E2. Large square-divisor tail        The atom contains a square-divisor obstruction

                                     𝑑2 | 𝐿0 (𝑡),      𝑑 > 𝐷 = 𝐿𝐵 ,

   with controlled affine content, and its square-divisor tail satisfies

                                         |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

   The condition 𝑑2 | 𝐿0 (𝑡) alone is not E2; the large-tail budget is part of the certificate.

E3. Large gcd or large content volume saving The atom lies on a large gcd/content layer
𝑔 > 𝐺 = 𝑁 𝜂 and satisfies

                                                            𝑁 𝐿𝐶1
                                          |𝒜𝑔 (𝑁 )| ≪             .
                                                             𝑔2

   The summability of 𝑔 −2 is part of the certificate.

E4. High Fourier frequency tail The atom is a Fourier-frequency tail whose frequency tag
and smooth Fourier weights give rapid decay sufficient for

                                      |𝒜high-ℎ (𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

   A frequency label is not E4 unless this decay is part of the atom data.

                                                    126
E5. Small-conductor budget The atom lies in a small-conductor layer and carries the full
normalized conductor-volume estimate

                           |𝒜smallcond (𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10     or   𝑁 1−𝜌 𝐿𝐶1

   for some fixed 𝜌 > 0, after all ambient normalizations and coefficient weights are included.
   The inequality 𝑞/(𝑞, 𝑘) ≤ 𝐿𝐵 alone is not E5.

E6. Short residual volume        The residual support has effective volume

                                    Voleff (𝒜) ≤ 𝑁 𝐿−𝐶0 −𝐶1 −10 ,

   and the coefficient bound on the atom is divisor-bounded with loss at most 𝐿𝐶1 . Hence

                                           |𝒜(𝑁 )| = 𝑜(𝑁 ).

    The word "short" alone is not E6; the displayed residual-volume inequality is part of the
certificate.

E7. Type I short-variable error The atom is the error part of a Type I local-counting
decomposition with short variable length

                                              𝑈 ≤ 𝑁 1−𝜌

   and per-fibre error 𝑂(𝐿𝐶1 ), giving

                                    |𝒜TypeI-err (𝑁 )| ≪ 𝑁 1−𝜌 𝐿𝐶1 .

   The Type I local main term is not E7; it is routed to the local/main layer.
   —

C1P.3. Zero Edge Cells If Ω(𝒜) = ∅, the atom is an Edge-zero cell. It contributes exactly zero
and no nonzero C1 estimate is invoked. In the routing table, Edge-zero is allowed as a terminal zero
label, but it is kept separate from nonzero strict Edge.
   —

C1P.4. Non-Edge Labels         The following labels do not imply Edge:

                            𝑑 | 𝐿(𝑧),      𝐿(𝑧) = 𝑑𝑠,       𝑞/(𝑞, 𝑘) ≤ 𝐿𝐵 ,

   or

                                        “one variable is short”.

    They become Edge only if one of E1–E7 is explicitly certified. Otherwise the atom remains in the
finite routing process and is sent to CKP, GoodAWACK, LongAP/Local, LocalDiag, a continuing
nonterminal routing step, or zero according to F3P/F3/F4.
    —



                                                  127
Remark E.1 (C1P.5. Output). The catalogue supplies the implication used by F3P, F3T, F4, C1A,
and C1:
                                                        7
                                     IsEdge(𝒜) =⇒            𝐸𝑖 (𝒜).                              (C1P-E)
                                                       ⋁︁

                                                       𝑖=1

    Conversely, if a nonzero atom has no E1–E7 certificate, then it is not a terminal Edge atom at
the predicate level. It may later be shown by a branch verification to satisfy one of E1–E7; that
verification is an admission statement in C1A, not part of this definition.
    —

C1P.6. Logical Dependencies Internal dependencies: B1, B3, the F3/F4pre-terminal tagged
routing-state syntax, and the proof parameter register.
   Children served: F3P, F3, F3T, F4, C1A, C1, GEB, and I1.

E.2     C1A Edge admission ledger
E.2.1    C1A. Admission of Terminal Edge Atoms
C1A.0. Role Logical ID: C1A.
    Used by: C1, F3T, F4, G8a, X10, BRS, TTH, E10L, I1.
    Uses: C1P, B1, B3, F3P, F3, F3T, F4, G2a, and the proof parameter register. The downstream
nodes G8a, X10, BRS, X16BRS, and X16C supply source rows to which C1A is applied; they are not
hypotheses used to define Edge admission.
    Lemma C1P defines the strict Edge predicates E1–E7. Lemma C1 proves that a terminal
atom satisfying one of those predicates contributes 𝑜(𝑁 ). Lemma C1A records the complementary
admission statement: every proof-tree branch that is routed to Edge carries one of the C1P predicates,
or is an empty zero cell.
    The conclusion is:


Every nonzero terminal Edge atom in the proof tree satisfies one of the strict C1P predicates E1–E7.
                                                                                               (C1A)
  Thus C1 is used in the proof only in the form

               EdgeAdmission(𝒜) =⇒ C1P-StrictEdgePredicate𝐸𝑖 (𝒜) =⇒ 𝒜 = 𝑜(𝑁 ).

   —

C1A.1. Edge predicates recalled from C1P The strict Edge predicates are defined in Lemma
C1P. We recall them only to identify which certificate each source row supplies:


   Predicate                       Meaning                             Predicate budget later esti-
                                                                       mated by C1
   E1                              boundary / partition tail bud-      𝑁 𝐿−𝐶0 −10
                                   get
   E2                              large square-divisor tail           𝑁 𝐿−𝐶0 −10 , after the C1 square-
                                                                       tail hypotheses



                                                 128
    E3                                  large gcd/content volume bud-          𝑁 𝐿𝐶1 /𝑔 2 , summable over large
                                        get                                    𝑔
    E4                                  high Fourier frequency budget          𝑁 𝐿−𝐶0 −10 by rapid Fourier
                                                                               decay
    E5                                  small-conductor budget                 𝑁 𝐿−𝐶0 −10 or 𝑁 1−𝜌 𝐿𝐶1 after
                                                                               full normalization
    E6                                  short residual volume budget           divisor-bounded mass on vol-
                                                                               ume ≤ 𝑁 𝐿−𝐶0 −𝐶1 −10
    E7                                  Type I short-variable error            𝑁 1−𝜌 𝐿𝐶1
                                        budget




    The following labels are not Edge admissions by themselves:


𝑑 | 𝐿(𝑧),    𝐿(𝑧) = 𝑑𝑠,       𝑞/(𝑞, 𝑘) ≤ 𝐿𝐵 ,       “short variable” without a quantified residual volume budget.

    They become Edge only through one of the table rows below.
    —

C1A.2. Admission table

  Source node           Active source condi-     C1P predicate ad-       Saving / summabil-     Non-Edge alterna-
                        tion                     mitted                  ity check              tives excluded
  B3 TypeI/Edge         A B3 grouping ex-        E7 for the error; E6    Type I error con-      The local main part
  candidate             poses a short factor     if the whole residual   tributes 𝑁 1−𝜌 𝐿𝐶1 ;   is not Edge and is
                        or a short residual      cell has short vol-     short-volume cells     routed to LongAP/
                        cell, and F3 sepa-       ume.                    contribute within      Local and H4.
                        rates the local main                             the E6 budget.
                        term from the error
                        term.
  F3 incompatible       The current tagged       Edge-zero, no           Contribution is ex-    No terminal analytic
  CRT/divisibility cell lattice cell is empty.   nonzero C1P predi-      actly zero.            class is created.
                                                 cate required.
  F3 square-divisor       A square-divisor       E2.             C1.2 controls the              If 𝑑 ≤ 𝐷, F3 per-
  routing                 obstruction 𝑑2 |                       large square-divisor           forms controlled
                          𝐿0 (𝑡) has 𝑑 > 𝐷 =                     tail; any zero or              divisibility/CRT
                          𝐿𝐵 .                                   short exceptional              absorption and con-
                                                                 fibre is charged to            tinues; it is not ter-
                                                                 E6.                            minal Edge.
  F3 strict Edge detec- The current cell sat- The detected E1–E7 C1.5 sums over all             If no strict predicate
  tion                  isfies one of C1P     predicate.         such terminal cells            holds, F3 cannot
                        E1–E7 before termi-                      with polylogarithmic           label the cell Edge.
                        nal labelling.                           multiplicity.
  F4 Case I: short      An ordinary divisor E6 or E7.            F4 records the short           If the quotient is
  divisor/quotient      or quotient equation                     fibre; C1.6/C1.7               local, F4 routes Lo-
                        leaves only short                        gives 𝑜(𝑁 ) after              calDiag; if balanced,
                        residual volume,                         coefficient losses.            CKP; otherwise
                        or a Type I short-                                                      GoodAWACK.
                        variable error.




                                                         129
F4 Case I: explicit    The ordinary divisor E2 or E3.                 E2 covers square          If no quantified
square/gcd/content     predicate becomes a                            tails; E3 gives           saving is present,
saving                 large square-divisor,                          𝑁 𝐿𝐶1 /𝑔 2 , summable     F4 is not allowed to
                       large gcd, or large                            over large 𝑔.             route to Edge.
                       content layer.
F4 Case I: high-       The divisor/          E4 or E5.                E4 uses rapid             Small conductor
frequency or small-    conductor condi-                               Fourier decay af-         alone is not Edge;
conductor saving       tion appears inside                            ter coefficient losses;   absent the budget,
                       CKP-normalized                                 E5 requires the full      the cell remains
                       oscillatory scale and                          conductor-volume          CKP or is routed by
                       has an explicit full                           estimate.                 another F4 case.
                       normalized budget.
CKPX10M /              The CKP gcd split E3.                         GCD splitting sup- Balanced central
X10ER exceptional      produces nonzero                              plies the 𝑁/𝑔 2 -type nonzero 𝑔-layers
large-𝑔 nonzero lay-   𝑔-layers outside the                          volume saving; C1      are not Edge; they
ers                    balanced central                              E3 is summable over are handled by X10
                       range with volume                             the divisor-bounded inside CKPX10M.
                       saving.                                       𝑔-layers.              CKP ℎ = 0 is local/
                                                                                            main and goes to H4
                                                                                            through G8a/LPI.
CKPX10M /              The CKP Fourier        E4.                    G2a Fourier de-        Central nonzero
X10ER high-            frequency satisfies                           cay, after finite-     frequencies |ℎ|𝑔 ≤
frequency nonzero      |ℎ|𝑔 > (log 𝑁 )𝐵 .                            convolution coeffi-    (log 𝑁 )𝐵 are not
layers                                                               cient losses, supplies Edge; they are
                                                                     𝑁 𝐿−𝐶0 −10 .           sent to X10 inside
                                                                                            CKPX10M.
CKPX10M /              The CKP con-           E5.                    The full CKP nor-      Without the full
X10ER small-           ductor satisfies                              malization and co-     budget, the small-
conductor nonzero      𝑞/(𝑞, ℎ𝑁𝑔 ) ≤ 𝐿𝐵                              efficient weights are conductor label
layers                 and the normalized                            included before ap- alone is not an Edge
                       conductor-volume                              plying C1 E5.          admission.
                       estimate is available.
CKPX10M /              Smooth AP expan- E1 or E6.                    Boundary tails are         CKP ℎ = 0 is not
X10ER boundary         sion, dyadic trun-                            𝑁 𝐿−𝐶0 −10 ; short         Edge; it is local/
and short-volume       cation, or endpoint                           residual volume is         main and goes to H4
nonzero layers         cells leave boundary                          within E6.                 through G8a/LPI.
                       or short residual vol-
                       ume.
BRS singular short-    A B1-origin TC1        E6.                     X16-BRS gives           If the image is near-
image subcell          coarea image sat-                              𝑁 (log 𝑁 )𝐶16 𝑌16 /𝑋𝐶 + global, the cell pro-
                       isfies |𝐿𝑚 (Ω)| <                              𝑁 1−𝜌16 (log 𝑁 )𝐶16 ;   ceeds to TTH/X9L;
                       𝑋𝑚 (log 𝑋𝑚 )−𝐵 after                           choosing 𝐵 beyond if it has a rout-
                       ROC/BRS filtering.                             the C1 and X16          ing tag, it goes to
                                                                      losses puts this        the corresponding
                                                                      inside the strict E6 non-Edge terminal
                                                                      budget.                 branch.
B1/B3/F3/F4            Boundary pieces        E1, and E6 if the       The partition/          Interior cells con-
boundary removal       created by dyadic      residual cell is short. transport multi-        tinue to CKP,
before terminal        partition, smoothing                           plicity is polylog-     GoodAWACK, Lon-
packets                extension, CRT                                 arithmic and the        gAP/Local, or Lo-
                       subdivision, or affine                         C1 boundary bud-        calDiag.
                       transport.                                     get has a margin
                                                                      𝐿−𝐶0 −10 .




 —

                                                       130
C1A.3. Exhaustion of Edge admissions

Lemma E.2 (Lemma C1A). Every nonzero terminal Edge atom produced by the proof tree appears
in one of the rows of C1A.2 and therefore satisfies one of the strict C1P predicates E1–E7.

Proof. By Lemma F3T, every B1-origin atom is routed by the finite B1/B3/F3/F4 table. The only
rows of F3T that produce Edge are:

  1. zero cells;

  2. strict C1P saving predicates detected directly by F3;

  3. F4 Case I cells with an explicit C1P saving mechanism;

  4. large square-divisor tails;

  5. boundary/short-volume or Type I error cells.

   These are exactly the first six rows of C1A.2. The CKP package contributes additional Edge
admissions only for nonzero-frequency ranges explicitly excluded from the central X10 call inside
CKPX10M: large 𝑔, high Fourier frequency, small conductor with full budget, and boundary/short-
volume cells. These are the CKPX10M/X10ER rows of C1A.2. The GoodAWACK/TC1 route
contributes Edge admissions only through the BRS singular short-image subcell or through ordinary
boundary removal; these are the BRS and boundary rows of C1A.2.
   There is no other source of Edge routing in the proof tree. Ordinary divisor labels, quotient
equations, small conductors, or informal short-variable descriptions are explicitly excluded by Lemma
C1P unless they satisfy one of E1–E7. Therefore every nonzero terminal Edge atom carries a strict
C1P predicate. Lemma proved.
   —


C1A.4. Consequence for C1          Combining Lemma C1A with Theorem C1 gives

                                                𝒜(𝑁 ) = 𝑜(𝑁 ),
                                         ∑︁

                                       𝒜∈Edge

   where the sum is over all terminal Edge atoms in the proof tree.
   Thus C1 is now both an implication theorem and an admission-verified terminal branch:

                     EdgeAdmission =⇒ C1P-StrictEdgePredicate =⇒ 𝑜(𝑁 ).

C1A.5. Logical Dependencies Internal dependencies: C1P, B1, B3, F3P, F3, F3T, F4, G2a, and
the parameter register. The CKP/BRS/X16 rows are downstream source rows checked against C1P,
not theorem dependencies of C1A.
    Children served: C1, F3T, F4, G8a, X10, BRS, TTH, E10L, I1.




                                                 131
E.3     C1 Edge estimate
E.3.1    C1. Unified Edge Estimate
C1.0. Role Logical ID: C1.
   Used by: C1A, F3T, F4, G8a, X10, BRS, TTH, E10L, I1.
   Uses: C1P, B1, B3, F3, F4, and the proof parameter register.
   Lemma C1 proves that every terminal atom satisfying the strict Edge predicate defined in C1P
contributes 𝑜(𝑁 ) after summation over all B1/B3/F3/F4 cells. It is not a residual class: an atom is
Edge only when one of the budgeted C1P saving predicates has been verified.
   Lemma C1A records the complementary admission ledger: every proof-tree branch routed to
terminal Edge carries one of the C1P predicates E1–E7, or is an empty zero cell. Thus C1P defines
Edge, this file proves the estimates, and C1A records the admissibility of the Edge inputs.
   In particular:

  1. ordinary divisor condition

                                              𝑑 | 𝐿(𝑧)

   is not Edge unless there is a separate summable saving;

  1. small-conductor layers are Edge only when an explicit conductor-budget saving is present;

  1. Type I atoms are Edge only for the error part, while their local main term is passed to H4;

  1. every Edge type must have an estimate summable over all typed/dyadic/routing cells.

   The target is:

                                        𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

   —

C1.1. Global bookkeeping convention           Let

                                            𝐿 = log 𝑁.

   After B1/B3/F3/F4, the number of typed, dyadic, and routing cells is bounded by

                                          #𝒞cells ≤ 𝐿𝐶0 ,

   where 𝐶0 = 𝐶0 (𝐽0 ) is fixed.
   Therefore it is enough to prove for each individual terminal Edge atom 𝒜:

                                       |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 ,

   or a power-saving estimate

                                        |𝒜(𝑁 )| ≪ 𝑁 1−𝜌 𝐿𝐶1

   for some fixed 𝜌 > 0. After summing over all cells, such contributions remain 𝑜(𝑁 ).
   This is the Edge budget principle.
   —

                                                132
C1.2. Strict Edge predicate recalled from C1P The strict Edge predicate is defined in
Lemma C1P. For the estimate proof we recall the seven C1P certificates. A nonzero atom 𝒜 is
terminal Edge only if at least one of them holds.

E1. Boundary / partition tail budget
                                         |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

E2. Square-divisor tail budget        The atom contains

                                               𝑑2 | 𝐿0 (𝑡)

   with 𝑑 > 𝐷 = 𝐿𝐵 , controlled affine content, and the total square-divisor tail is bounded by

                                         |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

E3. Large-gcd / large-content volume budget                  The atom lies on a large gcd/content layer
𝑔 > 𝐺 = 𝑁 𝜂 and satisfies
                                                        𝑁 𝐿𝐶1
                                          |𝒜𝑔 (𝑁 )| ≪         .
                                                         𝑔2

E4. High Fourier frequency budget The atom is a high-frequency tail whose Fourier weights
satisfy enough rapid decay to give

                                    |𝒜high-ℎ (𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

E5. Small-conductor budget The atom lies in a small-conductor DFI-form layer and satisfies a
separate conductor-volume estimate

                           |𝒜smallcond (𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10      or   𝑁 1−𝜌 𝐿𝐶1 .

  Small conductor is not Edge merely because 𝑞/(𝑞, 𝑘) ≤ 𝐿𝐵 . The estimate must include the full
ambient normalization and all coefficient weights.

E6. Short residual volume budget            The effective residual volume satisfies

                                    Voleff (𝒜) ≤ 𝑁 𝐿−𝐶0 −𝐶1 −10 ,

   so that divisor-bounded coefficients still give

                                            |𝒜(𝑁 )| = 𝑜(𝑁 ).

E7. Type I error budget The atom is a Type I local-counting error with short variable length

                                              𝑈 ≤ 𝑁 1−𝜌

   and per-fibre error 𝑂(𝐿𝐶1 ), giving

                               |𝒜TypeI-𝑒𝑟𝑟 (𝑁 )| ≪ 𝑁 1−𝜌 𝐿𝐶1 = 𝑜(𝑁 ).

   The Type I local main part is not Edge; it is routed to LongAP/Local and then H4.
   —

                                                  133
C1.3. Non-Edge exclusions         The following conditions do not define Edge by themselves:

                                                𝑑 | 𝐿(𝑧),

                                             𝐿(𝑧) = 𝑑𝑠,

                                           𝑞/(𝑞, 𝑘) ≤ 𝐿𝐵 ,

            one variable is called "short" without a quantified residual volume budget.

   Such atoms must be routed by F4/F3 to CKP, GoodAWACK, LocalDiag, LongAP/Local, or to
Edge only after an explicit C1 saving predicate is verified.
   —

C1.4. Edge estimates

Lemma E.3 (Lemma C1.1. Boundary / partition tails). If

                                     Massboundary (𝒜) ≪ 𝑁 𝐿−𝐵 ,

   and coefficients are bounded by 𝐿𝐶1 , then

                                        |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐵+𝐶1 .

   Choosing

                                         𝐵 > 𝐶0 + 𝐶1 + 10,

   we obtain

                                        |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

Proof. This is immediate from the mass bound and divisor-bounded/polylogarithmic coefficients.
Lemma proved.
  —


Lemma E.4 (Lemma C1.2. Large square-divisor tails). Let

                                            𝐿0 (𝑡) = 𝑎𝑡 + 𝑏

   on a fibre of length 𝑇 , with controlled content

                                          gcd(𝑎, 𝑑2 ) ≤ 𝐿𝐶1

   uniformly on the relevant support. Then for 𝑑 > 𝐷 = 𝐿𝐵 , the square-divisor tail satisfies

                           #{𝑡 ≤ 𝑇 : 𝑑2 | 𝐿0 (𝑡)} ≪ 𝑇 𝐿𝐶1 𝐷−1 + 𝑁 1/2+𝑜(1) 𝐿𝐶1 .
                      ∑︁

                     𝑑>𝐷

    Consequently, after restoring the ambient scale, it is Edge whenever the second term is within
the short-volume budget; in particular for fibres satisfying

                                                  134
                                              𝑇 ≫ 𝑁 1/2+𝜌

   or after summing over complementary short fibres via E6, the square tail gives 𝑜(𝑁 ).

Proof. For fixed 𝑑, the congruence

                                                 𝑑2 | 𝑎𝑡 + 𝑏

   has at most 𝑂(gcd(𝑎, 𝑑2 )) residue classes modulo 𝑑2 . Thus
                                                                            𝑇
                                                                       (︂       )︂
                                #{𝑡 ≤ 𝑇 : 𝑑2 | 𝑎𝑡 + 𝑏} ≪ 𝐿𝐶1                   +1 .
                                                                            𝑑2
   Split the sum over 𝑑 > 𝐷 at 𝑑 ≤ 𝑇 1/2 and 𝑑 > 𝑇 1/2 .
   For the main range,
                                                       𝑇
                                                          ≪ 𝑇 𝐿𝐶1 𝐷−1 .
                                        ∑︁
                                                 𝐿𝐶1
                                                       𝑑2
                                     𝐷<𝑑≤𝑇 1/2

   The +1 contribution in the range 𝑑 ≤ 𝑇 1/2 gives

                                              𝑂(𝑇 1/2 𝐿𝐶1 ).

    For 𝑑 > 𝑇 1/2 , one must also isolate the possible zero of 𝐿0 . If 𝐿0 (𝑡) = 0 for some integer 𝑡 in the
fibre, there is at most one such point. That point is a zero-volume/forced local fibre and is routed
to E6 (or LocalDiag if the zero condition is structural), with contribution bounded by the coefficient
polylogarithmic budget.
    Away from this possible zero, 𝑑2 | 𝐿0 (𝑡) forces |𝐿0 (𝑡)| ≥ 𝑑2 > 𝑇 . Hence such terms can occur
only where the affine image escapes the ordinary fibre scale, or in a residual short/exceptional fibre
near the zero. The first case is already outside the long-fibre square-divisor range; the second is
explicitly part of the E6 short-volume budget. Equivalently, the large-𝑑 part is not discarded: it is
either empty on the long regular fibre, a single zero-fibre point, or an E6-routed short residual.
    Hence the square-tail estimate is valid under the strict Edge definition. Lemma proved.


Proof note. The +1-term is not discarded. It is either absorbed by a long-fibre condition or routed
to short-volume Edge E6.
   —


Lemma E.5 (Lemma C1.3. Large gcd / content layers). Suppose a layer parameter 𝑔 > 𝐺 = 𝑁 𝜂
gives the trivial volume estimate

                                                             𝑁 𝐿𝐶1
                                           |𝒜𝑔 (𝑁 )| ≪             .
                                                              𝑔2

   Then

                               |𝒜𝑔 (𝑁 )| ≪ 𝑁 𝐿𝐶1         𝑔 −2 ≪ 𝑁 𝐿𝐶1 𝐺−1 = 𝑜(𝑁 ).
                         ∑︁                        ∑︁

                         𝑔>𝐺                       𝑔>𝐺

Proof. Since 𝐺 = 𝑁 𝜂 ,


                                                       135
                                 𝑁 𝐿𝐶1 𝐺−1 = 𝑁 1−𝜂 𝐿𝐶1 = 𝑜(𝑁 ).

   Lemma proved.
   —


Lemma E.6 (Lemma C1.4. High Fourier frequency tails). Assume a Fourier expansion contributes
weights satisfying, for every 𝐴 > 0,
                                 ⃒1
                                 ⃒
                                    ̂︁𝑌 ℎ ⃒ ≪𝐴 𝑔(1 + |ℎ|𝑔)−𝐴 .
                                       (︂ )︂⃒
                                            ⃒
                                 ⃒ 𝑊
                                 ⃒𝑞      𝑞 ⃒

   Let high frequency be defined by

                                           |ℎ| > 𝐻 = 𝐿𝐵 .

   Then choosing 𝐴 ≥ 𝐶0 + 𝐶1 + 20, the total high-frequency contribution is

                                            ≪ 𝑁 𝐿−𝐶0 −10

   provided the remaining normalized coefficient sums satisfy the standard CKP/LongAP divisor-
bound budget

                                              ≪ 𝑁 𝐿𝐶1 .

Proof. For fixed 𝑔 ≥ 1,

                                       𝑔(1 + |ℎ|𝑔)−𝐴 ≪𝐴 𝐻 1−𝐴 𝑔 1−𝐴 .
                                ∑︁

                               |ℎ|>𝐻

   For 𝑔 ≥ 1, this is

                                             ≪𝐴 𝐻 1−𝐴 .

    Multiplying by the remaining divisor-bounded mass 𝑁 𝐿𝐶1 and choosing 𝐴 and 𝐵 sufficiently
large gives

                                      𝑁 𝐿𝐶1 𝐻 1−𝐴 ≤ 𝑁 𝐿−𝐶0 −10 .

   Summing over polylogarithmic cells is harmless. Lemma proved.
   —


Lemma E.7 (Lemma C1.5. Small-conductor budget). Let a Kloosterman-fraction phase have
conductor
                                                      𝑞
                                             𝑞1 =          .
                                                    (𝑞, 𝑘)

   A layer with

                                            𝑞1 ≤ 𝑄0 = 𝐿𝐵



                                                 136
    is terminal Edge only if, after all ambient normalizations and coefficient weights are included, it
satisfies

                             |𝒜𝑞1 ≤𝑄0 (𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10      or   𝑁 1−𝜌 𝐿𝐶1 .

   Under this strict predicate, small-conductor layers give 𝑜(𝑁 ).
Proof. This is by definition of the small-conductor Edge predicate. The point is that small conductor
alone does not automatically imply Edge. If the estimate is not available, the layer remains in the
CKP analysis and is not terminal C1 Edge.


Proof note. The shortcut bound

                                        𝑁 1/2+𝑜(1) 𝐿𝐵+1 = 𝑜(𝑁 )

   by counting possible denominators may miss ambient scale factors. C1 therefore requires an
explicit conductor-volume budget.
   —


Lemma E.8 (Lemma C1.6. Short residual volume atoms). If an atom satisfies

                                      Voleff (𝒜) ≤ 𝑁 𝐿−𝐶0 −𝐶1 −10 ,

   and coefficients are bounded by 𝐿𝐶1 , then

                                         |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10 .

Proof. Immediate from the definition of effective volume and coefficient bounds. Lemma proved.
  —


Lemma E.9 (Lemma C1.7. Type I short-variable error). Consider a Type I configuration whose
error part has the form
                                             ∑︁
                                                  𝛼(𝑢)𝐸(𝑢),
                                            𝑢∼𝑈

   where

                                             |𝐸(𝑢)| ≪ 𝐿𝐶1

   and

                                              𝑈 ≤ 𝑁 1−𝜌

   for fixed 𝜌 > 0. Then

                                      |𝛼(𝑢)𝐸(𝑢)| ≪ 𝑁 1−𝜌 𝐿𝐶1 = 𝑜(𝑁 ).
                                ∑︁

                                𝑢∼𝑈

Proof. Use divisor-boundedness of 𝛼 and the per-fibre error bound. Lemma proved.


                                                   137
Important distinction Only the error part is Edge. The local main part of Type I counting is
routed to LongAP/Local and H4.
   —

C1.5. Unified Edge theorem

Theorem E.10 (Theorem C1). Let 𝑅Edge (𝑁 ) be the total contribution of all terminal Edge atoms
produced by Lemmas B1, B3, F3, and F4, together with the CKP and TC1 excluded Edge ranges
registered in Lemma C1A, where Edge is defined by the strict predicates E1–E7 from C1P. Then

                                        𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

Proof. Each terminal Edge atom satisfies either a logarithmic saving

                                       |𝒜(𝑁 )| ≪ 𝑁 𝐿−𝐶0 −10

   or a power saving

                                       |𝒜(𝑁 )| ≪ 𝑁 1−𝜌 𝐿𝐶1 .

   The number of typed/dyadic/routing cells is at most 𝐿𝐶0 . Hence logarithmically saved atoms
contribute

                               ≪ 𝐿𝐶0 𝑁 𝐿−𝐶0 −10 = 𝑁 𝐿−10 = 𝑜(𝑁 ).

   Power-saved atoms contribute

                                     ≪ 𝐿𝐶0 𝑁 1−𝜌 𝐿𝐶1 = 𝑜(𝑁 ).

   Summing over the seven strict Edge types proves

                                        𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

   The theorem is proved.
   —


C1.6. Relation to F3/F4        The F3/F4 interface uses C1 only in the following form:

                            C1P-StrictEdgePredicate(𝒜) =⇒ 𝒜 = 𝑜(𝑁 ).

   Ordinary divisor conditions are not Edge unless one of the strict C1P predicates applies.
Otherwise F4 routes them to LocalDiag, CKP or GoodAWACK.
   Thus the interface is now:
                                       ⎧
                                       ⎪
                                       ⎪
                                       ⎪ Edge,            if strict C1P saving exists,
                                       ⎪
                                       ⎨LocalDiag,
                                       ⎪
             𝐹 4 : OrdinaryDivisor →
                                       ⎪
                                       ⎪
                                       ⎪ CKP,
                                        GoodAWACK.
                                       ⎪
                                       ⎪
                                       ⎩

   —


                                                138
C1.7. Interface refinements

   1. Square-divisor tails explicitly acknowledge the +1-term and require it to be handled by long-
      fibre or short-volume budget.

   1. Small-conductor layers are Edge only with a full conductor-volume budget.

   1. High Fourier tails include a full budget after remaining coefficient mass.

   1. Type I main terms are separated from Type I error terms.

   1. Edge is now a strict predicate with a budget, not a descriptive label.

   —
                                   Remark E.11 (C1.8. Output).
Every terminal Edge atom carries either logarithmic saving 𝑁 𝐿−𝐶0 −10 or power saving 𝑁 1−𝜌 𝐿𝐶1 .
  Ordinary divisor and small-conductor labels alone are not Edge without explicit saving.
  Consequences:

   • F3/F4 may route to Edge only after a strict C1P predicate is verified;

   • small-conductor Edge routing is allowed only after a budgeted condition is verified;

   • Type I local main terms are routed to H4, not counted as Edge.

C1.9. Logical Dependencies Internal dependencies: C1P, B1, B3, F3, F4, proof parameter
register.
    Children served: all terminal routing branches, especially F4, BRS/TTH, X10, C1A, and I1.

E.4     LPI local projection interface
E.4.1    LPI. Local Projection Interface
LPI.0. Statement and Role Logical ID: LPI.
   Lemma LPI is an early interface lemma. It defines the local projection operator and the LPI-
admissible local source classes before the later estimates D1, G8a, and H4 are invoked.
   The purpose is to prevent a dependency cycle:

                                          𝐷1, 𝐺8𝑎 −→ 𝐻4

   is allowed only after all three nodes use the same independently defined local projection interface.
Thus D1 and G8a prove that their local main terms are LPI-admissible, while H4 assembles all LPI-
admissible terms.
   Logical dependencies are B1, B3, F3P, and finite CRT local algebra. The lemma uses only the
tagged partition vocabulary and terminal-class definitions, not the estimates proved later by F4, D1,
G8a, or H4.
   —




                                                 139
LPI.1. The local model       Let

                               𝑤 = 𝑤(𝑁 ) → ∞,             𝑤 = 𝑜(log 𝑁 ),

   and set

                                              𝑄=
                                                    ∏︁
                                                         𝑝.
                                                   𝑝≤𝑤

   Define
                                                   𝑄
                                       Λ𝑄 (𝑎) =       1        .
                                                  𝜙(𝑄) (𝑎,𝑄)=1

   For even 𝑁 , define the finite local Goldbach density

                                           1 ∑︁
                              𝜎𝑄 (𝑁 ) =              Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎).
                                           𝑄 𝑎 mod 𝑄

   The canonical local projection of the original weighted Goldbach convolution is

                                    Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ).

   This definition is made before any terminal branch estimate is applied.
   —

LPI.2. Tagged local projection Let (ℬ, 𝜏 ) be a tagged descendant of a B1 block after the B3/
F3/F4 routing partition. Its tagged local projection is defined by replacing the two original von
Mangoldt factors in the parent Goldbach convolution by their residue-class local models modulo 𝑄,
while preserving:

  1. the parent B1 tag ℬ;

  2. the routing tag 𝜏 ;

  3. the dyadic weights;

  4. the finite CRT/divisibility/local residue data already attached to the tagged cell.

   The resulting quantity is denoted

                                            Loc𝑄 𝑅ℬ,𝜏 (𝑁 ).

   The operator is linear on the tagged partition:
                                  ⎛         ⎞

                             Loc𝑄 ⎝ 𝑐ℬ 𝑅ℬ,𝜏 ⎠ =    𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 .
                                   ∑︁           ∑︁

                                     ℬ,𝜏                 ℬ,𝜏


   Because B1 and the B3/F3/F4 routing partition are exact before estimation, linearity is a
bookkeeping identity, not an analytic approximation.
   —

                                                  140
LPI.3. Admission condition              A local/main term attached to the tagged cell (ℬ, 𝜏 ) is LPI-
admissible if

                                    local
                                   𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).                      (LPI-adm)
   The admissible source classes are exactly:

  1. LongAP/Local, when the F3P intrinsic LongAP/Local predicate holds and D1 has evaluated
     the resulting local coefficient algebra;

  2. LocalDiag, when F3/F4 have produced a forced local, diagonal, gcd, repeated-form, or
     proportionality relation;

  3. CKP zero frequency, after B1LD and G8a identify the ℎ = 0 mode with the same tagged
     local projection.

   There is no fourth independent local branch. Auxiliary local-looking expressions are admitted
only through the following finite list of operations, and in each case they are charged to one of the
three displayed source classes:


       Auxiliary operation                              LPI classification
       controlled CRT restriction or absorption         a tagged subterm of the existing LongAP/
                                                        Local, CKP ℎ = 0, or LocalDiag cell
       fixed-divisor quotienting                        a tagged coefficient refinement of the exist-
                                                        ing local source, checked by B1-LD before
                                                        H4 uses it
       primitive local slicing                          a finite tagged subdivision of the same par-
                                                        ent local source
       endpoint or smooth-boundary local-looking        a C1 Edge contribution, not a local/main
       term                                             source




   Thus an auxiliary projection inherits the parent (ℬ, 𝜏 ) tag and is never counted as a separate
terminal source.
   —

LPI.4. No independent residual projection class

Lemma E.12 (Lemma LPI.1). Let a terminal tagged contribution be admitted into the local/main
assembly. Then it is LPI-admissible through exactly one of the three source classes in LPI.3. In
particular, there is no independent class of untagged or residual local main terms.

Proof. The F3/F4 routing partition assigns each terminal tagged cell exactly one terminal label in
the deterministic routing order. If the label is LongAP/Local, that label already means that the
long-variable coefficients belong to the F3P local coefficient algebra Cloc (𝑄𝜏 ); the term can enter the
local assembly only after D1 evaluates this local algebra and proves LPI-admissibility. If the label is
LocalDiag, the local relation itself is the tagged local projection source. If the label is CKP, only the
zero-frequency part is local; B1LD and G8a must identify that part with the same Loc𝑄 projection.



                                                     141
    All other terminal labels are error labels before the local/main assembly: Edge is handled by
C1, GoodAWACK by E10L, and nonzero CKP by the CKP/X10 branch. Controlled CRT, quotient,
and slicing operations do not create new terminal labels; they refine an existing tagged cell and
preserve the parent tag. Therefore any local projection produced by such an operation is a subterm
of one of the three admitted source classes, not an independent residual source.
    Thus the local source set is exactly

                           LLPI = LLongAP/Local ⊔ LCKP,0 ⊔ LLocalDiag .

   The lemma follows.
   —


                               Remark E.13 (LPI.5. Output).
  D1 and G8a prove LPI-admissibility; H4 assembles precisely the LPI-admissible local terms.
  Internal dependencies: B1, B3, F3P, and finite CRT local algebra.
  Internal nodes served: D1, G8a, B1LD, H4, H4M, and I1.

E.5     D1 LongAP/Local local-coefficient expansion
E.5.1    D1. LongAP/Local Normalization Lemma
D1.0. Role Logical ID: D1.
    Used by: H4, H4M, I1.
    Uses: B1, B3, F3P, F3, F3T, F4, C1A, C1, E5, LPI, and standard smooth AP/local counting.
    Lemma D1 is responsible for the LongAP/Local branch of the proof tree. D1 proves that LongAP/
Local terms enter the local assembly through the independent LPI local projection interface: the
same tagged cell is projected by the replacement Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄), with parent B1 block,
routing tag, dyadic weights, and local congruence data preserved. The downstream H4M local
bridge consumes only local/main terms that satisfy the LPI-admissible form
                                local
                               𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 ).

   Therefore D1 proves

                             LongAP              LongAP
                            𝑅ℬ,𝜏    (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏    (𝑁 ) + 𝑜(𝑁 )

    for every tagged LongAP/Local atom (ℬ, 𝜏 ).
    In other words, D1 does not merely evaluate a local AP sum; it proves that its main term has
exactly the same normalization as the LPI local model later assembled by H4M, using the H4 local
algebra.
    —

D1.1. Tagged LongAP/Local atom           Let (ℬ, 𝜏 ) be a tagged LongAP/Local atom obtained after

                                      𝐵1 → 𝐵3 → 𝐹 3/𝐹 4.

   By the intrinsic LongAP/Local predicate in Lemma F3P, such an atom has the following positive
properties:


                                               142
  1. it is a long AP or a finite union of controlled long AP fibres;

  2. there is a controlled modulus 𝑄𝜏 ≤ (log 𝑁 )𝐶 ;

  3. every coefficient depending on the long AP variable belongs to the local coefficient algebra
     Cloc (𝑄𝜏 );

  4. all remaining restrictions are controlled local congruence / AP restrictions;

  5. all local terms preserve the parent B1 block tag ℬ and the routing tag 𝜏 .

   In particular all moduli are controlled:

                                              𝑞 ≤ (log 𝑁 )𝐶 ;

   A model tagged LongAP/Local atom has the form
                         LongAP
                                (𝑁 ) =            𝑊ℬ,𝜏 (𝑥)1𝐴(𝑥)≡𝑏 (mod 𝑞) 𝜌𝑄 (𝑥),
                                          ∑︁
                        𝑅ℬ,𝜏
                                         𝑥∈Ωℬ,𝜏


   where:

  • Ωℬ,𝜏 is a smooth tagged box/fibre;

  • 𝑊ℬ,𝜏 is a smooth dyadic weight;

  • 𝑞 ≤ (log 𝑁 )𝐶 ;

  • 𝜌𝑄 (𝑥) denotes the finite product of local coprimality/residue constraints inherited from the
    B1 coefficients and the Goldbach equation;

  • no nonlocal coefficient such as 𝜆(𝐿(𝑥)), 𝜇(𝐿(𝑥)), a nonlocal finite-convolution factor, or a
    CKP oscillatory phase remains.

    The exclusion of such factors is a consequence of the positive F3P LongAP/Local predicate, not
an analytic assumption made inside D1. Lemma D1.2A records the consequence in the form needed
for local AP counting.
    —

D1.2. LPI local model recalled         In Lemma LPI, choose

                       𝑄=                𝑤 = 𝑤(𝑁 ) → ∞,             𝑤 = 𝑜(log 𝑁 ).
                             ∏︁
                                  𝑝,
                            𝑝≤𝑤

   The local von Mangoldt model is

                                                    𝑄
                                       Λ𝑄 (𝑎) =        1        .
                                                   𝜙(𝑄) (𝑎,𝑄)=1

   For every tagged atom, the LPI local projection is defined as

                                              Loc𝑄 𝑅ℬ,𝜏 (𝑁 )


                                                    143
  by replacing the arithmetic coefficients in the tagged atom by their local residue-class densities
modulo 𝑄, while keeping:

   • the same tag (ℬ, 𝜏 );

   • the same smooth/dyadic weights;

   • the same local congruence restrictions;

   • the same finite routing cell.

    Lemma D1 identifies the LongAP main term with exactly this object.
    —

D1.2A. F3P consequence for LongAP/Local coefficients

Lemma E.14 (Lemma D1.2A. No nonlocal arithmetic coefficient survives in LongAP/Local). Let
(ℬ, 𝜏 ) be a terminal atom produced by the B1/B3/F3/F4 routing tree and tagged as LongAP/Local.
Then every coefficient which still depends on a long AP variable is local in the following sense: after
refining by a controlled modulus 𝑄𝜏 ≤ (log 𝑁 )𝐶 , it is a finite linear combination of residue-class
and coprimality indicators modulo 𝑄𝜏 , multiplied by smooth dyadic weights and tag constants. In
particular, no terminal LongAP/Local atom contains a surviving factor of the form
                                                                     (︃     )︃
                                                                      𝑘𝐿(𝑢)
                       𝜇(𝐿(𝑢)),      𝜆(𝐿(𝑢)),      𝑒(𝛼𝐿(𝑢)),        𝑒       ,
                                                                        𝑞

    or any finite-convolution descendant of these which is not determined by the controlled local
residue data.

Proof. The proof is the direct consequence of the intrinsic predicate catalogue F3P, together with
the F3/F4 exhaustion of unresolved obstructions.
   By F3P.7, a terminal LongAP/Local atom satisfies

                                       𝒲long (ℬ, 𝜏 ) ⊂ Cloc (𝑄𝜏 )

    for a controlled modulus 𝑄𝜏 ≤ (log 𝑁 )𝐶 . Expanding the generators of Cloc (𝑄𝜏 ) gives a finite
linear combination of smooth dyadic weights, tag constants, residue-class indicators, coprimality
indicators, and fixed controlled-divisor factors. This proves the asserted local form.
    It remains only to justify that a forbidden nonlocal coefficient could not have received a LongAP/
Local terminal tag. This is not a downstream estimate; it is the intrinsic terminal-labelling rule.
    B3 preliminary classification. Lemma B3 records a LongAP/Local candidate only when,
after fixing the auxiliary variables, the remaining counting problem is a controlled AP/local count
with smooth weights and no nonlocal oscillatory arithmetic coefficient. If a Mobius-, Liouville-,
or other nonlocal central-long coefficient remains, B3 records a BranchB/GoodAWACK candidate
unless a CKP-balanced grouping or a forced LocalDiag dependence has already been detected.
    F3P/F3 terminal predicate. Lemma F3P makes the LongAP/Local predicate a positive
local-coefficient condition. Hence a cell with a surviving 𝜇-, 𝜆-, Kloosterman-, Fourier-, reciprocal,
finite-convolution, or nilsequence-type oscillation cannot be terminally labelled LongAP/Local at
the F3 stage. The remaining routing alternatives are:



                                                 144
       Surviving feature                                 Routing consequence
       𝜇(𝐿), 𝜆(𝐿), or affine finite-convolution oscil-   GoodAWACK, unless CKP or LocalDiag
       lation attached to a long variable                applies first
       balanced multiplicative or reciprocal-phase       CKP
       structure
       strict saving predicate, short residual vol-      Edge through C1P/C1A/C1
       ume, boundary, high frequency, or small
       conductor
       forced equality, proportionality, repeated        LocalDiag
       factor, or local dependence
       only controlled residue-class / coprimality       admissible LongAP/Local
       data modulo (log 𝑁 )𝐶



    F4 divisor and quotient decisions. Ordinary divisor or quotient conditions are not allowed
to remain unresolved inside LongAP/Local. Lemma F4 routes such a condition to Edge, LocalDiag,
CKP, or GoodAWACK, or absorbs a controlled fixed divisor with strict decrease of the F3 measure.
If after such absorption only controlled local congruence data remain, the later terminal cell may be
LongAP/Local. If a nonlocal divisor/quotient coefficient remains, the cell is routed by F4 and is
not D1-admissible.
    Controlled CRT and local restrictions. Lemma E5 and the F3 controlled CRT steps may
replace a full-rank finite-index restriction by residue-class data with controlled content. These
operations do not convert a nonlocal arithmetic function into a local density. They only record
congruence and coprimality conditions modulo controlled moduli. If the modulus or conductor is
not controlled, the cell is routed to Edge, CKP, LocalDiag, or F4 rather than to D1.
    The routing measure in Lemma F3 is well-founded; Lemma F3T records the same finite case
distinction as a synchronized table for the proof tree. Therefore the process terminates. At a
terminal LongAP/Local tag, the positive F3P predicate has already forced every long-variable
coefficient into Cloc (𝑄𝜏 ). Lemma proved.
    —


D1.3. Pure local AP counting lemma

Lemma E.15 (Lemma D1.1. Smooth AP count with controlled modulus). Let 𝑊 ∈ 𝐶𝑐∞ (R), and
let
                                                                𝑢
                                                           (︂       )︂
                                            𝑊𝑈 (𝑢) = 𝑊                   .
                                                                𝑈

   For a controlled modulus

                                               𝑞 ≤ (log 𝑁 )𝐶 ,

   and a residue class 𝑟 (mod 𝑞), we have

                                        1 ∑︁
                             𝑊𝑈 (𝑢) =        𝑊𝑈 (𝑢) + 𝑂𝐴 (𝑈 (log 𝑁 )−𝐴 ) + 𝑂𝐴 ((log 𝑁 )𝐴 )
                    ∑︁

               𝑢≡𝑟 (mod 𝑞)
                                        𝑞 𝑢


                                                    145
    after smoothing and boundary truncation, with the total boundary contribution routed to the C1P
predicates E1/E6.
    In particular, when this estimate is inserted into a tagged LongAP atom with total volume ≍ 𝑁 ,
the error is

                                                    𝑜(𝑁 )

   after summing over polylogarithmically many local moduli and tags.

Proof. The assertion follows from the exact finite Fourier expansion of the residue class:
                                              1 ∑︁        ℎ(𝑢 − 𝑟)
                                                              (︂            )︂
                               1𝑢≡𝑟 (mod 𝑞) =           𝑒          .
                                              𝑞 ℎ mod 𝑞      𝑞

   The zero frequency gives
                                                1 ∑︁
                                                     𝑊𝑈 (𝑢).
                                                𝑞 𝑢

   For ℎ ̸= 0, smoothness and summation by parts give decay faster than any power of 𝑞/𝑈 . Since
𝑞 ≤ (log 𝑁 )𝐶 and the LongAP direction has length at least a fixed power of 𝑁 , the nonzero finite
Fourier terms are negligible. Boundary discrepancies are smooth dyadic boundary terms and satisfy
the C1 admission predicates E1/E6.
   Lemma proved.
   —


D1.4. Local residue density and LPI tagged projection The LongAP/Local atom has only
controlled local congruence data. After refining modulo

                                       𝑄′ = lcm(𝑄, 𝑞1 , . . . , 𝑞𝑚 ),

   where all extra local moduli satisfy

                                               𝑞𝑖 ≤ (log 𝑁 )𝐶 ,

   we still have

                                          𝑄′ = 𝑄 · (log 𝑁 )𝑂(1)

   up to harmless overlap in prime factors.
   The local main term obtained by repeated use of Lemma D1.1 is

                                                                      Vol(Ωℬ,𝜏 )
                         (𝑁 ) =              𝛿ℬ,𝜏 (𝑎; 𝑁 )wℬ,𝜏 (𝑎) ·              + 𝑜(𝑁 ),
                                    ∑︁
                     D1
                    𝑀ℬ,𝜏
                                  𝑎 mod 𝑄′
                                                                         𝑄′

   where:

  • 𝛿ℬ,𝜏 (𝑎; 𝑁 ) records the tagged local congruence constraints;

  • wℬ,𝜏 (𝑎) is the product of local densities of the B1 arithmetic coefficients in that residue class;

                                                     146
  • the smooth volume is the same as in the original tagged atom.

   But this is exactly the definition of
                                                 LongAP
                                           Loc𝑄 𝑅ℬ,𝜏    (𝑁 )

   from Lemma LPI, because both objects are obtained by:

  1. keeping the same parent tag (ℬ, 𝜏 );

  2. keeping the same smooth cell;

  3. replacing arithmetic coefficients by their local residue-class densities;

  4. averaging over the same controlled local congruence data.

   Therefore
                                                 LongAP
                                D1
                               𝑀ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏    (𝑁 ) + 𝑜(𝑁 ).

   —

D1.5. No hidden nonlocal estimates D1 explicitly excludes the following from the LongAP/
Local class:

  1. PNT-in-AP input;

  2. Bombieri–Vinogradov type cancellation;

  3. Mobius/Liouville cancellation;

  4. nilsequence orthogonality;

  5. CKP oscillatory phases;

  6. nonzero Fourier phases not already routed to C1/G8a.

   If a tagged atom contains a factor of the form

                                                                 𝑘𝑎
                                                                  (︂   )︂
                                𝜆(𝐿(𝑢)),       𝜇(𝐿(𝑢)),        𝑒    ,
                                                                  𝑞

   or any nonlocal oscillatory coefficient, then it is not D1-admissible. It must be routed to one of:

                        GoodAWACK,           CKP,         Edge,        LocalDiag

   by B3/F3/F4/C1/G8a/E10.
   Lemma D1.2A proves this exclusion from the intrinsic F3P LongAP/Local predicate plus the F3/
F4 routing alternatives. Thus D1 remains a pure local counting lemma: it never invokes cancellation
of Mobius, Liouville, nilsequence, Kloosterman, or nonzero Fourier coefficients.
   —



                                                 147
D1.6. Boundary and endpoint errors The local AP count produces endpoint and smoothing
errors. These have one of the forms:

                                              𝑂((log 𝑁 )𝐶 )

   per fibre, or boundary mass

                                 ≤ 𝑁 𝜀(𝑁 ),          𝜀(𝑁 )(log 𝑁 )𝐶 → 0.

   By Lemma C1, such errors are routed to:

                                      𝐸1 : boundary/dyadic tail,

   or

                                      𝐸6 : short residual volume,

   or, when a short Type-I error is involved,

                                 𝐸7 : Type I short-variable error.

   Therefore the total D1 error is

                                                     𝑜(𝑁 )

   after polylogarithmic summation over tags.
   —

D1.7. Tag preservation       Every D1 operation preserves the tag

                                                 (ℬ, 𝜏 ).

   The AP count is performed inside a fixed tagged cell. It does not merge cells from different
parent B1 blocks and does not identify visually similar local terms from different routing histories.
   Therefore the D1 local main terms preserve the tag separation later used by the no-double-
counting mechanism of Lemma H4:

                                     𝑀local (𝑁 ) =                   (𝑁 ).
                                                     ∑︁
                                                               local
                                                           𝑐ℬ 𝑀ℬ,𝜏
                                                     ℬ,𝜏

   —




                                                     148
D1.8. D1 theorem

Theorem E.16 (Theorem D1). Let (ℬ, 𝜏 ) be a terminal LongAP/Local atom produced by the routing
tree. Then
                              LongAP              LongAP
                             𝑅ℬ,𝜏    (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏    (𝑁 ) + 𝑜(𝑁 ).

   Consequently, summing over all tagged LongAP/Local atoms,

                           𝑅LongAP/Local (𝑁 ) = 𝑀LongAP/Local (𝑁 ) + 𝑜(𝑁 ),

   where
                                                                         LongAP
                     𝑀LongAP/Local (𝑁 ) =                       𝑐ℬ Loc𝑄 𝑅ℬ,𝜏    (𝑁 ).
                                                  ∑︁

                                            ℬ,𝜏 ∈LongAP/Local

   Thus every D1 local term is LPI-admissible and therefore ready for the H4 assembly.

Proof. Fix a tagged terminal LongAP/Local atom (ℬ, 𝜏 ). By Lemma D1.2A, all remaining arithmetic
restrictions are controlled local AP/congruence conditions with moduli ≤ (log 𝑁 )𝐶 , and no nonlocal
oscillatory factor remains.
    Apply the smooth AP counting lemma to the long AP directions. The zero/local part gives the
average over the corresponding residue classes modulo the controlled local modulus. Refining these
                                                                                           LongAP
local conditions with the LPI modulus 𝑄 gives exactly the tagged LPI projection Loc𝑄 𝑅ℬ,𝜏         (𝑁 ),
namely the explicit Λ𝑄 -local replacement inside the same B1/F3 cell. Boundary and endpoint
discrepancies are C1 Edge errors and contribute 𝑜(𝑁 ).
    All operations are performed inside the fixed tag (ℬ, 𝜏 ), so no local terms are merged or double-
counted. Summing over the polylogarithmic number of tagged LongAP/Local atoms preserves the
𝑜(𝑁 ) error.
    The theorem is proved.
   —


D1.9. Interface refinements        The D1 statement used in the proof is:

                                               LongAP
                                 𝒜(𝑁 ) = Loc𝑄 𝑅ℬ,𝜏    (𝑁 ) + 𝑜(𝑁 )

   for every tagged LongAP/Local atom.
   Thus:

  1. D1 local terms satisfy the LPI admission condition consumed by H4;

  2. D1 does not hide nonlocal cancellation estimates;

  3. all local AP normalizations are compatible with Λ𝑄 (𝑎) = 𝑄/𝜙(𝑄)1(𝑎,𝑄)=1 ;

  4. all boundary/endpoint errors are routed to C1;

  5. parent B1 and routing tags are preserved.

   —

                                                  149
                                Remark E.17 (D1.10. Output).
     Every tagged LongAP/Local atom equals its LPI canonical local projection plus 𝑜(𝑁 ).
   D1 contains no hidden nonlocal cancellation; boundary and endpoint errors are routed to C1;
parent B1 and routing tags are preserved.
   Consequences:

  • the LPI admission condition is discharged for LongAP/Local terms;

  • together with G8a, all major local/main sources are LPI-admissible;

  • the remaining external checks are outside D1.

D1.11. Logical Dependencies Internal dependencies: B1, B3, F3, F3T, F4, C1A, C1, E5, LPI.
  Children served: H4, H4M, and I1.

E.6     B1 local-density compatibility
E.6.1    B1LD. Local Densities for B1-Inherited CKP Coefficients
B1LD.0. Role Logical ID: B1LD.
    Used by: G8a, H4, H4M.
    Uses: B1, C1, and finite CRT local algebra.
    This file supplies the local-density interface used in G8a.5. The issue is not that the B1
coefficients are literally the von Mangoldt function; they are finite-convolution pieces coming from
the exact Heath–Brown decomposition. The point is that the LPI canonical local projection Loc𝑄
is defined tagwise for those very same finite-convolution coefficients, and CRT local density algebra
is compatible with the exact B1 decomposition.
    —

B1LD.1. Local model of an elementary B1 coefficient Let 𝑄 =                    𝑝≤𝑤 𝑝 be squarefree. For
                                                                             ∏︀

an elementary coefficient sequence 𝑎(𝑛) of B1 type

                                     𝜇(𝑛)1𝑛≤𝑦 ,     1,      log 𝑛,

   with a fixed smooth dyadic cutoff, define its local residue model modulo 𝑄 by

                                            1
                  𝑎𝑄 (𝑟; 𝑄) =                                             𝑎(𝑛)𝑊𝑋 (𝑛),
                                                                ∑︁
                                |{𝑛 ∼ 𝑋 : 𝑛 ≡ 𝑟 (mod 𝑄)}|      𝑛∼𝑋
                                                            𝑛≡𝑟 (mod 𝑄)

    with the usual empty-class convention. Partial summation handles the log 𝑛 coefficient, the
constant coefficient is immediate, and the 𝜇-coefficient is local because on a squarefree modulus its
residue constraints factor prime-by-prime by CRT. Boundary errors from the smooth dyadic cutoff
are C1 Edge errors.
    Thus every elementary B1 coefficient has a well-defined finite local model modulo 𝑄, with errors
𝑜(1) after the standard choice 𝑤 = 𝑜(log 𝑁 ).
    —




                                                  150
B1LD.2. Finite convolution compatibility Let ℬ be a typed B1 dyadic block. Its coefficient
is a finite Dirichlet-convolution expression in elementary factors of the three types above. For a
residue class 𝑟 mod 𝑄, the local model of the product constraint is the finite CRT convolution

                        𝜌ℬ,𝑄 (𝑟) =                           𝑎1,𝑄 (𝑟1 ; 𝑄) · · · 𝑎𝑘,𝑄 (𝑟𝑘 ; 𝑄).   (B1-local)
                                              ∑︁

                                     𝑟1 ···𝑟𝑘 ≡𝑟   (mod 𝑄)

   Because 𝑄 is squarefree, CRT gives a product over primes 𝑝 ≤ 𝑤:

                                       𝜌ℬ,𝑄 (𝑟) =            𝜌ℬ,𝑝 (𝑟 mod 𝑝).
                                                      ∏︁

                                                      𝑝≤𝑤

   This is a finite algebraic identity for the local models. It does not require any prime distribution
theorem.
   —

B1LD.3. Compatibility with Loc𝑄 The LPI canonical local projection of a tagged atom is
defined by replacing each arithmetic coefficient in that tagged atom by its local residue-class model
modulo 𝑄, while keeping the same smooth dyadic weights, routing tag, and linear/congruence
constraints.
    For a tagged CKP atom (ℬ, 𝜏 ), the outer coefficient sequences 𝛼𝑔 (𝑎) and 𝛾𝑔 (𝑞) are restrictions,
gcd-splits, and dyadic localizations of B1 finite-convolution coefficients. Therefore their local models
are exactly the corresponding restrictions of 𝜌ℬ,𝑄 . The operations involved are finite CRT restriction,
gcd splitting, and fixed dyadic localization; each commutes with the finite local convolution (B1-
local), up to C1 boundary terms.
    Hence the local densities used in

                                                   Loc𝑄 𝑅ℬ,𝜏
                                                         CKP
                                                             (𝑁 )

are precisely the local densities of the B1-inherited CKP coefficients that appear in the ℎ = 0 term
of G8a.
    —

B1LD.4. Lemma Lemma B1-LD. For every tagged CKP atom (ℬ, 𝜏 ), the local-density
replacement used by LPI for Loc𝑄 𝑅ℬ,𝜏 CKP (𝑁 ) is the CRT finite-convolution local model of the B1-

inherited coefficient sequences 𝛼𝑔 (𝑎), 𝛾𝑔 (𝑞), and the fibre coefficients in G8a. Therefore the zero-
frequency CKP term computed in G8a.5 has the same arithmetic local density factors as the
canonical LPI projection, up to C1 boundary errors.

Proof. Elementary coefficient local models are defined in B1LD.1. B1LD.2 shows that finite
convolution and CRT localization commute. G1a gcd splitting and the tagged CKP dyadic
restrictions are finite refinements of the same coefficient support, so they preserve the local model
tagwise. LPI defines Loc𝑄 using exactly these tagwise local coefficient models; H4 later assembles the
LPI-admitted terms. Thus the arithmetic coefficient part of the G8a ℎ = 0 term and the arithmetic
coefficient part of the LPI canonical projection agree. Endpoint and smoothing discrepancies are C1
boundary errors by construction. Lemma proved.
   —




                                                        151
Remark E.18 (B1LD.5. Output). Every CKP zero-frequency term that enters the local/main
branch has the same tagged B1 local coefficient model as the LPI projection. Thus the ℎ = 0 CKP
contribution can be assembled by H4M, using the H4 local algebra, without changing normalization
and without double-counting any parent B1 block.

B1LD.6. Logical Dependencies           Internal dependencies: B1, C1, finite CRT local algebra.
  Children served: G8a, H4, H4M.

E.7     H4 local reconstruction and singular series
E.7.1    H4. Local/Main Compatibility Lemma
H4.0. Role Logical ID: H4.
    Used by: H4M and the local algebra/error-budget record.
    Uses: LPI, B1, B3, F3P, F3, F3T, F4, D1, G8a, B1LD, C1, C1A, finite CRT local algebra, and the
local model Λ𝑄 .
    Lemma H4 is the local algebra component used by H4M. After B3/F3/F4/C1 routing, all
terminal atoms are divided into error classes and local/main classes; H4 evaluates the admitted
tagged local projection and its finite local factors.
    The error classes are already handled by:

                  Edge → 𝐶1,         CKPℎ̸=0 → 𝐺8𝑎,       GoodAWACK → 𝐸10.

   The local/main contributions come from:

                          LongAP/Local,         LocalDiag,      CKPℎ=0 .

   H4 has to prove:

                                    𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   The local/main compatibility statement must prevent:

  1. double counting local terms;

  2. loss of some local terms;

  3. mismatched normalizations among LongAP, LocalDiag, and CKP zero-frequency terms;

  4. incorrect assembly of the Euler product.

   Lemma H4 handles this by consuming the LPI local projection/admission interface and the
parent B1 block tags. Thus H4 is an assembly lemma: it does not define the local projection
independently of D1 or G8a, but assembles the local terms after D1, G8a, and B1LD have proved
LPI-admissibility.
   —




                                                152
H4.1. Local modulus and local model The local model is the one defined in Lemma LPI. We
recall it here for the calculation of the singular series.
    Let

                               𝑤 = 𝑤(𝑁 ) → ∞,           𝑤 = 𝑜(log 𝑁 ),

   and set

                                             𝑄=
                                                   ∏︁
                                                        𝑝.
                                                  𝑝≤𝑤

   Define the local model of Λ modulo 𝑄 by
                                                  𝑄
                                      Λ𝑄 (𝑎) =       1        .
                                                 𝜙(𝑄) (𝑎,𝑄)=1
   This normalization gives average value one:
                                        1 ∑︁
                                                  Λ𝑄 (𝑎) = 1.
                                        𝑄 𝑎 mod 𝑄

   Define the local Goldbach density at modulus 𝑄:
                                          1 ∑︁
                              𝜎𝑄 (𝑁 ) =             Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎).
                                          𝑄 𝑎 mod 𝑄

   The canonical local main term is

                                   𝑀𝑄 (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ),

    where the 𝑜(𝑁 ) comes only from endpoint/smooth partition effects already routed to C1 or from
replacing exact interval length by 𝑁 + 𝑂(1).
    —

H4.2. Canonical local projection of the original problem                 Define the canonical local
projection of the original Goldbach sum by

                                      Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ).

   Equivalently,
                                              1 ∑︁
                         Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 ·             Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎).
                                              𝑄 𝑎 mod 𝑄

    This definition is independent of a particular branch of the proof tree.
    The purpose of H4 is to prove that the sum of all local/main pieces produced by D1, G8a zero-
frequency, and LocalDiag routing is exactly this canonical local projection, up to 𝑜(𝑁 ):

                                𝑀local (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).

   Then H4 computes the limit

                                          𝜎𝑄 (𝑁 ) → S(𝑁 )

   as 𝑤 → ∞.
   —

                                                 153
H4.3. Parent B1 block tags        From Lemma B1 we have the exact decomposition
                                     𝑅Λ (𝑁 ) =           𝑐ℬ 𝑅ℬ (𝑁 ).
                                                  ∑︁

                                                 ℬ∈B𝐽0

   Every atom produced later by B3/F3 carries a parent tag
                                           tag(𝒜) = (ℬ, 𝜏 ),
   where:
  • ℬ is the parent typed B1 block;
  • 𝜏 is the finite routing/grouping history inside B3/F3.
    B3/F3 guarantee that these tagged atoms form a finite partition of the parent block contribution;
this is Lemma F3.15:
                                     𝑅ℬ (𝑁 ) =              𝑅ℬ,𝜏 (𝑁 ),
                                                   ∑︁

                                                 𝜏 ∈𝒯 (ℬ)

   with no overlap at the tagged level.
   This is the bookkeeping mechanism preventing double counting.
   —

H4.4. Canonical local projection of tagged atoms For every tagged local atom (ℬ, 𝜏 ), define
its canonical local projection
                                            Loc𝑄 𝑅ℬ,𝜏 (𝑁 )
   as the contribution of that tagged cell to the local model obtained by replacing the arithmetic
coefficients by their residue-class local densities modulo 𝑄, while keeping the same smooth/dyadic
weights and the same tag.
   The definition is linear:
                                   ⎛         ⎞

                              Loc𝑄 ⎝ 𝑐ℬ 𝑅ℬ,𝜏 ⎠ =    𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 .
                                    ∑︁           ∑︁

                                     ℬ,𝜏                 ℬ,𝜏

     The local/main term assigned by D1, G8a zero frequency, or LocalDiag is admitted into H4 only
if it equals this canonical projection:
                               local
                              𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).
    This is the LPI admission condition as consumed by H4.
    Thus H4 does not accept arbitrary local-looking main terms. It accepts only tagged canonical
local projections.
    By Lemma LPI.1, the admitted local source set is exactly
                            LLPI = LLongAP/Local ⊔ LCKP,0 ⊔ LLocalDiag .
   There is no separate residual projection class. The only auxiliary local-looking operations
that can occur before H4 are controlled CRT restriction or absorption, fixed-divisor quotienting,
primitive local slicing, and endpoint or smooth-boundary localization. The first three are finite
tagged refinements of one of the three displayed source classes and must satisfy the same LPI
admission condition. Endpoint and smooth-boundary terms are C1 Edge contributions, not local/
main sources.
   —

                                                  154
H4.4A. Compatibility of the local projection with prior routing The following compatibility
table records the LPI admission compatibility consumed by H4. It records why the operation Loc𝑄
may be applied to the tagged terminal cells without changing the local algebra of the original
Goldbach convolution.

   Source operation before H4      Compatibility with Loc𝑄            Excluded failure mode
   B1 Heath–Brown expansion        B1 is an exact finite convolu-     treating an individual B1 sum-
                                   tion identity before estimation; mand as an independent local
                                   Λ𝑄 replaces the two original       model
                                   von Mangoldt factors after the
                                   identity is summed over all B1
                                   tags.
   B3 product grouping             B3 only partitions the finite      counting two different group-
                                   product-coordinate descriptions ings as two local main terms for
                                   and preserves the parent B1        the same tagged cell
                                   tag.
   F3/F4 terminal routing          F3/F4 refine the summation do- admitting a local term without
                                   main by exact tagged partitions its inherited B1/F3 tag
                                   or send the cell to a terminal
                                   class.
   Controlled CRT absorption       compatible finite-index restric- changing the normalization
                                   tion of residue classes mod-       from Λ𝑄 to a branch-specific
                                   ulo 𝑄; incompatible fibres are density
                                   empty.
   Fixed-divisor or quotient deci- admitted locally only after B1- quotient main term with un-
   sion                            LD identifies the quotient coeffi- matched arithmetic coefficient
                                   cient model with the same CRT
                                   local replacement.
   Gcd/local/proportional relation admitted only as a tagged Lo- untagged diagonal density or
                                   calDiag projection of a parent noncanonical specialization
                                   cell.
   LongAP/Local branch             F3P gives the intrinsic local-     branch-specific AP density with
                                   coefficient predicate, and D1.2A a nonlocal coefficient
                                   expands it into the tagged
                                   Loc𝑄 projection.
   CKP zero-frequency branch       G8a.5 and B1-LD identify the importing a nonzero Fourier
                                   ℎ = 0 mode with the tagged         mode into H4
                                   local projection.
   C1 Edge removal                 C1 contributes only 𝑜(𝑁 ) and is losing a local main term by
                                   not part of 𝑀local .               labelling it Edge without a
                                                                      strict C1P predicate
   GoodAWACK and CKP               these are error branches, al-      double-counting an error
   nonzero modes                   ready handled before H4.           branch as a local term
   Primitive local slicing         a finite tagged subdivision of     treating a slice as a new branch
                                   an already admitted LongAP/ without its parent tag
                                   Local, CKP ℎ = 0, or Local-
                                   Diag source.
   Endpoint or smooth-boundary routed to C1 and charged as            importing a boundary correc-
   localization                    𝑜(𝑁 ).                             tion into the main term




                                                  155
    Consequently H4 does not rely on the phrase "canonical projection" as an unproved convention.
A local/main contribution is admitted only after the source operation is compatible with the single
replacement rule

                                                Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄)

    on the tagged Goldbach convolution.
    —

H4.5. Local reconstruction from the B1 decomposition

Lemma E.19 (Lemma H4.1). The sum of the canonical local projections of all tagged descendants
of the exact B1 decomposition reconstructs the local Goldbach model:

                                        Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).               (H4-reconstruct)
                   ∑︁          ∑︁
                        𝑐ℬ
                    ℬ        𝜏 ∈𝒯 (ℬ)

     Equivalently, the LPI local projection of the full tagged proof tree is not a branch-specific surrogate.
It is the convolution of the single local model Λ𝑄 with itself along 𝑛1 + 𝑛2 = 𝑁 .

Proof. By Lemma B1, the Heath–Brown decomposition used in the proof is an exact finite identity
for the two von Mangoldt factors in 𝑅Λ (𝑁 ), after the fixed smooth dyadic partition of unity has
been inserted. Thus

                                               𝑅Λ (𝑁 ) =          𝑐ℬ 𝑅ℬ (𝑁 )
                                                             ∑︁

                                                              ℬ

    before any terminal estimate is applied. By Lemma F3.15, each parent block is then partitioned
into tagged descendants:

                                              𝑅ℬ (𝑁 ) =               𝑅ℬ,𝜏 (𝑁 ).
                                                             ∑︁

                                                           𝜏 ∈𝒯 (ℬ)

    The local replacement defined by LPI and evaluated by H4 is the finite CRT local replacement
of the same B1 coefficient factors and the same dyadic cells. For B1-inherited finite-convolution
coefficients this compatibility is the content of Lemma B1-LD: elementary B1 coefficient models,
finite convolution, CRT restriction, gcd splitting, and tagged dyadic localization commute with the
local replacement, up to C1 boundary terms.
    Therefore applying Loc𝑄 to the exact B1/F3 tagged partition gives the same result as applying
the local model directly to the original two von Mangoldt factors. On the original factors the local
model is by definition
                                                                         𝑄
                                    Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄) =                  1        .
                                                                        𝜙(𝑄) (𝑛,𝑄)=1
    Consequently the reconstructed local sum is

                                                           Λ𝑄 (𝑛1 )Λ𝑄 (𝑛2 )
                                                  ∑︁

                                               𝑛1 +𝑛2 =𝑁

   with the same smooth endpoint convention as the tagged cells. Counting by the residue class
𝑎 ≡ 𝑛1 (mod 𝑄) gives



                                                            156
                                           1 ∑︁
                  Λ𝑄 (𝑛1 )Λ𝑄 (𝑛2 ) = 𝑁 ·             Λ𝑄 (𝑎)Λ𝑄 (𝑁 − 𝑎) + 𝑜(𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).
         ∑︁

      𝑛1 +𝑛2 =𝑁
                                           𝑄 𝑎 mod 𝑄

   The 𝑜(𝑁 ) term is only the endpoint/smooth-boundary discrepancy already admitted by C1.
This proves (H4-reconstruct).
   —


H4.6. Dyadic and routing recombination

Lemma E.20 (Lemma H4.2). The dyadic partitions and routing tags used before H4 do not change
the local main term:

                                    𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).
                              ∑︁

                              ℬ,𝜏

Proof. The dyadic partition in B1 is an exact partition of unity on the active support. Hence
summing over dyadic scales recombines the original B1 finite-convolution expression exactly. The
later B3/F3/F4 operations are finite tagged partitions: they split summation domains by grouping
choices, CRT restrictions, fixed-divisor cases, quotient cases, terminal predicates, and boundary
alternatives. Each such operation is either an exact finite partition or a boundary/short-volume
removal already admitted by C1.
    The operator Loc𝑄 is linear and tag-preserving. Therefore it commutes with the finite recombi-
nation of dyadic cells and routing cells. The only discrepancy comes from endpoint and smooth-
boundary cells, which are C1 Edge contributions and hence 𝑜(𝑁 ). Lemma proved.
   —


H4.7. Admission of branch local terms

Lemma E.21 (Lemma H4.3). Every terminal local/main contribution entering I1 satisfies the LPI
admission condition consumed by H4

                                     local
                                    𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 ).                      (H4-adm)

Proof. There are three sources of local/main terms.

  1. LongAP/Local. Lemma F3P first states the positive local-coefficient

    predicate for this terminal class, and Lemma D1, including Lemma D1.2A, expands that local
algebra into controlled residue/coprimality data. Its local residue-density theorem then identifies
the zero/local part of the long AP count with Loc𝑄 𝑅ℬ,𝜏 (𝑁 ), with boundary terms routed to C1.

  1. CKP zero frequency. Lemma G8a separates ℎ = 0 from ℎ ̸= 0. The

    nonzero frequencies are not local terms. The ℎ = 0 term is identified in G8a.5 with the LPI
canonical local projection. Lemma B1-LD supplies the arithmetic compatibility of the B1-inherited
finite-convolution coefficients under gcd splitting, CRT localization, and tagged dyadic restriction.

  1. LocalDiag. In the F3/F4 routing, LocalDiag means forced equality,

                                                     157
   proportionality, gcd-local dependence, or collision that produces a canonical local tagged cell. If
a degeneracy is not a canonical local projection, it is not admitted by H4; the F3T routing table
sends it instead to Edge, CKP, GoodAWACK, impossible, or a continuing routed case. Thus every
LocalDiag term that reaches H4 is already a tagged canonical local projection.
   These are the only local/main terminal classes in the routing table. Hence every local/main
contribution entering I1 satisfies (H4-adm). Lemma proved.
   —


H4.8. No double counting lemma
Lemma E.22 (Lemma H4.4). The sum of all admitted local/main terms satisfies
                                  𝑀local (𝑁 ) =                               (𝑁 ),
                                                       ∑︁
                                                                        local
                                                                    𝑐ℬ 𝑀ℬ,𝜏
                                                  ℬ,𝜏 ∈𝒯local (ℬ)

    and no local term is counted twice.
Proof. Each term is indexed by its parent B1 block ℬ and a unique routing tag 𝜏 . By Lemma
F3.15, B3/F3/F4 routing produces a finite exact partition of every parent block into tagged cells.
Therefore two different tags correspond either to disjoint cells of the same B1 block or to summands
from different B1 blocks in the exact B1 decomposition.
    This includes LocalDiag atoms. A LocalDiag condition is detected by a structural predicate
such as forced equality, proportionality, gcd-local dependence, or collision of forms; two different
routing cells may therefore look locally identical. H4 does not identify local terms by visual form. It
sums the canonical projection of each tagged cell. Since the underlying tagged cells are disjoint by
(F3-partition), structurally identical LocalDiag expressions from different tags are complementary
summands, not duplicates.
    If (ℬ, 𝜏 ) ̸= (ℬ ′ , 𝜏 ′ ), then either ℬ ̸= ℬ ′ , in which case the two summands already occur separately
in the exact B1 expansion, or ℬ = ℬ ′ and 𝜏 =                ̸ 𝜏 ′ , in which case Lemma F3.15 gives disjoint
summation domains. Linearity of Loc𝑄 then preserves this separation.
    Thus no double counting occurs. Lemma proved.
   —


H4.9. Completeness of local terms
Lemma E.23 (Lemma H4.5). Every terminal local/main atom produced by B3/F3/F4 routing is
included in exactly one of:
                             LongAP/Local,            LocalDiag,             CKPℎ=0 .
Proof. By Lemma F3 and the LPI interface, every terminal atom belongs to exactly one termi-
nal routing class at the tagged level. Error classes are Edge, CKP nonzero-frequency error, and
GoodAWACK. Local/main classes are LongAP/Local, LocalDiag, and CKP zero-frequency. Con-
trolled CRT restrictions, quotients and local slicing may produce auxiliary local projection subterms,
but such subterms inherit the parent tag and are admitted only inside one of these three classes.
They are not a separate local/main source.
    Therefore every admitted local/main contribution is tagged once and all terminal local/main
contributions are included. Lemma proved.
   —


                                                       158
H4.10. Linearity of the local projection

Lemma E.24 (Lemma H4.6). The sum of canonical local projections over all tagged local/main
atoms equals the canonical local projection of the original Goldbach sum:

                                  𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).
                            ∑︁

                            ℬ,𝜏

Proof. The B1 decomposition is exact and B3/F3 tagged routing partitions each parent block into
finitely many cells. The operator Loc𝑄 is linear by definition. Therefore applying Loc𝑄 before or
after summing the tagged decomposition gives the same result.
    All nonlocal/error classes contribute either zero to the admitted local sum or are handled as
𝑜(𝑁 ) by C1/E10/G8a nonzero-frequency estimates. The remaining admitted local classes sum to
the full canonical local projection. Lemma proved.
   —


H4.11. Compatibility of LongAP, CKP zero-frequency and LocalDiag normalizations

Lemma E.25 (Lemma H4.7A). Every admitted local/main term from LongAP/Local, CKP zero-
frequency, or LocalDiag is normalized by the single LPI operation

                                         Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄)

    inside the original tagged Goldbach convolution. No branch is allowed to introduce a separate
local density convention.
    Equivalently, for every admitted tagged cell (ℬ, 𝜏 ),
                                   local
                                  𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 ).

   The verification is as follows.


   Source of the local term Branch expression be-     H4 normalization        Excluded alternative
                            fore H4
   LongAP/Local             a long arithmetic-   F3P forces all long-       if a nonlocal coefficient
                            progression main termvariable coefficients into survives, the atom is
                            after D1             Cloc (𝑄𝜏 ), and D1.2A      not F3P-LongAP/Local
                                                 identifies the remain-     and hence not LPI-
                                                 ing AP density with        admitted
                                                 Loc𝑄 𝑅ℬ,𝜏
   CKP ℎ = 0               the zero Fourier mode G8a.5 identifies the       ℎ= ̸ 0 is never local and
                           in the G8a CKP expan- ℎ = 0 mode with the        is sent to CKP/X10
                           sion                  same Loc𝑄 𝑅ℬ,𝜏 , using
                                                 B1-LD for coefficient
                                                 compatibility




                                                    159
   LocalDiag              a forced equality, pro-  admitted only when       noncanonical degen-
                          portionality, gcd-local  the diagonal cell is the eracies are routed by
                          dependence, or collision canonical local projec- F3T to Edge, CKP,
                          cell                     tion of a tagged B1/F3/ GoodAWACK, impos-
                                                   F4 cell                  sible, or a continuing
                                                                            routed case
  Controlled CRT restric- a finite-index refine-   admitted only when the branch-specific CRT
  tion or absorption      ment of a tagged source parent tag and the Λ𝑄 - densities are not H4
                          class                    replacement rule are     inputs
                                                   preserved
  Fixed-divisor quotient- a coefficient refinement admitted only through    quotient main terms
  ing                     of a tagged source class the B1-LD compatibility with unmatched coeffi-
                                                   check                    cient models are not H4
                                                                            inputs
  Primitive local slicing a finite subdivision of  admitted only as part    a slice is not a fourth
                          the same tagged local    of its parent LongAP/ local branch
                          source                   Local, CKP ℎ = 0, or
                                                   LocalDiag class
  Endpoint or smooth-     boundary correction      routed to C1 as 𝑜(𝑁 ), boundary terms are not
  boundary localization                            not admitted into H4M local main terms




Proof. The operator Loc𝑄 was defined before any branch-specific terminal analysis: it replaces the
arithmetic coefficients in the original tagged Goldbach convolution by their residue-class local model
modulo 𝑄, while keeping the same dyadic weights, summation domain, and routing tag. Therefore it
is a property of the parent tagged cell, not of the branch that later recognizes the local contribution.
     For LongAP/Local cells, F3P and D1.2A first exclude exactly the obstruction that would leave
a branch-specific arithmetic coefficient. The remaining main term is the residue-density projection
of the same tagged cell, hence it is Loc𝑄 𝑅ℬ,𝜏 .
     For CKP cells, G8a separates the zero Fourier mode from all nonzero modes. The nonzero
modes are analytic error terms. The zero mode is a local projection only after G8a.5 and B1-LD
identify its finite-convolution coefficients with the B1-inherited local density; hence its admitted
form is again Loc𝑄 𝑅ℬ,𝜏 .
     For LocalDiag cells, the routing tag records the exact forced relation that created the diagonal
cell. H4 admits such a cell only when the diagonal specialization is a canonical tagged projection. If
the specialization is not canonical, it never reaches H4 and is routed elsewhere by F3T.
    Thus the equality of normalizations is not assumed branch-by-branch; it is enforced by the
definition of admissible local/main term.
    —


H4.12. Calculation of local factors         By the definition of Λ𝑄 ,

                                     1 ∑︁       𝑄             𝑄
                         𝜎𝑄 (𝑁 ) =                  1(𝑎,𝑄)=1     1           .
                                     𝑄 𝑎 mod 𝑄 𝜙(𝑄)          𝜙(𝑄) (𝑁 −𝑎,𝑄)=1

   Since 𝑄 is squarefree, CRT gives



                                                  160
                                             𝜎𝑄 (𝑁 ) =                 𝜎𝑝 (𝑁 ),
                                                                ∏︁

                                                                𝑝≤𝑤

   where
                            1
                                           )︂2
                                      𝑝
                                (︂
                  𝜎𝑝 (𝑁 ) =                      #{𝑎 mod 𝑝 : (𝑎, 𝑝) = 1, (𝑁 − 𝑎, 𝑝) = 1}.
                            𝑝        𝑝−1

   For 𝑝 = 2 and even 𝑁 , the only unit residue is 1, and 𝑁 − 1 ≡ 1 (mod 2), so

                                                        𝜎2 (𝑁 ) = 2.

    For odd 𝑝:
    If 𝑝 | 𝑁 , then the forbidden residues 𝑎 ≡ 0 and 𝑎 ≡ 𝑁 coincide. Hence the number of admissible
residues is

                                                           𝑝 − 1,

   and
                                          1
                                                                )︂2
                                                         𝑝                             𝑝
                                                   (︂
                                𝜎𝑝 (𝑁 ) =                             (𝑝 − 1) =           .
                                          𝑝             𝑝−1                           𝑝−1

   If 𝑝 ∤ 𝑁 , the two forbidden residues are distinct, so the number of admissible residues is

                                                           𝑝 − 2,

   and
                               1                                       𝑝(𝑝 − 2)         1
                                                 )︂2
                                           𝑝
                                     (︂
                     𝜎𝑝 (𝑁 ) =                         (𝑝 − 2) =                 =1−          .
                               𝑝          𝑝−1                          (𝑝 − 1) 2     (𝑝 − 1)2

   Therefore
                                             ∏︁ (︂                  1                    𝑝
                                                                             )︂ ∏︁
                          𝜎𝑄 (𝑁 ) = 2                     1−                                .
                                           3≤𝑝≤𝑤
                                                                 (𝑝 − 1)2         3≤𝑝≤𝑤
                                                                                        𝑝−1
                                            𝑝∤𝑁                                    𝑝|𝑁

   Letting 𝑤 → ∞, we get
                                                                      ∏︁ 𝑝 − 1
                                           𝜎𝑄 (𝑁 ) → 2𝐶2                          ,
                                                                      𝑝|𝑁
                                                                            𝑝−2
                                                                      𝑝>2

   where
                                                       ∏︁ (︂         1
                                                                              )︂
                                           𝐶2 =                1−          .
                                                   𝑝>2
                                                                  (𝑝 − 1)2

   Thus

                                                  𝜎𝑄 (𝑁 ) → S(𝑁 ).

   —

                                                               161
H4.13. Local/Main compatibility theorem

Theorem E.26 (Theorem H4). In the proof tree, the sum of all terminal local/main contributions
satisfies

                                      𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

Proof. By the compatibility table H4.4A and Lemma H4.3, every LongAP/Local, CKP zero-
frequency, and LocalDiag term that reaches H4 satisfies the LPI admission condition
                               local
                              𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).

   By Lemma H4.4, local/main terms are summed by unique B1 parent tags and routing tags, so
there is no double counting. By Lemma H4.5, all terminal local/main atoms are included.
   Therefore

                            𝑀local (𝑁 ) =         𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 ).
                                            ∑︁

                                            ℬ,𝜏

   By Lemmas H4.1, H4.2, and H4.6, the tagged sum of canonical local projections reconstructs
the full local Goldbach model:

                      𝑀local (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).

   By the local factor calculation,

                                            𝜎𝑄 (𝑁 ) → S(𝑁 ).

   Hence

                                      𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   The theorem is proved.
   —


                                  Remark E.27 (H4.14. Output).
           All admitted local/main terms assemble as Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ).
    Every local term is a canonical local projection, carries parent B1 and routing tags, and is
admitted only when its branch-specific normalization matches Loc𝑄 . CKP zero-frequency, LongAP
local terms, and LocalDiag terms are combined by tagged linearity. The singular series is computed
from the explicit finite local model Λ𝑄 .
    —

H4.15. Logical Dependencies Internal dependencies: LPI, B1, B3, F3P, F3, F3T, F4, D1, G8a,
B1LD, C1, C1A, and the local model Λ𝑄 .
   Children served: H4M and the local algebra/error-budget record.




                                                    162
E.8     H4M master local bridge theorem
E.8.1    H4M. Master Local Bridge Theorem
H4M.0. Statement and Role Logical ID: H4M.
    Lemma H4M is the reader-facing master theorem for the local/main handoff. It packages the
local projection interface, the branch-specific admission lemmas, and the H4 local algebra into one
autonomous bridge:

           F3F4M + LPI + 𝐷1 + 𝐺8𝑎 + 𝐵1𝐿𝐷 + 𝐻4 =⇒ 𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   The theorem has two purposes.

  1. It identifies exactly which terminal contributions are allowed to enter the local/main assembly.

  2. It proves that those admitted contributions, and no hidden residual local terms, reconstruct
     the singular-series main term.

    In particular, 𝑀other local , whenever used as bookkeeping notation, denotes only explicitly LPI-
admitted tagged local projection subterms. It is not an independent branch and contains no untagged
local main term.
    —

H4M.1. Setup Let the exact B1/B3/F3/F4 routing partition be fixed. A terminal tagged cell is
written

                                               (ℬ, 𝜏 ),

    where ℬ is the parent B1 dyadic block and 𝜏 records the finite routing tag.
    Let 𝑄 = 𝑝≤𝑤(𝑁 ) 𝑝, with 𝑤(𝑁 ) → ∞ and 𝑤(𝑁 ) = 𝑜(log 𝑁 ), and let Loc𝑄 be the LPI tagged
              ∏︀

local projection. A terminal local contribution is admitted only if

                               local
                              𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).                 (H4M-adm)
   The LPI source classes are exactly

                            LLPI = LLongAP/Local ⊔ LCKP,0 ⊔ LLocalDiag .                  (H4M-src)
    Auxiliary local-looking operations are not new source classes. Controlled CRT restriction, fixed-
divisor quotienting, and primitive local slicing inherit the parent tag and are counted inside one of
the three classes in (H4M-src). Endpoint and smooth-boundary terms are Edge terms, not local
main terms.
    —

H4M.2. Master Theorem

Theorem E.28 (Theorem H4M. Local bridge and singular-series assembly). After the F3F4M
terminal routing partition is applied, the sum of all terminal local/main contributions satisfies

                                   𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).                        (H4M-main)
   Moreover:

                                                163
   1. every local/main summand in 𝑀local is LPI-admissible in the sense of (H4M-adm);

   2. the admitted source set is exactly the disjoint union (H4M-src);

   3. there is no fourth residual local projection class and no untagged local-main contribution;

   4. the tagged summation has no double counting.

   —

H4M.3. Proof Step 1: source exhaustion. Lemma F3F4M proves that the terminal routing
partition has exactly five terminal structural classes:

                  Edge,   CKP,     GoodAWACK,            LocalDiag,   LongAP/Local,

    and no sixth mixed terminal class. Lemma LPI.1 then proves the local-source identity (H4M-
src) inside these terminal classes. The proof uses only the pre-terminal B1/B3/F3P local-source
vocabulary and finite CRT local algebra. It does not use H4 as a definition of local admissibility.
Therefore the phrase "local/main term" has a fixed meaning before the H4M bridge is applied.
    Controlled CRT restrictions, fixed-divisor quotienting, and primitive local slicing preserve the
parent tag (ℬ, 𝜏 ). Hence any local-looking term produced by such an operation is a subterm of an
already admitted LongAP/Local, CKP zero-frequency, or LocalDiag source. Boundary and endpoint
pieces are C1-admitted Edge terms.
    Define the residual local-source class
                                   (︁                                      )︁
              Lother local := LLPI ∖ LLongAP/Local ⊔ LCKP,0 ⊔ LLocalDiag .              (H4M-other)
   The F3F4M terminal partition excludes a sixth terminal class, and LPI.1 says that every LPI-
admissible local source is one of the three displayed classes. Therefore

                                             Lother local = ∅.                        (H4M-no-other)
   This is the precise sense in which 𝑀other local is not an independent branch. Any notation of
that form can only denote a bookkeeping subsum inside one of the three classes in (H4M-src).
   Step 2: branch admission. Each class in (H4M-src) satisfies the same LPI normalization.
   For LongAP/Local cells, Lemma D1 proves that the F3P positive local-coefficient predicate
expands into controlled local residue data and gives
                             LongAP/Local
                           𝑀ℬ,𝜏             (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ).

    For CKP zero-frequency cells, Lemma G8a separates the ℎ = 0 mode from all nonzero frequencies.
Lemma B1LD verifies that the B1 inherited finite convolution coefficients have the same CRT local
model used by Loc𝑄 . Therefore the CKP zero-frequency contribution also satisfies (H4M-adm).
    For LocalDiag cells, the F3/F4 terminal tag records a forced local, diagonal, gcd-local, repeated-
form, or proportional relation. Lemma H4 admits such a cell only as the canonical LPI projection of
the tagged parent cell; a noncanonical degeneracy is routed away before H4M by the finite routing
table. Hence every LocalDiag term reaching the assembly satisfies (H4M-adm).
    Step 3: no double counting. The exact B1 decomposition and the B3/F3/F4 routing
partition give disjoint tagged cells. Lemma H4.4 proves that H4 sums local terms by the ordered tag
(ℬ, 𝜏 ), not by their visual local formula. If two LocalDiag expressions look algebraically identical


                                                   164
but come from different tags, they are distinct summands of the exact partition. If the tags differ
inside a fixed parent block, their summation domains are disjoint. Linearity of Loc𝑄 preserves this
separation.
    Thus the local/main sum is

                 𝑀local (𝑁 ) =                   𝑐ℬ,𝜏 Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 ),                     (H4M-tag-sum)
                                     ∑︁

                                  (ℬ,𝜏 )∈LLPI

    with each admitted tagged cell counted exactly once.
    Step 4: reconstruction of the full local model. Lemma H4.1 reconstructs the canonical
local projection of a parent B1 block from the projections of its tagged descendants. Lemma H4.2
shows that dyadic partitions and routing tags commute with the LPI projection up to C1-admitted
boundary errors. Lemma H4.6 then gives

                            𝑐ℬ,𝜏 Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).                         (H4M-reconstruct)
                 ∑︁

              (ℬ,𝜏 )∈LLPI

   Combining (H4M-tag-sum) and (H4M-reconstruct),

                                    𝑀local (𝑁 ) = Loc𝑄 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).                              (H4M-local)
   Step 5: finite local factor calculation. By the definition of Λ𝑄 in LPI/H4,

                                     Loc𝑄 𝑅Λ (𝑁 ) = 𝑁 𝜎𝑄 (𝑁 ) + 𝑜(𝑁 ),

   where

                                            𝜎𝑄 (𝑁 ) =                𝜎𝑝 (𝑁 ).
                                                             ∏︁

                                                           𝑝≤𝑤(𝑁 )

   The CRT calculation in H4.12 gives the standard Goldbach local factors:

                                            0,              𝑝 = 2, 𝑁 ̸≡ 0 (mod 2),
                                          ⎧
                                          ⎪
                                          ⎪
                                          ⎪
                                          ⎪2,               𝑝 = 2, 𝑁 ≡ 0 (mod 2),
                                          ⎪
                                          ⎪
                                          ⎪
                                            𝑝(𝑝 − 2)
                                          ⎨
                              𝜎𝑝 (𝑁 ) =              ,      𝑝 > 2, 𝑝 ∤ 𝑁,
                                            (𝑝 − 1)2
                                          ⎪
                                          ⎪
                                          ⎪
                                          ⎪ 𝑝
                                          ⎪
                                                            𝑝 > 2, 𝑝 | 𝑁.
                                          ⎪
                                          ⎪      ,
                                            𝑝−1
                                          ⎩


   For even 𝑁 , this product converges as 𝑤(𝑁 ) → ∞ to
                                          ∏︁ 𝑝 − 1                     ∏︁ (︂          1
                                                                                          )︂
                        S(𝑁 ) = 2𝐶2                    ,      𝐶2 =              1−          .
                                          𝑝|𝑁
                                                 𝑝−2                   𝑝>2
                                                                                   (𝑝 − 1)2
                                          𝑝>2

   Therefore (H4M-local) implies (H4M-main). The theorem is proved.
   —

Parameter check E.29 (H4M.4. Parameter Check). The bridge introduces no new asymptotic
estimate and no new external theorem. All losses are inherited from already fixed inputs:

  1. the number of tagged B1/B3/F3/F4 descendants is polylogarithmic;

                                                           165
    2. every branch error in the local admission statements is 𝑜(𝑁 ); GEB records the same summa-
       bility downstream, but is not a proof input to H4M;

    3. 𝑤(𝑁 ) → ∞ and 𝑤(𝑁 ) = 𝑜(log 𝑁 ) are the only requirements for the finite local factor limit;

    4. boundary and endpoint terms are C1-admitted before the local assembly.

    Thus H4M is a logical assembly theorem, not an additional analytic estimate.
    —

H4M.5. Interface Corollary

Corollary E.30 (Corollary H4M.1. Local/main input for I1). In the final weighted assembly, the
entire local/main contribution may be replaced by

                                       𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ),

    with no hidden residual local class. Equivalently, by (H4M-other) and (H4M-no-other),

               Lother local = ∅,       𝑀other local = 0   as an independent quotient class.

    Consequently, I1 imports the local/main branch through the single bridge

                                   H4M =⇒ 𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

    —

H4M.6. Logical Dependencies Internal dependencies: F3F4M, LPI, B1, B3, F3P, F3, F3T,
F4, D1, G8a, B1LD, H4, C1P, C1A, C1, and finite CRT local algebra.
    External dependencies: none beyond the local finite CRT algebra and the standard Euler-product
limit already recorded in H4.
    Children served: GEB, I1, and the full proof assembly.


F     CKP/X10 Package
The external DFI theorem used by this appendix is stated once in Appendix B.

F.1     G1a CKP gcd splitting
F.1.1    G1a. CKP GCD Splitting Lemma
G1a.0. Role Logical ID: G1a.
   Used by: G2a, G3a, G4a, CKPX10M, G8a, X10.
   Uses: B3, F3, F4, and the CKP terminal predicate.
   Lemma G1a is the first technical step in the CKP package. It transforms the balanced finite-
convolution equation

                                               𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁




                                                    166
   into the coprime form required for the later Fourier/AP expansion and for the application of
Kloosterman-fraction estimates.
   The main output is

                       𝑔 = gcd(𝑢, 𝑢′ ),          𝑢 = 𝑔𝑎,        𝑢′ = 𝑔𝑞,          (𝑎, 𝑞) = 1.

   If 𝑔 ∤ 𝑁 , there are no solutions. If 𝑔 | 𝑁 , the equation becomes

                                                                            𝑁
                                         𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,       𝑁𝑔 =          .
                                                                            𝑔
   —

G1a.1. Initial CKP block          Consider a balanced CKP atom of the form
                                                                          𝑢 𝑢′ 𝑦 𝑦 ′
                                                                     (︂                )︂
             ℛ(𝑁 ) =                   𝛼(𝑢)𝛼′ (𝑢′ )𝛽(𝑦)𝛽 ′ (𝑦 ′ )𝑊
                           ∑︁
                                                                           , , ,     1𝑢𝑦+𝑢′ 𝑦′ =𝑁 ,
                                                                          𝑈 𝑈′ 𝑌 𝑌 ′
                       𝑢∼𝑈, 𝑢′ ∼𝑈 ′
                       𝑦∼𝑌, 𝑦 ′ ∼𝑌 ′

    where 𝑊 is a smooth compactly supported weight and the coefficients are finite-convolution/
divisor-bounded:

                            |𝛼(𝑢)|, |𝛼′ (𝑢′ )|, |𝛽(𝑦)|, |𝛽 ′ (𝑦 ′ )| ≪ (log 𝑁 )𝐶(𝐽0 ) .

   The balanced CKP range means that, after grouping variables,

                           𝑈 ≍ 𝑈 ′ ≍ 𝑁 1/2+𝑂(𝜅) ,             𝑌 ≍ 𝑌 ′ ≍ 𝑁 1/2+𝑂(𝜅) .

   The exact shape of the ranges is not important for G1a. The only point needed here is that 𝑢
and 𝑢′ are the two grouped convolution variables to which gcd splitting is applied.
   —

G1a.2. GCD splitting         For each pair (𝑢, 𝑢′ ), set

                                                  𝑔 = gcd(𝑢, 𝑢′ ).

   Then there are unique positive integers 𝑎, 𝑞 such that

                                  𝑢 = 𝑔𝑎,           𝑢′ = 𝑔𝑞,         (𝑎, 𝑞) = 1.

   Substituting into

                                                 𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁,

   gives

                                                𝑔𝑎𝑦 + 𝑔𝑞𝑦 ′ = 𝑁.

   If

                                                       𝑔 ∤ 𝑁,

                                                        167
   there are no solutions. If

                                                     𝑔 | 𝑁,

   then, writing
                                                            𝑁
                                                    𝑁𝑔 =      ,
                                                            𝑔
   we obtain the reduced equation

                                     𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,             (𝑎, 𝑞) = 1.

   —

G1a.3. Exact reparametrization of the block                       The decomposition by 𝑔 is exact:

                                             ℛ(𝑁 ) =            ℛ𝑔 (𝑁 ),
                                                         ∑︁

                                                         𝑔|𝑁

   where

              ℛ𝑔 (𝑁 ) =                     𝛼(𝑔𝑎)𝛼′ (𝑔𝑞)                   𝛽(𝑦)𝛽 ′ (𝑦 ′ )𝑊𝑔 (𝑎, 𝑞, 𝑦, 𝑦 ′ ),
                                ∑︁                               ∑︁

                          𝑎∼𝑈/𝑔, 𝑞∼𝑈 ′ /𝑔                  𝑦∼𝑌, 𝑦 ′ ∼𝑌 ′
                             (𝑎,𝑞)=1                       𝑎𝑦+𝑞𝑦 ′ =𝑁𝑔

   and
                                                                𝑔𝑎 𝑔𝑞 𝑦 𝑦 ′
                                                           (︂                     )︂
                                 𝑊𝑔 (𝑎, 𝑞, 𝑦, 𝑦 ′ ) = 𝑊           , , ,     .
                                                                𝑈 𝑈′ 𝑌 𝑌 ′
   If 𝑔 ∤ 𝑁 , the corresponding layer is empty. Therefore the sum is only over 𝑔 | 𝑁 .
   —

G1a.4. Ranges after splitting          Define

                                                   𝑈                   𝑈′
                                            𝐴𝑔 =     ,          𝑄𝑔 =      .
                                                   𝑔                   𝑔
   Then

                                             𝑎 ∼ 𝐴𝑔 ,           𝑞 ∼ 𝑄𝑔 .

   In the balanced symmetric case 𝑈 ≍ 𝑈 ′ ≍ 𝑁 1/2 , this gives

                                                                𝑁 1/2
                                              𝐴𝑔 ≍ 𝑄𝑔 ≍               .
                                                                 𝑔

   This is the form used later in G3a/G4a:

                                                           𝑁 1/2
                                                   𝑆𝑔 :=         .
                                                            𝑔
   —

                                                         168
G1a.5. Coefficient preservation          Define the new coefficients

                                𝛼𝑔 (𝑎) = 𝛼(𝑔𝑎),           𝛾𝑔 (𝑞) = 𝛼′ (𝑔𝑞).

   If the original coefficients are divisor-bounded, then

                                    |𝛼𝑔 (𝑎)|, |𝛾𝑔 (𝑞)| ≪ (log 𝑁 )𝐶(𝐽0 ) .

   Moreover, on dyadic intervals,

                                                𝑔 (log 𝑁 )
                                                           𝐶(𝐽0 )
                                      ‖𝛼𝑔 ‖2 ≪ 𝐴1/2               ,

                                                𝑔 (log 𝑁 )
                                                           𝐶(𝐽0 )
                                      ‖𝛾𝑔 ‖2 ≪ 𝑄1/2               .

   These estimates are needed for the later DFI/Kloosterman-fraction matching.
   —

G1a.6. Local meaning of the condition (𝑎, 𝑞) = 1                 The condition

                                                (𝑎, 𝑞) = 1

    is not an additional restriction; it is part of the exact gcd parametrization. It guarantees the
existence of the inverse class

                                               𝑎 (mod 𝑞),

   which appears when solving the congruence

                                           𝑎𝑦 ≡ 𝑁𝑔 (mod 𝑞).

   This condition matches the coprimality condition in the DFI Kloosterman-fraction estimate
used by X10.
   —

G1a.7. Lemma G1a
Lemma F.1 (Lemma G1a). Suppose a CKP atom contains the equation

                                             𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁.

   Then exact gcd splitting gives a disjoint decomposition by

                                              𝑔 = gcd(𝑢, 𝑢′ ),

   and on every nonzero layer 𝑔 | 𝑁 the equation becomes

                                                           𝑁
                           𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,        𝑁𝑔 =      ,          (𝑎, 𝑞) = 1.
                                                           𝑔

   The coefficients remain finite-convolution/divisor-bounded, and the new dyadic ranges are

                                        𝑎 ∼ 𝑈/𝑔,          𝑞 ∼ 𝑈 ′ /𝑔.

                                                    169
   In the balanced range this gives

                                                              𝑁 1/2
                                                     𝑎, 𝑞 ≍         .
                                                               𝑔
Proof. All assertions follow from the uniqueness of the decomposition

                                𝑢 = 𝑔𝑎,               𝑢′ = 𝑔𝑞,          (𝑎, 𝑞) = 1,

   where 𝑔 = gcd(𝑢, 𝑢′ ), and from substitution into the original equation. If 𝑔 ∤ 𝑁 , the equation

                                                   𝑔(𝑎𝑦 + 𝑞𝑦 ′ ) = 𝑁

    is impossible. If 𝑔 | 𝑁 , division by 𝑔 gives the reduced equation. The coefficient and range
statements follow immediately from dyadic support and divisor-boundedness.
    The lemma follows.
    —


              Remark F.2 (G1a.8. Output). G1a gives the exact CKP gcd splitting.
     Nonzero layers require 𝑔 | 𝑁 , and every such layer has reduced equation 𝑎𝑦 + 𝑞𝑦 ′ = 𝑁/𝑔 with
(𝑎, 𝑞) = 1.

G1a.9. Logical Dependencies Internal dependencies: B3, F3, F4, and the CKP terminal
predicate.
   Children served: G2a, G3a, G4a, CKPX10M, G8a, X10.

F.2     G2a smooth AP Fourier expansion
F.2.1    G2a. Weighted Smooth AP Fourier Expansion for CKP
G2a.0. Role Logical ID: G2a.
  Used by: G3a, G4a, CKPX10M, G8a, X10, C1A, C1.
  Uses: G1a, C1A, C1, and the CKP terminal predicate.
  Lemma G2a is the second step of the CKP package after gcd splitting in Lemma G1a.
  The CKP fibre contains not only the smooth weight 𝑊𝑌 (𝑦), but the full tagged fibre weight:
                                                   𝑁𝑔 − 𝑎𝑦                         𝑁𝑔 − 𝑎𝑦
                                              (︂              )︂              (︂         )︂
                                          ′
                       𝐹𝑎,𝑞 (𝑦) = 𝛽(𝑦)𝛽                    𝑊𝑌 (𝑦)𝑊𝑌 ′                      .
                                                      𝑞                               𝑞
   Lemma G2a turns

                                   𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,                (𝑎, 𝑞) = 1

   into a smooth AP Fourier expansion in which:

   • ℎ = 0 gives the local/main zero-frequency term;
   • ℎ ̸= 0 gives the oscillatory Kloosterman-fraction input for Lemma G3a;
   • Fourier weights satisfy rapid decay sufficient for the high-frequency Edge routing in C1P/C1A/
     C1and for the CKP assembly in G8a.

   —

                                                         170
G2a.1. Reduced CKP equation             On a fixed 𝑔-layer after Lemma G1a, we have

                           𝑢 = 𝑔𝑎,        𝑢′ = 𝑔𝑞,               (𝑎, 𝑞) = 1,           𝑔 | 𝑁.

   Set
                                                               𝑁
                                                      𝑁𝑔 =       .
                                                               𝑔

   Then the CKP equation becomes

                                                   𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 .

   Eliminate 𝑦 ′ :

                                                          𝑁𝑔 − 𝑎𝑦
                                                   𝑦′ =           .
                                                             𝑞

   The condition 𝑦 ′ ∈ Z is equivalent to

                                              𝑎𝑦 ≡ 𝑁𝑔 (mod 𝑞).

   Since (𝑎, 𝑞) = 1, this is equivalent to the congruence

                                              𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞).

   —

G2a.2. Tagged weighted fibre         For fixed (𝑔, 𝑎, 𝑞), define the tagged fibre contribution

                                     𝒮𝑎,𝑞 =                          𝐹𝑎,𝑞 (𝑦),
                                                          ∑︁

                                                   𝑦≡𝑁𝑔 𝑎 (mod 𝑞)

   where
                                                   𝑁𝑔 − 𝑎𝑦                            𝑁𝑔 − 𝑎𝑦
                                              (︂            )︂                   (︂             )︂
                                          ′
                       𝐹𝑎,𝑞 (𝑦) = 𝛽(𝑦)𝛽                    𝑊𝑌 (𝑦)𝑊𝑌 ′                         .
                                                      𝑞                                  𝑞

   Here:

  • 𝑊𝑌 , 𝑊𝑌 ′ are smooth dyadic weights inherited from the fixed tag (ℬ, 𝜏 );

  • 𝛽, 𝛽 ′ are divisor-bounded finite-convolution coefficient weights;

  • the summand is defined as zero unless (𝑁𝑔 − 𝑎𝑦)/𝑞 ∈ Z and lies in the tagged dyadic support.

    For Fourier expansion, the smooth part is expanded directly. If some finite-convolution coefficient
is not smooth enough to enter the transform, it remains in the outer coefficient sequence and is
treated as a divisor-bounded weight in G3a/G4a. In either convention, the resulting Fourier
coefficient has the rapid-decay bound recorded below, with at most a polylogarithmic loss.
    —



                                                          171
G2a.3. Smooth AP Fourier identity Let 𝐹 be a smooth compactly supported tagged fibre
weight on Z. For a residue class 𝑟 (mod 𝑞),

                                                          1 ∑︁ ̂︀       ℎ   ℎ𝑟
                                                                       (︂ )︂ (︂    )︂
                                            𝐹 (𝑦) =
                                  ∑︁
                                                                𝐹         𝑒    ,
                              𝑦≡𝑟 (mod 𝑞)
                                                          𝑞 ℎ∈Z         𝑞   𝑞

   where, in discrete normalization,

                                         𝐹̂︀ (𝜉) =         𝐹 (𝑦)𝑒(−𝑦𝜉).
                                                     ∑︁

                                                     𝑦∈Z

   Applying this with

                                            𝑟 = 𝑁𝑔 𝑎 (mod 𝑞),

   we get

                                        1 ∑︁ ̂︀                 ℎ   ℎ𝑁𝑔 𝑎
                                                           (︂ )︂ (︂           )︂
                                 𝒮𝑎,𝑞 =       𝐹𝑎,𝑞                𝑒       .
                                        𝑞 ℎ∈Z                   𝑞    𝑞

    This identity is exact for the tagged smooth fibre after the standard smooth extension convention.
Boundary errors caused by compact support truncation are C1A-admitted C1 boundary/short-
volume errors.
    —

G2a.4. Zero-frequency term             The zero-frequency term is

                                         1             1 ∑︁
                                    (0)
                                   𝒮𝑎,𝑞 = 𝐹̂︀𝑎,𝑞 (0) =      𝐹𝑎,𝑞 (𝑦).
                                         𝑞             𝑞 𝑦

   It has no oscillatory phase

                                                          ℎ𝑁𝑔 𝑎
                                                     (︂           )︂
                                                 𝑒
                                                           𝑞

    with ℎ ̸= 0. Therefore it is the CKP local/main contribution.
    In Lemma G8a, this term is further identified with the LPI-admissible canonical local projection
later assembled by H4M:
                                 (0)
                              𝑀CKP,ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                    CKP
                                                        (𝑁 ) + 𝑜(𝑁 ).

    Thus Lemma G2a supplies the AP/Fourier identity, while Lemma G8a supplies the LPI normal-
ization check consumed by H4M.
    —




                                                          172
G2a.5. Nonzero-frequency oscillatory terms                 The nonzero-frequency contribution is

                                                       1             ℎ   ℎ𝑁𝑔 𝑎
                                                                   (︂ )︂ (︂       )︂
                        𝒪𝑔 =               𝛼𝑔 (𝑎)𝛾𝑔 (𝑞) 𝐹̂︀𝑎,𝑞
                               ∑︁     ∑︁
                                                                       𝑒       .
                             ℎ̸=0 𝑎∼𝐴 ,𝑞∼𝑄
                                                       𝑞             𝑞    𝑞
                                       𝑔       𝑔
                                     (𝑎,𝑞)=1

   This is the precise input for Lemma G3a: a weighted bilinear Kloosterman fraction sum with
parameters

                                 𝑀 = 𝐴𝑔 ,          𝑄 = 𝑄𝑔 ,        𝑘 = ℎ𝑁𝑔 .

   The finite-convolution coefficients and the tagged fibre transform are absorbed into divisor-
bounded weighted coefficient sequences, with polylogarithmic losses only.
   —

G2a.6. Fourier-weight decay           Assume the CKP balanced range:

                                                                   𝑁 1/2+𝑂(𝜂)
                              𝑌 ≍ 𝑁 1/2+𝑂(𝜂) ,         𝑞 ≍ 𝑄𝑔 ≍               .
                                                                        𝑔

   For the smooth fibre weight we have, for every 𝐴 > 0,
                                                              )︂−𝐴
                               ⃒1
                               ⃒ 𝐹̂︀𝑎,𝑞 ℎ ⃒ ≪𝐴 𝐿𝐶𝐹 𝑌 1 + |ℎ|𝑌
                               ⃒       (︂ )︂⃒        (︂
                                            ⃒
                               ⃒𝑞                                  .
                                         𝑞 ⃒       𝑞       𝑞

   Since
                                                    𝑌
                                                      ≍𝑔
                                                    𝑞

   up to fixed dyadic constants, this gives
                                  ⃒1
                                  ⃒
                                  ⃒ 𝐹̂︀𝑎,𝑞 ℎ ⃒ ≪𝐴 𝐿𝐶𝐹 𝑔(1 + |ℎ|𝑔)−𝐴 .
                                          (︂ )︂⃒
                                               ⃒
                                  ⃒𝑞        𝑞 ⃒

    The polylogarithmic factor 𝐿𝐶𝐹 records derivative bounds and finite-convolution coefficient losses.
It is harmless in C1P/C1A/C1and G8a because all those estimates have arbitrary polylogarithmic
saving margins.
    —

G2a.7. Lemma G2a

Lemma F.3 (Lemma G2a). For each fixed balanced CKP 𝑔-layer after G1a, the reduced equation

                                     𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,         (𝑎, 𝑞) = 1,

   is equivalent, after eliminating 𝑦 ′ , to

                                               𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞).

   For the tagged weighted fibre

                                                     173
                                                𝑁𝑔 − 𝑎𝑦                      𝑁𝑔 − 𝑎𝑦
                                           (︂            )︂             (︂             )︂
                      𝐹𝑎,𝑞 (𝑦) = 𝛽(𝑦)𝛽 ′                𝑊𝑌 (𝑦)𝑊𝑌 ′                   ,
                                                   𝑞                            𝑞

   we have the exact smooth AP expansion

                                                      1 ∑︁ ̂︀       ℎ   ℎ𝑁𝑔 𝑎
                                                                   (︂ )︂ (︂       )︂
                                         𝐹𝑎,𝑞 (𝑦) =
                             ∑︁
                                                            𝐹𝑎,𝑞      𝑒       .
                        𝑦≡𝑁𝑔 𝑎 (mod 𝑞)
                                                      𝑞 ℎ∈Z         𝑞    𝑞

   The zero-frequency term

                                                    1 ̂︀
                                                      𝐹𝑎,𝑞 (0)
                                                    𝑞

   is the CKP local/main term, and the nonzero frequencies produce the DFI/Kloosterman input

                                                     1                ℎ   ℎ𝑁𝑔 𝑎
                                                                    (︂ )︂ (︂           )︂
                      𝒪𝑔 =               𝛼𝑔 (𝑎)𝛾𝑔 (𝑞) 𝐹̂︀𝑎,𝑞
                             ∑︁     ∑︁
                                                                        𝑒       .
                           ℎ̸=0 𝑎∼𝐴 ,𝑞∼𝑄
                                                     𝑞                𝑞    𝑞
                                     𝑔          𝑔
                                  (𝑎,𝑞)=1

   Moreover, in the balanced range,
                               ⃒1
                               ⃒
                               ⃒ 𝐹̂︀𝑎,𝑞 ℎ ⃒ ≪𝐴 𝐿𝐶𝐹 𝑔(1 + |ℎ|𝑔)−𝐴 .
                                       (︂ )︂⃒
                                            ⃒
                               ⃒𝑞        𝑞 ⃒

Proof. The congruence follows from

                              𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ⇐⇒ 𝑎𝑦 ≡ 𝑁𝑔 (mod 𝑞),

    and from (𝑎, 𝑞) = 1. The smooth AP expansion is the standard additive-character/Poisson
identity for a smooth residue-class sum. The decomposition into zero and nonzero frequencies
                                   ̸ 0. The decay estimate follows from rapid decay of the Fourier
follows by separating ℎ = 0 from ℎ =
transform of the tagged smooth fibre weight and from 𝑌 /𝑞 ≍ 𝑔 in the balanced CKP range. Lemma
proved.
    —


 Remark F.4 (G2a.8. Output). G2a gives the weighted smooth AP Fourier expansion for CKP.
    After G1a, the reduced equation is converted to the congruence 𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞). The full
tagged fibre weight is expanded into zero and nonzero frequencies. The zero frequency is the CKP
local/main term later normalized in G8a; the nonzero frequencies are routed to the Kloosterman
input in G3a. Fourier-weight decay carries only harmless polylogarithmic loss.

G2a.9. Logical Dependencies Internal dependencies: G1a, C1A, C1, and the CKP terminal
predicate.
   Children served: G3a, G4a, CKPX10M, G8a, X10, C1A, C1.




                                                      174
F.3     G3a CKP-to-DFI conversion
F.3.1    G3a. CKP to Kloosterman-Fraction Reduction
G3a.0. Role Logical ID: G3a.
    Used by: G4a, CKPX10M, X10.
    Uses: G1a, G2a.
    Lemma G3a converts the nonzero-frequency part of CKP after smooth AP Fourier expansion
into bilinear Kloosterman-fraction form. It is the direct bridge between G2a and G4a, and then
into CKPX10M.
    The target is:

                                                            1 ̂︁              ℎ   ℎ𝑁𝑔 𝑎
                                                                            (︂ )︂ (︂          )︂
                                                𝛽𝑔 (𝑎)𝛾𝑔 (𝑞) 𝑊
                                 ∑︁         ∑︁
                   𝒪𝑔      ⇝                                    𝑌               𝑒       ,
                                 ℎ̸=0 𝑎∼𝐴 , 𝑞∼𝑄
                                                            𝑞                 𝑞    𝑞
                                            𝑔        𝑔
                                           (𝑎,𝑞)=1

   that is, to a Kloosterman-fraction sum with parameters

                                 𝑀 = 𝐴𝑔 ,             𝑄 = 𝑄𝑔 ,           𝑘 = ℎ𝑁𝑔 .

   In the balanced CKP case,

                                                               𝑁 1/2
                                                𝐴𝑔 ≍ 𝑄𝑔 ≍            .
                                                                𝑔
   —

G3a.1. Input from G1a and G2a                   After G1a we have the exact gcd splitting

                                                                                       𝑁
                          𝑢 = 𝑔𝑎,          𝑢′ = 𝑔𝑞,       (𝑎, 𝑞) = 1,          𝑁𝑔 =      .
                                                                                       𝑔

   After G2a, the reduced equation

                                                  𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔

   gives the congruence

                                                𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞).

   The smooth AP Fourier expansion gives the nonzero-frequency part

                                                          1 ̂︁             ℎ   ℎ𝑁𝑔 𝑎
                                                                         (︂ )︂ (︂        )︂
                     𝒪𝑔 =                     𝛼𝑔 (𝑎)𝛾𝑔 (𝑞) 𝑊
                               ∑︁      ∑︁
                                                              𝑌              𝑒       ,
                               ℎ̸=0 𝑎∼𝐴 , 𝑞∼𝑄
                                                          𝑞                𝑞    𝑞
                                       𝑔         𝑔
                                      (𝑎,𝑞)=1

   up to harmless smooth weights in 𝑎, 𝑞 inherited from the dyadic decomposition.
   —




                                                         175
G3a.2. Incorporating smooth dyadic weights The coefficients after gcd splitting and dyadic
localization can be written as

                          𝛽𝑔 (𝑎) = 𝛼(𝑔𝑎)𝜔𝐴 (𝑎),                𝛾𝑔 (𝑞) = 𝛼′ (𝑔𝑞)𝜔𝑄 (𝑞),

   where 𝜔𝐴 , 𝜔𝑄 are smooth dyadic cutoffs supported on

                                             𝑎 ≍ 𝐴𝑔 ,          𝑞 ≍ 𝑄𝑔 .

    All smooth weights depending only on 𝑎 or only on 𝑞 may be absorbed into 𝛽𝑔 or 𝛾𝑔 . The CKP
fibre weight from G8a is slightly more general: after eliminating 𝑦 ′ , the factor
                                                     𝐹̂︀𝑎,𝑞 (ℎ/𝑞)
depends smoothly on both 𝑎 and 𝑞. This two-variable weight is not replaced by a separated surrogate.
It is kept as a normalized smooth DFI weight 𝑊𝑔,ℎ (𝑎, 𝑞). The derivative admissibility of 𝑊𝑔,ℎ ,
including the chain-rule terms from 𝛽 ′ ((𝑁𝑔 − 𝑎𝑦)/𝑞) and 𝑊𝑌 ′ ((𝑁𝑔 − 𝑎𝑦)/𝑞), is proved in CKPD and
the X10 external input.
    Separated Taylor/localization is used only for genuinely one-variable dyadic factors. This avoids
the earlier overcompressed statement that every mild multi-variable weight can simply be absorbed
into 𝛽𝑔 and 𝛾𝑔 .
    Thus it is enough to treat sums of the form
                                                                                     ℎ𝑁𝑔 𝑎
                                                                                (︂            )︂
                        𝒪𝑔,ℎ =                   𝛽𝑔 (𝑎)𝛾𝑔 (𝑞)𝑊𝑔,ℎ (𝑎, 𝑞)𝑒
                                        ∑︁
                                                                                           .
                                   𝑎∼𝐴𝑔 , 𝑞∼𝑄𝑔
                                                                                      𝑞
                                     (𝑎,𝑞)=1

   —

G3a.3. Weighted DFI form               In the separated model one may define the weighted coefficient
                                                            1 ̂︁         ℎ
                                                                     (︂ )︂
                                         𝛾̃︀𝑔,ℎ (𝑞) = 𝛾𝑔 (𝑞) 𝑊  𝑌          .
                                                            𝑞            𝑞
   Then
                                                                  ℎ𝑁𝑔 𝑎
                                                                          (︂             )︂
                               𝒪𝑔,ℎ =           𝛽𝑔 (𝑎)𝛾̃︀𝑔,ℎ (𝑞)𝑒
                                             ∑︁
                                                                        .
                                      𝑎∼𝐴 , 𝑞∼𝑄
                                                                   𝑞
                                             𝑔       𝑔
                                           (𝑎,𝑞)=1

   For the actual CKP fibre, the equivalent DFI form is

                                                         ℎ𝑁𝑔 𝑎
                                                                    (︂         )︂
               𝒪𝑔,ℎ =           𝛽𝑔 (𝑎)𝛾𝑔 (𝑞)𝑊𝑔,ℎ (𝑎, 𝑞)𝑒                                           (G3a-DFI-weight)
                           ∑︁
                                                               ,
                      𝑎∼𝐴 , 𝑞∼𝑄
                                                          𝑞
                           𝑔       𝑔
                         (𝑎,𝑞)=1

   where 𝑊𝑔,ℎ is a smooth two-variable weight satisfying the DFI derivative conditions with a
polylogarithmic parameter by CKPD. X10 is invoked for this weighted form; the separated display
above is only the simpler model used for norm bookkeeping.
   This is exactly a bilinear Kloosterman fraction sum of the form
                                                             𝑘𝑚
                                                                          (︂        )︂
                                𝐵𝑘 (𝑀, 𝑄) =
                                                         ∑︁
                                                     𝛼𝑚 𝛽𝑛 𝑒    ,
                                            𝑚∼𝑀, 𝑛∼𝑄
                                                              𝑛
                                                     (𝑚,𝑛)=1


                                                         176
   with the dictionary
                   𝑚 = 𝑎,       𝑛 = 𝑞,       𝑀 = 𝐴𝑔 ,          𝑄 = 𝑄𝑔 ,      𝑘 = ℎ𝑁𝑔 .
   —

G3a.4. Coefficient norms From G1a coefficient preservation, finite-convolution divisor-
boundedness gives

                                               𝑔 (log 𝑁 )
                                                          𝐶(𝐽0 )
                                     ‖𝛽𝑔 ‖2 ≪ 𝐴1/2               ,

                                               𝑔 (log 𝑁 )
                                                          𝐶(𝐽0 )
                                     ‖𝛾𝑔 ‖2 ≪ 𝑄1/2               .
   The exponent 𝐶(𝐽0 ) is uniform in 𝑔. Indeed, the B1 elementary coefficients are bounded by fixed
powers of log 𝑁 on every dyadic block; after the exact substitution 𝑢 = 𝑔𝑎, 𝑢′ = 𝑔𝑞, the bounds
become |𝛽𝑔 (𝑎)|, |𝛾𝑔 (𝑞)| ≪ (log 𝑁 )𝐶(𝐽0 ) on supports of lengths 𝐴𝑔 and 𝑄𝑔 . Thus

                               𝑔 (log 𝑁 )                            𝑔 (log 𝑁 )
                                          𝐶(𝐽0 )                                𝐶(𝐽0 )
                     ‖𝛽𝑔 ‖2 ≪ 𝐴1/2               ,         ‖𝛾𝑔 ‖2 ≪ 𝑄1/2
with the same structural exponent for every admissible 𝑔-layer. Summing over 𝑔 | 𝑁 later uses the 𝑔-
decay in G4a and the excluded-range routing in X10ER, and only adds another fixed polylogarithmic
loss.
    For the weighted coefficient, using the Fourier decay from G2a:
                                                          )︃−𝐴
                               ⃒1
                                                (︃
                                                     |ℎ|𝑌
                               ⃒
                                  ̂︁𝑌 ℎ ⃒ ≪𝐴 𝑌
                                     (︂ )︂⃒
                                                  1+
                                          ⃒
                               ⃒ 𝑊                             ,
                               ⃒𝑞      𝑞 ⃒   𝑄𝑔       𝑄𝑔

   and in balanced range 𝑌 /𝑄𝑔 ≍ 𝑔, we obtain

                            ‖𝛾̃︀𝑔,ℎ ‖2 ≪𝐴 𝑔(1 + |ℎ|𝑔)−𝐴 𝑄1/2
                                                         𝑔 (log 𝑁 )
                                                                    𝐶(𝐽0 )
                                                                           .
   For the nonseparated weighted form (G3a-DFI-weight), the same coefficient norms are used,
while the supremum and derivative losses of 𝑊𝑔,ℎ are charged to the smooth-weight parameter in
X10. These are precisely the hypotheses used in G4a/X10.
   —

G3a.5. Balanced parameter matching              In balanced CKP range,
                                                                   𝑁 1/2
                                   𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,           𝑆𝑔 =         .
                                                                    𝑔
   The DFI external parameter is
                                                           ℎ𝑁
                                           𝑘 = ℎ𝑁𝑔 =          .
                                                            𝑔
   Then
                                              |ℎ|𝑁  𝑁   𝑁
                             |𝑘| + 𝐴𝑔 𝑄𝑔 ≍         + 2 = 2 (1 + |ℎ|𝑔).
                                                𝑔   𝑔   𝑔
   This is exactly the expression used in G4a:
                             (|𝑘| + 𝐴𝑔 𝑄𝑔 )3/8 = 𝑁 3/8 𝑔 −3/4 (1 + |ℎ|𝑔)3/8 .
   —

                                                     177
G3a.6. Lemma G3a

Lemma F.5 (Lemma G3a). The nonzero-frequency contribution produced by G2a on each CKP
gcd layer 𝑔 is a finite sum of weighted bilinear Kloosterman fraction sums

                                                                 ℎ𝑁𝑔 𝑎
                                                                        (︂     )︂
                              𝒪𝑔,ℎ =           𝛽𝑔 (𝑎)𝛾̃︀𝑔,ℎ (𝑞)𝑒
                                           ∑︁
                                                                       ,
                                     𝑎∼𝐴 , 𝑞∼𝑄
                                                                  𝑞
                                           𝑔       𝑔
                                         (𝑎,𝑞)=1

   where
                                                          1 ̂︁         ℎ
                                                                  (︂ )︂
                                       𝛾̃︀𝑔,ℎ (𝑞) = 𝛾𝑔 (𝑞) 𝑊  𝑌          .
                                                          𝑞            𝑞

   This matches the DFI Kloosterman-fraction form with

                                𝑀 = 𝐴𝑔 ,           𝑄 = 𝑄𝑔 ,            𝑘 = ℎ𝑁𝑔 .

   In balanced range,

                                                             𝑁 1/2
                                               𝐴𝑔 ≍ 𝑄𝑔 ≍           ,
                                                              𝑔

   and the coefficient norms satisfy

                                                   𝑔 (log 𝑁 ) ,
                                                             𝐶
                                         ‖𝛽𝑔 ‖2 ≪ 𝐴1/2

                               ‖𝛾̃︀𝑔,ℎ ‖2 ≪𝐴 𝑔(1 + |ℎ|𝑔)−𝐴 𝑄1/2
                                                            𝑔 (log 𝑁 ) .
                                                                      𝐶


Proof. The congruence and Fourier expansion are G2a. Absorbing all one-variable smooth dyadic
weights into 𝛽𝑔 and 𝛾𝑔 , and then absorbing the Fourier factor into 𝛾̃︀𝑔,ℎ , gives exactly the displayed
bilinear Kloosterman fraction sum. The dictionary (𝑚, 𝑛, 𝑘) = (𝑎, 𝑞, ℎ𝑁𝑔 ) is immediate. The
coefficient norm estimates follow from finite-convolution divisor-boundedness and Fourier decay
from G2a. Lemma proved.
   —


                               Remark F.6 (G3a.7. Output).
         G3a converts CKP nonzero frequencies to weighted Kloosterman-fraction sums.
   The parameters are 𝑀 = 𝐴𝑔 , 𝑄 = 𝑄𝑔 , 𝑘 = ℎ𝑁𝑔 , and the coefficient norms match the hypotheses
used by G4a and CKPD.

G3a.8. Logical Dependencies Internal dependencies: G1a and G2a. CKPD and X10 are
downstream smooth-weight and external-estimate inputs consumed by CKPX10M, not inputs to
the algebraic conversion in G3a.
   Children served: G4a, CKPX10M, and X10.




                                                       178
F.4     CKP/X10 smooth-weight derivative appendix
F.4.1     CKPD. CKP/X10 Smooth-Weight Derivative Check
CKPD.0. Role Logical ID: CKPD.
   Used by: G4a, CKPX10M, X10, GEB, I1.
   Uses: G1a, G2a, G3a, C1P/C1A/C1, and DFI Theorem 2. The notation is the CKP interface
notation later consumed by G4a and G8a; CKPD does not use either theorem as an input.
   This appendix supplies the derivative check for the smooth two-variable weight sent from the
CKP branch to the DFI/X10 Kloosterman-fraction external theorem.
   This appendix proves that check in the CKP interface. It should be read together with:

  1. Lemma G2a, which gives the weighted AP Fourier expansion;

  2. Lemma G3a, which keeps the Fourier fibre as a nonseparated two-variable weight;

  3. Lemma G4a, Lemma CKPX10M, and the X10 external input, which consume this derivative
     verification when invoking DFI Theorem 2.

   The conclusion is:


the CKP nonzero-frequency weight is DFI-admissible with only polylogarithmic derivative parameter.

   —

CKPD.1. DFI theorem used by X10 The external input used in this appendix is Theorem 2
of
   W. Duke, J. B. Friedlander, H. Iwaniec, "Bilinear forms with Kloosterman fractions", Invent.
Math. 128 (1997), 23–43, DOI 10.1007/s002220050135,
   together with the smooth-weight formulation stated around formulas (1.7) and (1.8) of that
paper. The X10 input records the same statement. No alternative Kloosterman-fraction estimate is
used as a substitute for this input.
   We use it in the following dyadic form. Let 𝑀, 𝑄 ≥ 1, 𝑟 ≥ 1, and let 𝛼𝑚 , 𝛽𝑞 be arbitrary complex
sequences supported on 𝑚 ≍ 𝑀 , 𝑞 ≍ 𝑄. Let 𝐹 (𝑚, 𝑞) be a smooth weight supported on the same
dyadic box and satisfying

             |𝐹 (𝑚, 𝑞)| ≤ 1,      𝜕𝑞 𝐹 (𝑚, 𝑞) ≪ 𝜂 𝑖+𝑗 𝑀 −𝑖 𝑄−𝑗 ,
                                𝑖 𝑗
                               𝜕𝑚                                       0 ≤ 𝑖, 𝑗 ≤ 2.        (DFI-wt)
   Then, for every 𝜀 > 0,


                                 𝑟𝑚
                               (︂     )︂
                 𝛼𝑚 𝛽𝑞 𝐹 (𝑚, 𝑞)𝑒           ≪𝜀 𝜂 2 ‖𝛼‖2 ‖𝛽‖2 (𝑟 + 𝑀 𝑄)3/8 (𝑀 + 𝑄)11/48+𝜀 .   (DFI-X10)
           ∑︁

        𝑚≍𝑀, 𝑞≍𝑄
                                  𝑞
         (𝑚,𝑞)=1


   In the CKP application, 𝜂 is a fixed power of log 𝑁 . Thus the 𝜂 2 factor is part of the existing
polylogarithmic loss. The purpose of the remaining sections is exactly to prove (DFI-wt) for the
actual nonseparated CKP fibre weight, not for a model separated weight.
   For reference, the CKP substitution into (DFI-X10) is:



                                                     179
   DFI quantity                      CKP quantity                        Verified in
   𝑚∼𝑀                               𝑎 ∼ 𝐴𝑔                              G1a/G8a
   𝑞∼𝑄                               𝑞 ∼ 𝑄𝑔                              G1a/G8a
   (𝑚, 𝑞) = 1                        (𝑎, 𝑞) = 1                          G1a
   𝑟≥1                               𝑟 = |ℎ|𝑁𝑔 , ℎ ̸= 0                  G2a/G3a
   𝛼𝑚 , 𝛽𝑞                           finite-convolution coefficient      G3a/G4a
                                     sequences
   𝐹 (𝑚, 𝑞)                          normalized 𝑊 ̃︁𝑔,ℎ (𝑎, 𝑞)           CKPD.3–CKPD.6



    All noncentral, high-frequency, small-conductor, large-𝑔, and boundary ranges are excluded
before this table is used; they are routed through C1P/C1A/C1 through the excluded-range routing
statement X10ER.
    —

CKPD.2. Parameter and citation check The preceding table is the internal substitution. For
publication use, the following checklist separates what is proved inside the proof package from the
single remaining external citation check.

  DFI hypothesis or pa- CKP realization                 Verification locus        Verification type
  rameter
  dyadic support 𝑚 ≍ 𝑀 , 𝑎 ≍ 𝐴𝑔 , 𝑞 ≍ 𝑄𝑔 after          G1a/G8a                   internal
  𝑞≍𝑄                      the 𝑔-split
  coprimality (𝑚, 𝑞) = 1   (𝑎, 𝑞) = 1 after the         G1a                       internal
                           CKP gcd normalization
  integer parameter 𝑟 ≥ 1 𝑟 = |ℎ|𝑁𝑔 with ℎ ̸= 0         G2a/G3a                   internal
  arbitrary ℓ2 coefficient finite-convolution CKP       G3a/G4a and X10           internal
  sequences                coefficient sequences
  smooth two-variable      ̃︁𝑔,ℎ (𝑎, 𝑞)
                           𝑊                            CKPD.3–CKPD.6             internal
  weight
  derivative order re-     all 𝜕𝑎𝑖 𝜕𝑞𝑗 , 0 ≤ 𝑖, 𝑗 ≤ 2   CKPD.4–CKPD.6             internal
  quired by DFI
  derivative parameter 𝜂 (log 𝑁 )𝐶DFI                   CKPD.6                    internal, charged to the
                                                                                  polylog budget
  excluded ℎ = 0 term     local CKP contribution G8a/LPI, then H4 as-             internal
                                                 sembly
  high frequency, noncen- not sent to DFI/X10    X10ER, C1P/C1A/C1                internal
  tral, boundary, small
  conductor
  exact agreement with    the displayed dyadic   external DFI paper /             external theorem check
  DFI Theorem 2 and       statement (DFI-X10)    X10
  formulas (1.7)–(1.8)



    Thus CKPD proves the smooth-weight and parameter part of the X10 application. It does not
remove X10 as an external dependency: the exact DFI theorem matching remains the external
citation point in the CKP branch.
    —

                                                    180
CKPD.3. Setup: central CKP notation Fix one tagged central balanced CKP layer after the
G1a gcd split. We use the CKP interface notation that is later assembled in G8a:

                                     𝑁
                           𝑁𝑔 =        ,        𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,               (𝑎, 𝑞) = 1,
                                     𝑔

   with

                           𝑎 ≍ 𝐴𝑔 ,           𝑞 ≍ 𝑄𝑔 ,          𝑦 ≍ 𝑌,           𝑦′ ≍ 𝑌 ′,

   and central balance

                                                                          𝑌
                                𝐴𝑔 ≍ 𝑄𝑔 ,                𝑌 ≍ 𝑌 ′,            ≍ 𝑔.              (CB)
                                                                          𝑄𝑔
   The noncentral ranges where any of these relations fails are not sent to X10; they are routed
through X10ER and C1P/C1A/C1 as recorded in G8a and X10.
   Let
                                                                 𝑁𝑔 − 𝑎𝑦
                                       𝑧 = 𝑧(𝑎, 𝑞, 𝑦) :=                 .
                                                                    𝑞

    On this support, 𝑧 ≍ 𝑌 ′ . Let 𝜔𝐴 , 𝜔𝑄 , 𝑊𝑌 , 𝑊𝑌 ′ be the smooth dyadic cutoffs belonging to the
fixed tag. They satisfy, for every fixed 𝑟 ≥ 0,
                                (𝑟)                              (𝑟)
                               𝜔𝐴 (𝑎) ≪𝑟 𝐴−𝑟
                                          𝑔 ,                   𝜔𝑄 (𝑞) ≪𝑟 𝑄−𝑟
                                                                           𝑔 ,


                               (𝑟)                               (𝑟)
                            𝑊𝑌 (𝑦) ≪𝑟 𝑌 −𝑟 ,                 𝑊𝑌 ′ (𝑧) ≪𝑟 (𝑌 ′ )−𝑟 .              (S)
   Any nonsmooth finite-convolution coefficient inherited from B1 is kept in the outer coefficient
sequences 𝛼𝑔 (𝑎), 𝛾𝑔 (𝑞). Thus the smooth object differentiated below is

                           Φ𝑎,𝑞 (𝑦) = 𝜔𝐴 (𝑎)𝜔𝑄 (𝑞)𝑊𝑌 (𝑦)𝑊𝑌 ′ (𝑧(𝑎, 𝑞, 𝑦)).                    (Phi)
   If one chooses to include a smooth coefficient cutoff inside Φ, it satisfies the same derivative
bounds and only changes the final logarithmic constant.
   —

CKPD.4. Exact formula for the DFI weight                         For ℎ ̸= 0, define

                                                         1 ̂︀     ℎ
                                                                       (︂ )︂
                                           𝒲𝑔,ℎ (𝑎, 𝑞) := Φ   𝑎,𝑞   ,
                                                         𝑞        𝑞

   where
                                                    ∫︁
                                     ̂︀ 𝑎,𝑞 (𝜉) =
                                     Φ                    Φ𝑎,𝑞 (𝑦)𝑒(−𝑦𝜉) 𝑑𝑦.                   (FT)
                                                     R
    This is the smooth representative of the discrete transform used in G2a. The standard smooth-
extension convention in G2a routes endpoint discrepancies to C1 boundary errors, so the DFI
derivative check is performed on (FT).
    The nonzero CKP contribution is therefore of the form

                                                          181
                                                                                    ℎ𝑁𝑔 𝑎
                                                                               (︂       )︂
                        𝒪𝑔,ℎ =                 𝛼𝑔 (𝑎)𝛾𝑔 (𝑞)𝒲𝑔,ℎ (𝑎, 𝑞)𝑒                       (CKP-X10)
                                    ∑︁
                                                                                          .
                                 𝑎∼𝐴𝑔 , 𝑞∼𝑄𝑔
                                                                                     𝑞
                                   (𝑎,𝑞)=1

   For DFI, set

                                 𝒜𝑔,ℎ,𝑅 := (log 𝑁 )𝐶* 𝑔(1 + |ℎ|𝑔)−𝑅 ,                             (Agh)
    where 𝐶* is a fixed constant large enough to dominate the dyadic smoothness constants in the
estimates below.
    The normalized DFI weight is

                                   ̃︁𝑔,ℎ (𝑎, 𝑞) := 𝒜−1 𝒲𝑔,ℎ (𝑎, 𝑞),
                                   𝑊                                                            (Wtilde)
                                                    𝑔,ℎ,𝑅

    with 𝑅 chosen later, larger than the fixed number of derivatives and the summation losses.
    Equations (Phi), (FT), (Agh), and (Wtilde) are the complete two-variable weight formula
used in the DFI invocation. The factor 𝒜𝑔,ℎ,𝑅 is not absorbed into the coefficient sequence and
is not discarded; it remains outside the normalized DFI weight and is accounted for in the final
𝑔, ℎ-summation.
    —

CKPD.5. Elementary chain-rule bounds                     On the central support,
                                             𝑦                     𝑧
                                     𝜕𝑎 𝑧 = − ,            𝜕𝑞 𝑧 = − .
                                             𝑞                     𝑞

   Using (CB), (S), and 𝑧 ≍ 𝑌 ′ , we get

                                                          𝑦              𝑌
                                                    (︂        )︂
                           𝜕𝑎 𝑊𝑌 ′ (𝑧) = 𝑊𝑌′ ′ (𝑧) −               ≪           ≪ 𝐴−1
                                                                                  𝑔 ,              (A1)
                                                          𝑞             𝑄𝑔 𝑌 ′
   because 𝐴𝑔 ≍ 𝑄𝑔 and 𝑌 ≍ 𝑌 ′ . Similarly,

                                                                   𝑧
                                                           (︂          )︂
                                 𝜕𝑞 𝑊𝑌 ′ (𝑧) = 𝑊𝑌′ ′ (𝑧) −                  ≪ 𝑄−1
                                                                               𝑔 .                 (Q1)
                                                                   𝑞
   Repeated derivatives are no worse. More precisely, for 0 ≤ 𝑖, 𝑗 ≤ 𝑅0 with fixed 𝑅0 ,

                                 𝜕𝑎𝑖 𝜕𝑞𝑗 𝑊𝑌 ′ (𝑧(𝑎, 𝑞, 𝑦)) ≪𝑅0 𝐴−𝑖 −𝑗
                                                                𝑔 𝑄𝑔 .                            (Zder)
    The proof is by induction and Faa di Bruno. Every 𝑎-derivative of 𝑧 contributes 𝑂(𝑌 /𝑄𝑔 ),
                                             (𝑟)
                                                          𝑔 ). Every 𝑞-derivative of 𝑧 is 𝑂(𝑌 𝑄𝑔 )
and after division by one 𝑌 ′ -scale from 𝑊𝑌 ′ this is 𝑂(𝐴−1                                  ′ −𝑗

at order 𝑗, and the corresponding derivative of 𝑊𝑌 ′ contributes (𝑌 ′ )−1 for each 𝑧-factor, giving
𝑂(𝑄−𝑗𝑔 ). Mixed derivatives combine the two estimates.
    Including the explicit 𝜔𝐴 , 𝜔𝑄 derivatives gives

                                   𝜕𝑎𝑖 𝜕𝑞𝑗 Φ𝑎,𝑞 (𝑦) ≪𝑅0 𝐴−𝑖 −𝑗
                                                         𝑔 𝑄𝑔 1𝑦≍𝑌                              (Phider)
    for 0 ≤ 𝑖, 𝑗 ≤ 𝑅0 , with the same statement after applying any bounded number of 𝑦-derivatives,
at the cost of the expected powers of 𝑌 −1 .
    —



                                                    182
CKPD.6. Fourier decay with parameter derivatives                              For every fixed 𝐵, 𝑖, 𝑗 ≥ 0,
                                                                                 (︃           )︃−𝐵
                                                               −𝑗 𝑌                    |ℎ|𝑌
                  𝜕𝑎𝑖 𝜕𝑞𝑗 𝒲𝑔,ℎ (𝑎, 𝑞) ≪𝐵,𝑖,𝑗 (1 + |ℎ|𝑔)𝑖+𝑗 𝐴−𝑖
                                                            𝑔 𝑄𝑔                    1+               .
                                                                 𝑄𝑔                     𝑄𝑔

   Equivalently, after increasing the constant and using 𝑌 /𝑄𝑔 ≍ 𝑔,

                          𝜕𝑎𝑖 𝜕𝑞𝑗 𝒲𝑔,ℎ (𝑎, 𝑞) ≪𝐵,𝑖,𝑗 𝐴−𝑖 −𝑗
                                                      𝑔 𝑄𝑔 𝑔(1 + |ℎ|𝑔)
                                                                      −𝐵+𝑖+𝑗
                                                                             .                           (Wder-raw)

Proof. Write
                                                     1                       ℎ𝑦
                                                         ∫︁            (︂      )︂
                                𝒲𝑔,ℎ (𝑎, 𝑞) =                 Φ𝑎,𝑞 (𝑦)𝑒 −       𝑑𝑦.
                                                     𝑞                        𝑞
   When |ℎ|𝑔 ≤ 1, no oscillatory integration is needed. The trivial bound
                                     1                                𝑌
                                         ∫︁
                                              |Φ𝑎,𝑞 (𝑦)| 𝑑𝑦 ≪            ≍𝑔
                                     𝑞                                𝑄𝑔
    gives the 𝑖 = 𝑗 = 0 case, and the differentiated version follows from (Phider), together with the
harmless derivatives of 𝑞 −1 and of the phase. Since 1 + |ℎ|𝑔 ≍ 1 in this range, this gives precisely
the right-hand side of (Wder-raw).
    It remains to consider |ℎ|𝑔 > 1, where oscillation is available. First ignore 𝑎, 𝑞-derivatives.
Integrating by parts 𝐵 times in 𝑦, using that every 𝑦-derivative of Φ costs 𝑌 −1 , gives
                                                (︃                  )︃−𝐵
                                       𝑌            |ℎ|𝑌
                      |𝒲𝑔,ℎ (𝑎, 𝑞)| ≪𝐵           1+                        ≪𝐵 𝑔(1 + |ℎ|𝑔)−𝐵 .                  (FD)
                                       𝑄𝑔            𝑄𝑔
    Now differentiate in 𝑎, 𝑞. Derivatives falling on Φ are controlled by (Phider), giving the expected
         𝑔 𝑄𝑔 . Derivatives falling on 𝑞
factors 𝐴−𝑖  −𝑗                           −1 also give powers of 𝑄−1 . Derivatives falling on the phase
                                                                    𝑔
contribute powers of
                                                 |ℎ|𝑌   |ℎ|𝑔
                                                      ≍      ,
                                                  𝑄2𝑔    𝑄𝑔

               𝑔 (1 + |ℎ|𝑔). Thus 𝑖 + 𝑗 total 𝑎, 𝑞-derivatives can lose at most (1 + |ℎ|𝑔)
   which are 𝑄−1                                                                           𝑖+𝑗 from the

Fourier-decay exponent. Repeating the integration-by-parts argument after these differentiations
proves (Wder-raw).
   —


Parameter check F.7 (CKPD.7. Parameter check: DFI-admissibility in the X10 range). Let
𝑅DFI = 2, matching the derivative order required in X10. Choose 𝑅 ≥ 𝑅DFI + 10 in the Fourier
decay step above. In the central X10 range,

                                              |ℎ|𝑔 ≤ (log 𝑁 )𝐵HF .                                             (HF)
   Combining (Wder-raw), (Agh), and (HF), we obtain

                                               ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ 1,
                                               𝑊                                                             (DFI-0)
   and, for 1 ≤ 𝑖 + 𝑗 ≤ 2,


                                                              183
                                       ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ (log 𝑁 )𝐶DFI 𝐴−𝑖 𝑄−𝑗 .
                               𝜕𝑎𝑖 𝜕𝑞𝑗 𝑊                                                   (DFI-der)
                                                                    𝑔   𝑔

    Also 𝑊 ̃︁𝑔,ℎ is supported on the same dyadic box 𝑎 ≍ 𝐴𝑔 , 𝑞 ≍ 𝑄𝑔 , because of 𝜔𝐴 𝜔𝑄 . Thus it
satisfies the smooth-weight hypotheses of the DFI-X10 statement with

                                           𝜂 = (log 𝑁 )𝐶DFI .

   The unnormalized factor 𝒜𝑔,ℎ,𝑅 is not lost. It is kept outside the normalized DFI weight and
charged to the ℎ-summation:

                                 𝒜𝑔,ℎ,𝑅 = (log 𝑁 )𝐶* 𝑔(1 + |ℎ|𝑔)−𝑅 .                         (A-loss)
    Choosing 𝑅 larger than the DFI derivative loss, the 3/8 growth from (|ℎ|𝑁𝑔 + 𝐴𝑔 𝑄𝑔 )3/8 , and
the fixed divisor summation losses leaves an absolutely summable (1 + |ℎ|𝑔)−2 -type tail. This is the
decay used in G4a/CKPX10M/G8a.
    —

CKPD.8. Output for X10 For each central CKP layer and each nonzero frequency in the X10
range, Lemma G3a supplies the weighted Kloosterman form (CKP-X10) with

                               𝑀 = 𝐴𝑔 ,        𝑄 = 𝑄𝑔 ,         𝑟 = |ℎ|𝑁𝑔 .

                                                     ̃︁𝑔,ℎ is DFI-admissible with polylogarithmic
    By (DFI-der), the normalized two-variable weight 𝑊
parameter. Therefore the DFI-X10 invocation in X10 applies to the actual CKP fibre, not merely
to a separated model weight.
    The excluded ranges are unchanged:

  1. ℎ = 0 is the CKP local term handled by G8a/LPI and then assembled by H4M;

  2. |ℎ|𝑔 > (log 𝑁 )𝐵HF is high-frequency Edge;

  3. noncentral balance failures route through X10ER and C1P/C1A/C1;

  4. small-conductor and boundary ranges route to C1P/C1A/C1as recorded in X10.

   Thus the CKP/X10 smooth-weight derivative obligation is discharged.
   —
                              Remark F.8 (CKPD.9. Output).
                CKP/X10 smooth-weight derivative check is proved by CKPD.
   The remaining checks around X10 are ordinary citation verification of the external DFI theorem
and parameter substitution. The internal smooth-weight derivative condition is discharged here.

CKPD.10. Logical Dependencies External dependency: X10 / DFI.
  Internal dependencies: G1a, G2a, G3a, and C1P/C1A/C1.
  Children served: X10, G4a, CKPX10M, GEB, I1.




                                                   184
F.5     G4a DFI matching
F.5.1    G4a. Exact Kloosterman Black-Box Matching
G4a.0. Role Logical ID: G4a.
   Used by: CKPX10M.
   Uses: G1a, G2a, G3a, CKPD, X10, X10ER, C1A, C1.
   Lemma G4a belongs to the CKP branch. Its task is to verify rigorously that the oscillatory
part of a balanced CKP block, after gcd splitting and smooth Fourier expansion, has the exact form
required for the external bilinear Kloosterman-fraction estimate of Duke–Friedlander–Iwaniec.
   In other words, G4a does not prove the external DFI estimate itself. It proves the matching:
our sum has the correct phase, parameters, admissible coefficients, and total contribution

                                                𝑜(𝑁 )

   after summing over 𝑔 and ℎ.
   —

G4a.1. External analytic theorem We use an external estimate for bilinear forms with
Kloosterman fractions. Let
                                                            𝑘𝑚
                                                             (︂    )︂
                               𝐵𝑘 (𝑀, 𝑄) =
                                               ∑︁
                                                    𝛼𝑚 𝛽𝑞 𝑒    ,
                                           𝑚∼𝑀, 𝑞∼𝑄
                                                             𝑞
                                             (𝑚,𝑞)=1

   where 𝑚 is the inverse of 𝑚 modulo 𝑞, and

                                            𝑒(𝑥) = 𝑒2𝜋𝑖𝑥 .

   The working form needed here is

                      𝐵𝑘 (𝑀, 𝑄) ≪𝜀 ‖𝛼‖2 ‖𝛽‖2 (|𝑘| + 𝑀 𝑄)3/8 (𝑀 + 𝑄)11/48+𝜀 .

    Here 𝛼 and 𝛽 are arbitrary complex coefficients. This is important because our coefficients are
finite-convolution coefficients built from 𝜇, 1, and log, and are controlled through their 𝐿2 -norms.
    —

G4a.2. Exact external theorem and formulation check For G4a we fix the concrete external
theorem.
    DFI Theorem 2. In Duke–Friedlander–Iwaniec, "Bilinear forms with Kloosterman fractions",
Invent. Math. 128 (1997), 23–43, DOI 10.1007/s002220050135, Theorem 2 states that, for the
bilinear form
                                                           𝑎𝑚
                                                             (︂   )︂
                               𝐵(𝑀, 𝑁 ) =
                                               ∑︁
                                                  𝛼𝑚 𝛽 𝑛 𝑒    ,
                                          𝑀 <𝑚≤2𝑀
                                                            𝑛
                                            𝑁 <𝑛≤2𝑁
                                            (𝑚,𝑛)=1

   where 𝑚 is the inverse of 𝑚 modulo 𝑛, and 𝛼𝑚 , 𝛽𝑛 are arbitrary complex coefficients, one has

                       𝐵(𝑀, 𝑁 ) ≪𝜀 ‖𝛼‖2 ‖𝛽‖2 (𝑎 + 𝑀 𝑁 )3/8 (𝑀 + 𝑁 )11/48+𝜀 .


                                                 185
   This formulation is compatible with G4a for the following reasons.
   1. The phase matches. In DFI the phase has the form
                                                   𝑎𝑚
                                                  (︂        )︂
                                                 𝑒    .
                                                    𝑛
   In our CKP sum the phase has the form
                                                      ℎ𝑁𝑔 𝑎
                                                 (︂          )︂
                                             𝑒              .
                                                       𝑞
   The parameter correspondence is

                            𝑚 ↔ 𝑎,       𝑛 ↔ 𝑞,             𝑎DFI ↔ |𝑘| = |ℎ|𝑁𝑔 .

   For ℎ < 0, the phase is the complex conjugate of the corresponding positive parameter phase, so
the external DFI theorem is applied with the positive integer parameter |ℎ|𝑁𝑔 .
   2. The coprimality condition matches. DFI requires

                                              (𝑚, 𝑛) = 1.

   In our sum this is exactly

                                              (𝑎, 𝑞) = 1,

    which is required for the existence of 𝑎 (mod 𝑞).
    3. The coefficients are admissible. DFI allows arbitrary complex coefficients 𝛼𝑚 , 𝛽𝑛 and
estimates the sum in terms of their 𝐿2 -norms. Our 𝛽𝑔 (𝑎) and 𝛾𝑔 (𝑞) are finite-convolution coefficients
built from 𝜇, 1, and log, with smooth dyadic weights. They are therefore admissible once their
𝐿2 -norms are estimated.
    4. The ranges are dyadic. DFI works on dyadic intervals. After gcd splitting we have

                                         𝑎 ∼ 𝑆𝑔 ,           𝑞 ∼ 𝑆𝑔 ,

   so

                                            𝑀 = 𝑄 = 𝑆𝑔 .

   5. A large external parameter is allowed. The DFI bound contains the factor

                                            (𝑎 + 𝑀 𝑁 )3/8 ,

   so the external parameter may be large. In our problem
                                                                 ℎ𝑁
                                           𝑘 = ℎ𝑁𝑔 =
                                                                  𝑔
   may exceed
                                                         𝑁
                                                 𝑆𝑔2 =      .
                                                         𝑔2
   This is why the computation produces the factor

                                                      186
                                               (1 + |ℎ|𝑔)3/8 ,

    which is then compensated by the smooth Fourier weight.
    6. The weighted coefficient is admissible. In the separated model we apply DFI not to
𝛾𝑔 (𝑞), but to

                                                          1 ̂︁     ℎ
                                                                  (︂ )︂
                                       𝛾̃︀𝑔,ℎ (𝑞) = 𝛾𝑔 (𝑞) 𝑊  𝑌      .
                                                          𝑞        𝑞

    This is still an arbitrary complex sequence in 𝑞, so DFI applies after estimating its 𝐿2 -norm. In
the CKP interface, the actual weight may be the more general nonseparated weight 𝑊𝑔,ℎ (𝑎, 𝑞). Its
DFI-admissible derivative bounds are proved in CKPD, so the same X10 call applies to the actual
CKP fibre.
    Thus the G4a input is the concrete application of DFI Theorem 2, together with the DFI smooth-
weight formulation recorded in X10 and CKPD.
    —

G4a.3. CKP sum after gcd splitting              A balanced CKP block has the base form

                                              𝑢𝑦 + 𝑢′ 𝑦 ′ = 𝑁.

   Write

                       𝑔 = gcd(𝑢, 𝑢′ ),       𝑢 = 𝑔𝑎,       𝑢′ = 𝑔𝑞,          (𝑎, 𝑞) = 1.

   If 𝑔 ∤ 𝑁 , there are no solutions. If 𝑔 | 𝑁 , write

                                                         𝑁
                                                 𝑁𝑔 =      .
                                                         𝑔

   Then the equation becomes

                                              𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 .

   Solve the congruence in 𝑦:

                                             𝑎𝑦 ≡ 𝑁𝑔 (mod 𝑞).

   Since (𝑎, 𝑞) = 1, this gives

                                             𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞).

   For the smooth weight 𝑊𝑌 , use the Fourier expansion for counting points in an arithmetic
progression:

                                                      1 ∑︁ ̂︁       ℎ   ℎ𝑟
                                                                   (︂ )︂ (︂     )︂
                                           𝑊𝑌 (𝑦) =
                                  ∑︁
                                                            𝑊𝑌        𝑒    .
                             𝑦≡𝑟 (mod 𝑞)
                                                      𝑞 ℎ∈Z         𝑞   𝑞

   The term ℎ = 0 gives the local/main term. The terms ℎ ̸= 0 give the oscillatory contribution.
   For fixed 𝑔 this gives

                                                      187
                                                        1 ̂︁                    ℎ   ℎ𝑁𝑔 𝑎
                                                                              (︂ )︂ (︂         )︂
                      𝒪𝑔 =                  𝛽𝑔 (𝑎)𝛾𝑔 (𝑞) 𝑊
                             ∑︁       ∑︁
                                                            𝑌                     𝑒       .
                             ℎ̸=0 𝑎∼𝐴 , 𝑞∼𝑄
                                                        𝑞                       𝑞    𝑞
                                      𝑔       𝑔
                                   (𝑎,𝑞)=1

   In the balanced range,

                                                                              𝑁 1/2
                                  𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,             𝑆𝑔 =                   .
                                                                               𝑔
   —

G4a.4. Why the Fourier weight must be included in the coefficient The external phase
parameter is
                                                             ℎ𝑁
                                             𝑘 = ℎ𝑁𝑔 =          .
                                                              𝑔
   It can be larger than
                                                            𝑁
                                                  𝐴𝑔 𝑄𝑔 ≍      .
                                                            𝑔2
   Therefore it is not safe to first estimate the bare Kloosterman-fraction sum and then multiply
separately by the Fourier weight. The correct matching is performed directly for the weighted DFI-
form sum.
   Define the new coefficient in the 𝑞-variable by
                                                         1 ̂︁             ℎ
                                                                     (︂ )︂
                                      𝛾̃︀𝑔,ℎ (𝑞) = 𝛾𝑔 (𝑞) 𝑊  𝑌              .
                                                         𝑞                𝑞
   Then
                                                                                   ℎ𝑁𝑔 𝑎
                                                                              (︂        )︂
                             𝒪𝑔,ℎ =                 𝛽𝑔 (𝑎)𝛾̃︀𝑔,ℎ (𝑞)𝑒
                                           ∑︁
                                                                                         .
                                      𝑎∼𝑆𝑔 , 𝑞∼𝑆𝑔
                                                                                    𝑞
                                        (𝑎,𝑞)=1

   This is exactly the form 𝐵𝑘 (𝑀, 𝑄) with parameters

                                                                                        ℎ𝑁
                             𝑀 = 𝑆𝑔 ,         𝑄 = 𝑆𝑔 ,           𝑘 = ℎ𝑁𝑔 =                 .
                                                                                         𝑔
   —

G4a.5. 𝐿2 -norms of the coefficients Finite-convolution coefficients built from 𝜇, 1, and log
are divisor-bounded. Hence

                                          ‖𝛽𝑔 ‖2 ≪ 𝑆𝑔1/2 (log 𝑁 )𝐶 ,

                                          ‖𝛾𝑔 ‖2 ≪ 𝑆𝑔1/2 (log 𝑁 )𝐶 .

   Now estimate the Fourier weight. Let
                                                                 𝑦
                                                            (︂       )︂
                                            𝑊𝑌 (𝑦) = 𝑊                    ,
                                                                 𝑌

                                                     188
  where 𝑊 ∈ 𝐶𝑐∞ . Then
                                                      ̂︁ (𝑌 𝜉).
                                          ̂︁𝑌 (𝜉) = 𝑌 𝑊
                                          𝑊

  Therefore
                                   ⃒1
                                   ⃒              ⃒ (︂
                                      ̂︁𝑌 ℎ ⃒ = 𝑌 ⃒𝑊̂︁ ℎ𝑌 ⃒ .
                                         (︂ )︂⃒          )︂⃒
                                              ⃒   ⃒        ⃒
                                   ⃒ 𝑊
                                   ⃒𝑞      𝑞  ⃒ 𝑞 ⃒     𝑞 ⃒
  On a balanced 𝑔-layer,

                                               𝑁 1/2
                                  𝑞 ≍ 𝑆𝑔 =           ,             𝑌 ≍ 𝑁 1/2 ,
                                                𝑔
  and therefore
                                                    𝑌
                                                      ≍ 𝑔.
                                                    𝑞
                     ̂︁ gives, for every 𝑅 > 0,
  The rapid decay of 𝑊
                                  ⃒1
                                  ⃒
                                     ̂︁𝑌 ℎ ⃒ ≪𝑅 𝑔(1 + |ℎ|𝑔)−𝑅 .
                                        (︂ )︂⃒
                                             ⃒
                                  ⃒ 𝑊
                                  ⃒𝑞      𝑞 ⃒
  Hence

                             ‖𝛾̃︀𝑔,ℎ ‖2 ≪𝑅 𝑔(1 + |ℎ|𝑔)−𝑅 𝑆𝑔1/2 (log 𝑁 )𝐶 .

  —

G4a.6. Application of the DFI theorem                    By the external DFI estimate,
                                                                      )︂3/8
                                                         |ℎ|𝑁
                                                    (︂
                      |𝒪𝑔,ℎ | ≪ ‖𝛽𝑔 ‖2 ‖𝛾̃︀𝑔,ℎ ‖2             + 𝑆𝑔2           (2𝑆𝑔 )11/48+𝜀 .
                                                           𝑔
  Substitute the coefficient norms:

                            ‖𝛽𝑔 ‖2 ‖𝛾̃︀𝑔,ℎ ‖2 ≪ 𝑔(1 + |ℎ|𝑔)−𝑅 𝑆𝑔 (log 𝑁 )𝐶 .

  Since

                                                          𝑁 1/2
                                               𝑆𝑔 =             ,
                                                           𝑔
  we have

                                              𝑔𝑆𝑔 = 𝑁 1/2 .

  Moreover,
                                                            𝑁
                                                𝑆𝑔2 =          ,
                                                            𝑔2
  and

                                                         189
                                         |ℎ|𝑁        𝑁
                                              + 𝑆𝑔2 = 2 (1 + |ℎ|𝑔).
                                           𝑔         𝑔

  Therefore
                                               )︂3/8
                                |ℎ|𝑁
                           (︂
                                     + 𝑆𝑔2             = 𝑁 3/8 𝑔 −3/4 (1 + |ℎ|𝑔)3/8 .
                                  𝑔

  Also

                                          𝑆𝑔11/48 = 𝑁 11/96 𝑔 −11/48 .

  Thus

               |𝒪𝑔,ℎ | ≪ 𝑁 1/2 𝑁 3/8 𝑁 11/96 𝑔 −3/4 𝑔 −11/48 (1 + |ℎ|𝑔)−𝑅+3/8 (log 𝑁 )𝐶 .

  The exponents of 𝑁 add to

                                  1 3 11   48 36 11  95
                                   + +   =   +  +   = .
                                  2 8 96   96 96 96  96
  The exponents of 𝑔 add to

                                     3 11   36 11   47
                                    − −   =− −    =− .
                                     4 48   48 48   48
  We obtain

                           |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝜀 𝑔 −47/48 (1 + |ℎ|𝑔)−𝑅+3/8 .

  The logarithmic factors are absorbed into 𝑁 𝜀 .
  —

G4a.7. Summation over ℎ           Take

                                                        𝑅 = 2.

  Then
                                                          3   13
                                                −𝑅 +        =− .
                                                          8   8
  Also

                                             (1 + |ℎ|𝑔)−13/8 ≪ 𝑔 −13/8 .
                                        ∑︁

                                        ℎ̸=0

  Therefore

                                        |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝜀 𝑔 −47/48 𝑔 −13/8 .
                                 ∑︁

                                 ℎ̸=0

  Since


                                                         190
                                                    13  78
                                                       = ,
                                                     8  48
   we get
                                               47 78    125
                                           −     −   =−     .
                                               48 48     48
   Thus

                                         |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝜀 𝑔 −125/48 .
                                  ∑︁

                                  ℎ̸=0

   —

G4a.8. Summation over 𝑔         Since
                                                    125
                                                        > 1,
                                                    48
   the series

                                                        𝑔 −125/48
                                                  ∑︁

                                                  𝑔≥1

   converges. Hence
                                    ∑︁ ∑︁
                                                  |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝜀 .
                                         𝑔 ℎ̸=0

   Choose
                                                           1
                                                    𝜀<        .
                                                           96
   Then

                                            𝑁 95/96+𝜀 = 𝑜(𝑁 ).

   Consequently,

                                                    𝒪𝑔 = 𝑜(𝑁 ).
                                               ∑︁

                                                𝑔

   —

G4a.9. The term ℎ = 0 The term ℎ = 0 is not an error. It is the zero Fourier frequency and
gives the CKP local/main contribution:

                                    ℎ=0             =⇒        𝑀CKP (𝑁 ).

   All terms with ℎ ̸= 0 contribute 𝑜(𝑁 ). Thus, at the oscillatory analysis level,

                                    CKP = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ).

   —

                                                        191
G4a.10. Coprimality and conductor issue                        The condition

                                                     (𝑎, 𝑞) = 1

    is present in our sum and is required to define 𝑎 (mod 𝑞). It matches the coprimality condition
in the external DFI Kloosterman-fraction estimate.
    The external numerator

                                                     𝑘 = ℎ𝑁𝑔

    may have a common divisor with 𝑞. This does not break the matching, because the DFI theorem
estimates phases of the form

                                                               𝑘𝑎
                                                          (︂        )︂
                                                      𝑒
                                                                𝑞

     with arbitrary integer 𝑘; coprimality is required between the inverted variable 𝑎 and the modulus
𝑞.
     If one further decomposes by conductor
                                                             𝑞
                                                 𝑞1 =               ,
                                                          gcd(𝑞, 𝑘)

    then small-conductor layers are already covered by the C1 Edge estimate, while large-conductor
layers remain in the same DFI form. Thus conductor splitting does not create a new unresolved
class.
    —

G4a.11. Final statement of Lemma G4a Suppose that after CKP reduction one obtains an
oscillatory weighted Kloosterman-fraction sum of the DFI form

                                                        1 ̂︁                  ℎ   ℎ𝑁𝑔 𝑎
                                                                             (︂ )︂ (︂   )︂
                        𝒪=                  𝛽𝑔 (𝑎)𝛾𝑔 (𝑞) 𝑊
                             ∑︁ ∑︁      ∑︁
                                                            𝑌                   𝑒       ,
                           𝑔 ℎ̸=0 𝑎∼𝑆 , 𝑞∼𝑆             𝑞                     𝑞    𝑞
                                        𝑔        𝑔
                                       (𝑎,𝑞)=1

     where

                                                 𝑁 1/2                      𝑁
                                       𝑆𝑔 =            ,             𝑁𝑔 =     ,
                                                  𝑔                         𝑔

                                     ‖𝛽𝑔 ‖2 , ‖𝛾𝑔 ‖2 ≪ 𝑆𝑔1/2 (log 𝑁 )𝐶 ,

     and 𝑊 ∈ 𝐶𝑐∞ . Then, using the DFI theorem for bilinear Kloosterman fractions,

                                                     𝒪 = 𝑜(𝑁 ).

     More precisely,

                                         |𝒪| ≪ 𝑁 95/96+𝜀 = 𝑜(𝑁 )

     for every sufficiently small fixed 𝜀 > 0.
     —

                                                          192
                              Remark F.9 (G4a.12. Output).
         G4a matches the central CKP nonzero-frequency sums to the X10/DFI input.
   This gives:
  1. the matching with the DFI Kloosterman-fraction form succeeds;
  2. the coefficients satisfy the required 𝐿2 -norm bounds;
  3. the large parameter ℎ𝑁/𝑔 is correctly compensated by the smooth Fourier weight;
  4. summation over ℎ ̸= 0 and 𝑔 gives 𝑜(𝑁 );
  5. ℎ = 0 remains a local/main term;
  6. the only deep external dependency is the DFI bilinear Kloosterman-fraction estimate recorded
     as X10.
   The central CKP nonzero-frequency sums satisfy the DFI/X10 hypotheses after the parameter
matching in X10. The actual nonseparated smooth fibre weight is DFI-admissible by CKPD,
and all excluded ranges route through X10ER, C1P/C1A/C1, and G2a. The CKP/X10 master
theorem CKPX10M consumes this matching result and packages it with the excluded-range routing
and 𝑔, ℎ-summation; G8a then consumes CKPX10M after the local zero-frequency mode has been
separated.

G4a.13. Logical Dependencies External dependency: X10 / DFI.
   Internal dependencies: G1a, G2a, G3a, CKPD, X10ER, C1A, and C1. The theorem CKPX10M is
the immediate downstream consumer; G8a consumes CKPX10M, not G4a as a separate premise.
   Children served: CKPX10M and the CKP branch closure.

F.6     CKPX10M master CKP/X10 nonzero-frequency theorem
F.6.1    CKPX10M. Master CKP/X10 Nonzero-Frequency Theorem
CKPX10M.0. Statement and Role Logical ID: CKPX10M.
    Lemma CKPX10M is the reader-facing master theorem for the CKP/X10 nonzero-frequency
interface. It packages the structural CKP reductions
                                        G1a,    G2a,   G3a,
   the actual smooth-weight derivative verification
                                               CKPD,
   the DFI/X10 theorem matching
                                           G4a + X10,
   and the excluded-range routing record
                                     X10ER + C1P/C1A/C1
   into one autonomous nonzero-frequency statement.
   The theorem does not handle the zero Fourier frequency. The ℎ = 0 CKP term is the local CKP
contribution and is normalized separately by G8a through the LPI tagged local projection before
H4 assembles it.
   —

                                                193
CKPX10M.1. Setup Fix a tagged CKP atom produced by the B1/B3/F3/F4 routing interface.
After the G1a gcd split one has

                              𝑢 = 𝑔𝑎,           𝑢′ = 𝑔𝑞,         (𝑎, 𝑞) = 1,              𝑔 | 𝑁,

   and
                                                                                𝑁
                                            𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,           𝑁𝑔 =       .
                                                                                𝑔

   In the central balanced CKP range,

                                                                       𝑁 1/2+𝑂(𝜂)
                   𝑎 ≍ 𝐴𝑔 ,        𝑞 ≍ 𝑄𝑔 ,         𝐴𝑔 ≍ 𝑄𝑔 ≍                     .                       (CKPX10M-CB)
                                                                            𝑔
    The weighted smooth AP expansion of G2a separates the frequency ℎ = 0 from the nonzero
frequencies ℎ ̸= 0. For ℎ ̸= 0, G3a writes the relevant oscillatory contribution in the form

                                                            ℎ𝑁𝑔 𝑎
                                                                        (︂           )︂
                  𝒪𝑔,ℎ =           𝛼𝑔 (𝑎)𝛾𝑔 (𝑞)𝒲𝑔,ℎ (𝑎, 𝑞)𝑒                                               (CKPX10M-raw)
                               ∑︁
                                                                  .
                         𝑎∼𝐴 , 𝑞∼𝑄
                                                             𝑞
                               𝑔        𝑔
                              (𝑎,𝑞)=1

   Here 𝛼𝑔 , 𝛾𝑔 are finite-convolution coefficient sequences inherited from B1 and the CKP routing,
and 𝒲𝑔,ℎ is the actual two-variable smooth Fourier fibre weight, not a separated surrogate.
   CKPD gives the exact formula

                                                           1 ̂︀     ℎ
                                                                        (︂ )︂
                                              𝒲𝑔,ℎ (𝑎, 𝑞) = Φ   𝑎,𝑞   ,                                   (CKPX10M-W)
                                                           𝑞        𝑞
   where

                                                                      𝑁𝑔 − 𝑎𝑦
                                                                 (︂             )︂
                    Φ𝑎,𝑞 (𝑦) = 𝜔𝐴 (𝑎)𝜔𝑄 (𝑞)𝑊𝑌 (𝑦)𝑊𝑌 ′                         .                           (CKPX10M-Phi)
                                                                         𝑞
   For a fixed large 𝑅, CKPD normalizes this weight by


         𝒲𝑔,ℎ (𝑎, 𝑞) = 𝒜𝑔,ℎ,𝑅 𝑊
                              ̃︁𝑔,ℎ (𝑎, 𝑞),         𝒜𝑔,ℎ,𝑅 = (log 𝑁 )𝐶* 𝑔(1 + |ℎ|𝑔)−𝑅 .                    (CKPX10M-A)

                         ̃︁𝑔,ℎ is supported on the same dyadic box and satisfies the DFI smooth-
   The normalized weight 𝑊
weight derivative bounds


   ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ 1,
   𝑊                               ̃︁𝑔,ℎ (𝑎, 𝑞) ≪ (log 𝑁 )𝐶DFI 𝐴−𝑖 𝑄−𝑗 ,
                           𝜕𝑎𝑖 𝜕𝑞𝑗 𝑊                                                      0 ≤ 𝑖, 𝑗 ≤ 2.   (CKPX10M-der)
                                                                𝑔   𝑔

    The amplitude 𝒜𝑔,ℎ,𝑅 remains outside the normalized DFI weight and is included in the final
𝑔, ℎ-summation.
    —




                                                           194
CKPX10M.2. Master Theorem
Theorem F.10 (Theorem CKPX10M. CKP/X10 nonzero-frequency cancellation). For every tagged
central CKP atom, the nonzero-frequency contribution satisfies

                                                   𝒪𝑔,ℎ = 𝑜(𝑁 ).                     (CKPX10M-NZ)
                                        ∑︁ ∑︁

                                        𝑔|𝑁 ℎ̸=0

   Moreover, every CKP nonzero-frequency layer outside the central X10 range is routed before the
DFI estimate is invoked:


       CKPℎ̸=0 = CentralDFI ⊔ HighFreq ⊔ SmallConductor ⊔ LargeG ⊔ Boundary/Short,
                                                                        (CKPX10M-split)
   with

                                        CentralDFI → 𝑋10,

   and all other displayed classes routed through X10ER and the strict Edge interface C1P/C1A/
C1, or declared empty by the CKP gcd and dyadic support conditions.
   Thus X10 leaves no residual CKP terminal class.
   —

CKPX10M.3. Proof Step 1: exact CKP normalization. G1a gives the exact gcd split
𝑢 = 𝑔𝑎, 𝑢′ = 𝑔𝑞, the condition 𝑔 | 𝑁 for nonempty CKP layers, and the coprimality (𝑎, 𝑞) = 1. G2a
applies the weighted smooth AP Fourier expansion to the full tagged CKP fibre. The zero frequency
                                                 ̸ 0, G3a keeps the actual nonseparated Fourier
ℎ = 0 is separated and is not sent to X10. For ℎ =
fibre and obtains (CKPX10M-raw).
    Step 2: DFI dictionary. The DFI Kloosterman-fraction variables are

          𝑚 = 𝑎,       𝑛 = 𝑞,      𝑀 = 𝐴𝑔 ,         𝑄 = 𝑄𝑔 ,       𝑟 = |ℎ|𝑁𝑔 .      (CKPX10M-dict)
   The phase
                                                    ℎ𝑁𝑔 𝑎
                                               (︂           )︂
                                              𝑒
                                                     𝑞
   therefore matches the DFI phase 𝑒(𝑟𝑚/𝑛). If ℎ < 0, the phase is the complex conjugate of the
positive-parameter phase, so the same external estimate is applied with 𝑟 = |ℎ|𝑁𝑔 . The coprimality
condition required by DFI is exactly (𝑎, 𝑞) = 1, already supplied by G1a.
   Step 3: actual smooth weight. The DFI input is not applied to a separated surrogate. CKPD
proves (CKPX10M-W), (CKPX10M-Phi), and the normalized derivative bounds (CKPX10M-der)
by direct chain-rule estimates for
                                                       𝑁𝑔 − 𝑎𝑦
                                        𝑧(𝑎, 𝑞, 𝑦) =           .
                                                          𝑞

   On the central support 𝑧 ≍ 𝑌 ′ , 𝐴𝑔 ≍ 𝑄𝑔 , and 𝑌 ≍ 𝑌 ′ . Thus differentiating 𝑊𝑌 ′ (𝑧) in 𝑎 gives an
                                                    𝑔 -scale factor. Fourier decay gives the amplitude
𝐴𝑔 -scale factor, and differentiating in 𝑞 gives a 𝑄−1
 −1

𝒜𝑔,ℎ,𝑅 , which remains visible in (CKPX10M-A).
   Step 4: central DFI estimate. In the central range G4a applies the DFI theorem recorded
as X10 to the normalized weighted sum. With

                                                    195
                                                              𝑁 1/2+𝑂(𝜂)
                                𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑆𝑔 ,         𝑆𝑔 =              ,
                                                                   𝑔

   and with the amplitude in (CKPX10M-A), the one-layer estimate is

            |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝜀+𝑂(𝜂) (log 𝑁 )𝐶 𝑔 −47/48 (1 + |ℎ|𝑔)−𝑅+3/8 .      (CKPX10M-layer)
   The exponent 95/96 is the DFI exponent produced by

                                              1 3 11  95
                                               + +   = .
                                              2 8 96  96

  The factor (1 + |ℎ|𝑔)−𝑅+3/8 is the Fourier-amplitude decay after paying for the DFI 𝑟-dependence.
  **Step 5: summation over ℎ and 𝑔.** Choose 𝑅 larger than the fixed derivative, divisor, and
DFI 𝑟-growth losses. Then

                                              (1 + |ℎ|𝑔)−𝑅+3/8 ≪ 1
                                         ∑︁

                                         ℎ̸=0

   uniformly with room to spare. Since 𝑔 | 𝑁 , the number of possible 𝑔-layers is divisor-bounded
and hence contributes 𝑁 𝑜(1) . Therefore

                                       |𝒪𝑔,ℎ | ≪ 𝑁 95/96+𝑂(𝜂)+𝜀+𝑜(1) = 𝑜(𝑁 ),
                            ∑︁ ∑︁

                            𝑔|𝑁 ℎ̸=0


   after fixing 𝜂 and the DFI 𝜀 sufficiently small, for example so that 𝑂(𝜂) + 𝜀 < 1/100.
   Step 6: excluded ranges. The preceding DFI invocation is deliberately restricted to the
central balanced nonzero-frequency range. X10ER records the complementary partition:

  • high Fourier frequencies are Edge by Fourier decay;

  • small-conductor layers are Edge in the CKP-normalized oscillatory scale;

  • large-𝑔 or large-content layers have gcd/content savings or are Edge;

  • boundary and short-volume layers satisfy strict Edge predicates;

  • empty layers remain empty after the exact G1a gcd split.

   Each noncentral nonzero-frequency layer is therefore checked against C1P, admitted by C1A,
and estimated by C1 before X10 is invoked. No noncentral class is left for DFI and no new CKP
terminal residual is created.
   Combining the central estimate with the excluded-range routing proves (CKPX10M-NZ) and
(CKPX10M-split). The theorem is proved.
   —

Parameter check F.11 (CKPX10M.4. Parameter Check). No new external theorem is introduced.
The only external analytic input is X10, the Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction
estimate in the dyadic smooth-weight form stated in X10 and CKPD.
    The internal parameter requirements are:

  1. the B1 depth and divisor losses are fixed in PAR;


                                                    196
  2. the CKP central balance 𝐴𝑔 ≍ 𝑄𝑔 ≍ 𝑁 1/2+𝑂(𝜂) /𝑔 is fixed before X10 is invoked;

  3. the derivative parameter in (CKPX10M-der) is polylogarithmic and charged to the global
     error budget;

  4. 𝑅 in (CKPX10M-A) is chosen once, larger than all fixed derivative and summation losses;

  5. 𝜂 and the DFI 𝜀 are chosen so that 95/96 + 𝑂(𝜂) + 𝜀 < 1.

   All excluded-range losses are charged to C1 through C1P/C1A, and all local ℎ = 0 terms are
outside CKPX10M and are handled by G8a/LPI/H4.
   —

CKPX10M.5. Interface Corollary
Corollary F.12 (Corollary CKPX10M.1. Nonzero-frequency input for G8a). G8a may import the
CKP nonzero-frequency conclusion through the single statement
                                                     (ℎ)
                             CKPX10M =⇒             𝑅CKP (𝑁 ) = 𝑜(𝑁 ),
                                             ∑︁

                                             ℎ̸=0

   with all central DFI hypotheses, actual smooth-weight derivative checks, 𝑔, ℎ-loss accounting,
and excluded-range routing already included.
   Together with the G8a zero-frequency local normalization,

                             ℎ=0      =⇒    Loc𝑄 𝑅ℬ,𝜏
                                                  CKP
                                                      (𝑁 ) + 𝑜(𝑁 ),

   this gives the CKP branch output

                                 𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ),

   where 𝑀CKP (𝑁 ) is LPI-admissible and is later assembled by H4M.
   —

CKPX10M.6. Logical Dependencies External dependency: X10.
   Internal dependencies: B1, B3, F3P, F3, F4, G1a, G2a, G3a, CKPD, G4a, X10ER, C1P, C1A,
C1, and PAR.
   Children served: G8a, GEB, I1, and the full proof assembly.

F.7     G8a CKP theorem
F.7.1    G8a. CKP Theorem and Zero-Frequency Normalization
G8a.0. Role Logical ID: G8a.
   Used by: H4, H4M, I1, CKP branch closure.
   Uses: G1a, G2a, CKPX10M, C1A, C1, B1LD, and LPI.
   Lemma G8a closes the CKP branch of the proof tree. For compatibility with the LPI local
projection interface later assembled by H4M, the schematic formulation

                                 𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 )

   is no longer sufficient. One must prove the sharper statement

                                              197
                               𝑀CKP,ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 )

   for every tagged CKP atom (ℬ, 𝜏 ).
   Otherwise the local/main assembly is not entitled to accept the CKP zero-frequency term.
   Thus G8a proves two things:

  1. zero-frequency normalization:

                                ℎ=0          =⇒        Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜(𝑁 );

  1. nonzero-frequency cancellation:

                                                       𝒪𝑔,ℎ = 𝑜(𝑁 )
                                                ∑︁

                                                ℎ̸=0

   after summing over all relevant CKP layers, using CKPX10M.
   —

G8a.1. Tagged CKP atom             Let (ℬ, 𝜏 ) be a tagged CKP atom produced by

                                             𝐵1 → 𝐵3 → 𝐹 3/𝐹 4.

   It has the schematic form

                 (𝑁 ) =                  𝛼(𝑢)𝛼′ (𝑢′ )𝛽(𝑦)𝛽 ′ (𝑦 ′ )𝑊𝑈 (𝑢)𝑊𝑈 ′ (𝑢′ )𝑊𝑌 (𝑦)𝑊𝑌 ′ (𝑦 ′ ),
                              ∑︁
             CKP
            𝑅ℬ,𝜏
                          𝑢𝑦+𝑢′ 𝑦 ′ =𝑁

   where:

  • 𝑢, 𝑢′ are balanced finite-convolution grouped variables;

  • 𝑦, 𝑦 ′ are complementary variables;

  • coefficients are divisor-bounded finite-convolution sequences inherited from B1;

  • all weights and ranges are tagged by (ℬ, 𝜏 );

  • the CKP balance regime gives

                          𝑈 ≍ 𝑈 ′ ≍ 𝑁 1/2+𝑂(𝜂) ,               𝑌 ≍ 𝑌 ′ ≍ 𝑁 1/2+𝑂(𝜂) .

    The tag (ℬ, 𝜏 ) is fixed throughout. This ensures compatibility with the LPI local projection
interface.
    —




                                                         198
G8a.2. GCD splitting           By Lemma G1a, write

                                    𝑢 = 𝑔𝑎,            𝑢′ = 𝑔𝑞,            (𝑎, 𝑞) = 1.

   Then a necessary condition for a nonempty layer is

                                                            𝑔 | 𝑁,

   and after putting

                                                                𝑁
                                                        𝑁𝑔 =      ,
                                                                𝑔

   we obtain the reduced equation

                                       𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 ,                (𝑎, 𝑞) = 1.

   Thus

                                              (𝑁 ) =                        (𝑁 ).
                                                              ∑︁
                                          CKP                         CKP
                                         𝑅ℬ,𝜏                        𝑅ℬ,𝜏,𝑔
                                                              𝑔|𝑁

    If a gcd layer with 𝑔 ∤ 𝑁 appears during the formal gcd split, its equation 𝑔𝑎 𝑦 + 𝑔𝑞 𝑦 ′ = 𝑁 has
empty support. The layer is not silently discarded: it carries the inherited tag (ℬ, 𝜏, 𝑔), contributes
zero, and is terminal Edge of zero effective volume. Thus the B3 CKP predicate remains a scale-
structural predicate; divisibility by 𝑁 is handled inside the exact G1a/G8a gcd decomposition.
    Large 𝑔-layers outside the balanced CKP range are routed by X10ER to C1P/C1A/C1and
contribute 𝑜(𝑁 ). Hence it suffices to treat the balanced range.
    —

G8a.3. Weighted smooth AP expansion                           For fixed 𝑔, 𝑎, 𝑞, the reduced equation is

                                                     𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔 .

   Eliminate 𝑦 ′ :

                                                             𝑁𝑔 − 𝑎𝑦
                                                     𝑦′ =            ,
                                                                𝑞

   and impose

                                                𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞).

   The weighted fibre is

                                                            𝑁𝑔 − 𝑎𝑦                            𝑁𝑔 − 𝑎𝑦
                                                       (︂             )︂                  (︂           )︂
                     𝒮𝑎,𝑞 =                  𝛽(𝑦)𝛽 ′                𝑊𝑌 (𝑦)𝑊𝑌 ′
                                  ∑︁
                                                                                                       .
                              𝑦≡𝑁𝑔 𝑎 mod 𝑞
                                                               𝑞                                  𝑞

   Define the smooth tagged fibre weight

                                                     𝑁𝑔 − 𝑎𝑦                             𝑁𝑔 − 𝑎𝑦
                                                (︂             )︂                   (︂            )︂
                        𝐹𝑎,𝑞 (𝑦) = 𝛽(𝑦)𝛽 ′                   𝑊𝑌 (𝑦)𝑊𝑌 ′                          ,
                                                        𝑞                                   𝑞

                                                             199
    with the convention that the summand is zero unless (𝑁𝑔 − 𝑎𝑦)/𝑞 ∈ Z and lies in the dyadic
support.
    The dependence of 𝐹𝑎,𝑞 on both 𝑎 and 𝑞 is part of the object sent to G3a/X10. The derivative
check for the normalized smooth Fourier weight is supplied by CKPD; the local chain-rule calculation
is summarized here. On the dyadic support (𝑁𝑔 − 𝑎𝑦)/𝑞 ≍ 𝑌 ′ ; hence
                                     𝑁𝑔 − 𝑎𝑦           𝑁𝑔 − 𝑎𝑦 ′ 𝑁𝑔 − 𝑎𝑦
                                (︂             )︂                            (︂          )︂
                      𝜕𝑞 𝑊𝑌 ′                       =−        𝑊𝑌 ′                            ≪ 𝑄−1
                                                                                                 𝑔 ,
                                        𝑞                𝑞2         𝑞
              (1)
after using 𝑊𝑌 ′ ≪ (𝑌 ′ )−1 and 𝑞 ≍ 𝑄𝑔 . Similarly, 𝜕𝑎 produces (𝑦/𝑞)𝑊𝑌′ ′ , which is admissible in the
central balanced CKP range 𝑌 ≍ 𝑌 ′ , 𝐴𝑔 ≍ 𝑄𝑔 . Noncentral ranges are not sent to X10; they are
among the X10ER and C1P/C1A/C1 routed exclusions. Thus the smooth weight may be treated as
a genuine two-variable DFI weight, not as a separated one-variable factor.
    For the local/Fourier splitting, the smooth part is expanded by additive characters:

                                              1 ∑︁ ̂︀                 ℎ   ℎ𝑁𝑔 𝑎
                                                                 (︂ )︂ (︂           )︂
                                       𝒮𝑎,𝑞 =       𝐹𝑎,𝑞                𝑒       ,
                                              𝑞 ℎ∈Z                   𝑞    𝑞

   where the smooth Fourier transform satisfies rapid decay inherited from the dyadic weights. Any
nonsmooth bounded finite-convolution coefficient that cannot be included into 𝐹𝑎,𝑞 is kept in the
outer coefficient sequence and is handled in G3a/G4a as divisor-bounded weight.
   This is the weighted version of the G2a step. It treats the full tagged CKP fibre rather than a
bare sum over 𝑊𝑌 (𝑦) only.
   —

G8a.4. Zero-frequency term               The zero-frequency contribution is

                                              1             1 ∑︁
                                         (0)
                                        𝒮𝑎,𝑞 = 𝐹̂︀𝑎,𝑞 (0) =      𝐹𝑎,𝑞 (𝑦).
                                              𝑞             𝑞 𝑦

   Therefore the tagged CKP zero-frequency contribution is

                             CKP,0                                                1
                                   (𝑁 ) =                             𝛼𝑔 (𝑎)𝛾𝑔 (𝑞) 𝐹̂︀𝑎,𝑞 (0),
                                                     ∑︁ ∑︁
                            𝑀ℬ,𝜏
                                                     𝑔|𝑁     𝑎,𝑞                  𝑞
                                                           (𝑎,𝑞)=1


   with all dyadic weights and tags inherited from (ℬ, 𝜏 ).
   This expression is local because it contains no oscillatory phase

                                                                ℎ𝑁𝑔 𝑎
                                                           (︂           )︂
                                                       𝑒
                                                                 𝑞

   with ℎ ̸= 0.
   However, LPI-admission requires more: this local term must equal the canonical tagged local
projection that H4 later assembles.
   —




                                                                200
G8a.5. CKP zero-frequency equals the LPI tagged local projection

Lemma F.13 (Lemma G8a.1). For every tagged CKP atom (ℬ, 𝜏 ),
                                CKP,0
                               𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                   CKP
                                                       (𝑁 ) + 𝑜(𝑁 ).

Proof. The LPI tagged local projection Loc𝑄 𝑅ℬ,𝜏 CKP (𝑁 ) is the explicit tagwise operation of Lemma

LPI: keep the same parent block, the same routing tag, the same smooth dyadic cells, and replace
only the arithmetic coefficient factors by their local residue-class model modulo

                                                𝑄=
                                                      ∏︁
                                                             𝑝.
                                                      𝑝≤𝑤

   In the CKP tagged atom, after gcd splitting and local residue decomposition modulo 𝑄, all
congruence restrictions are local. The smooth variables remain distributed over the same tagged
dyadic cells. The fibre part for fixed (𝑔, 𝑎, 𝑞) is exactly

                                      𝒮𝑎,𝑞 =                   𝐹𝑎,𝑞 (𝑦).
                                                    ∑︁

                                                𝑦≡𝑁𝑔 𝑎 mod 𝑞

   The finite AP identity gives
                                        1 ∑︁ ̂︀        ℎ   ℎ𝑁𝑔 𝑎
                                                           (︂ )︂ (︂          )︂
                                 𝒮𝑎,𝑞 =           𝐹𝑎,𝑞   𝑒       ,
                                        𝑞 ℎ mod 𝑞      𝑞    𝑞

   up to the already routed endpoint smoothing error. Its ℎ = 0 term is
                                       1 ̂︀         1 ∑︁
                                         𝐹𝑎,𝑞 (0) =      𝐹𝑎,𝑞 (𝑦).
                                       𝑞            𝑞 𝑦

   Therefore the full zero-frequency CKP term is the explicitly tagged sum
                           CKP,0                                         1 ∑︁
                                 (𝑁 ) =                   𝛼𝑔 (𝑎)𝛾𝑔 (𝑞)        𝐹𝑎,𝑞 (𝑦),
                                          ∑︁ ∑︁
                          𝑀ℬ,𝜏
                                          𝑔|𝑁     𝑎,𝑞                    𝑞 𝑦
                                                (𝑎,𝑞)=1

    with the same tag (ℬ, 𝜏 ).
    The arithmetic coefficient local densities in this expression are the B1-inherited finite-convolution
local densities. By Lemma B1-LD in Lemma G8A-LOCAL-DENSITY, finite B1 convolution, CRT
localization, gcd splitting, and tagged dyadic restriction commute with the LPI local replacement
operation. Thus the local coefficient factors in the displayed ℎ = 0 term are exactly the coefficient
factors used by Loc𝑄 𝑅ℬ,𝜏CKP (𝑁 ).

    The equation

                                            𝑎𝑦 + 𝑞𝑦 ′ = 𝑁𝑔

    has, for fixed (𝑔, 𝑎, 𝑞), a local solution density equal to the zero additive-character component of
the AP expansion. Indeed, the additive-character expansion separates the congruence condition
into frequencies. The component ℎ = 0 is precisely the average over the residue class

                                          𝑦 ≡ 𝑁𝑔 𝑎 (mod 𝑞),




                                                    201
    with density factor 1/𝑞. This is exactly the local projection of the tagged fibre after the same
local congruence data are imposed.
    All endpoint and smooth partition discrepancies are boundary errors satisfying C1A admission
and C1 Edge predicate E1/E6 and therefore contribute 𝑜(𝑁 ). The parent tag (ℬ, 𝜏 ) is preserved
throughout the gcd splitting, AP expansion and zero-frequency extraction. Therefore no local term
is moved between different tags.
    Hence
                                  CKP,0
                                 𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                     CKP
                                                         (𝑁 ) + 𝑜(𝑁 ).

   Lemma proved.
   The zero-frequency term is therefore not merely local-looking; it is explicitly identified with the
LPI tagged Λ𝑄 -projection of the same CKP cell.
   —


G8a.6. Nonzero frequencies and DFI reduction                        For ℎ ̸= 0, the contribution is

                                                        1                 ℎ   ℎ𝑁𝑔 𝑎
                                                                      (︂ )︂ (︂      )︂
                        𝒪𝑔,ℎ =            𝛽𝑔 (𝑎)𝛾𝑔,ℎ (𝑞) 𝐹̂︀𝑎,𝑞
                                    ∑︁
                                                                            𝑒       .
                                 𝑎∼𝐴 ,𝑞∼𝑄
                                                        𝑞                 𝑞    𝑞
                                     𝑔       𝑔
                                   (𝑎,𝑞)=1


   By Lemma CKPX10M, the full nonzero-frequency CKP/X10 package reduces this sum to
weighted bilinear Kloosterman fractions with parameters

                                 𝑀 = 𝐴𝑔 ,            𝑄 = 𝑄𝑔 ,        𝑘 = |ℎ|𝑁𝑔 .

    For negative ℎ, CKPX10M applies the same DFI estimate to the conjugate phase, with external
parameter 𝑟 = |ℎ|𝑁𝑔 . The theorem includes the actual two-variable smooth-weight derivative check,
the DFI/X10 matching, and the summation over 𝑔 and ℎ. In particular, the Fourier decay from the
weighted G2a step gives, for every 𝐴 > 0,
                                  ⃒1
                                  ⃒
                                  ⃒ 𝐹̂︀𝑎,𝑞 ℎ ⃒ ≪𝐴 𝑔(1 + |ℎ|𝑔)−𝐴 𝐿𝐶 .
                                          (︂ )︂⃒
                                               ⃒
                                  ⃒𝑞        𝑞 ⃒

   The extra factor 𝐿𝐶 absorbs the finite-convolution coefficient losses and derivatives of the tagged
smooth fibre weight.
   Thus the nonzero-frequency contribution satisfies

                                                          𝒪𝑔,ℎ = 𝑜(𝑁 ).
                                             ∑︁ ∑︁

                                                 𝑔 ℎ̸=0


   Large-𝑔, high-frequency, small-conductor, and boundary ranges are excluded from the central
DFI range inside CKPX10M and are routed through X10ER and C1P/C1A/C1 before X10 is
invoked.
   —




                                                           202
G8a.7. Large-g and boundary layers The CKP decomposition produces possible exceptional
layers:

  1. large gcd/content layers;

  2. high Fourier frequency tails;

  3. small-conductor DFI-form layers;

  4. boundary/short-volume layers.

   These are not counted inside the central DFI nonzero-frequency estimate. They are routed
through the C1A admission ledger to C1:

                                            large 𝑔 → 𝐸3,

                                            high ℎ → 𝐸4,

                                        small conductor → 𝐸5,

                                 boundary/short volume → 𝐸1/𝐸6.

   Therefore they contribute 𝑜(𝑁 ).
   —

G8a.8. CKP theorem
Theorem F.14 (Theorem G8a). For every tagged CKP atom (ℬ, 𝜏 ),
                                  CKP
                                 𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                   CKP
                                                       (𝑁 ) + 𝑜(𝑁 ).

   Consequently, summing over all CKP tags,

                                   𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ),

   where

                             𝑀CKP (𝑁 ) =               𝑐ℬ Loc𝑄 𝑅ℬ,𝜏 (𝑁 ).
                                              ∑︁
                                                                CKP

                                            ℬ,𝜏 ∈CKP

   Thus the CKP main term is LPI-admissible and can be assembled by Lemma H4.
Proof. Apply G1a to split gcd layers:

                                 𝑢 = 𝑔𝑎,     𝑢′ = 𝑔𝑞,        (𝑎, 𝑞) = 1.

    For each balanced layer, apply the weighted smooth AP expansion. Separating the frequency
ℎ = 0 gives the zero-frequency term. By Lemma G8a.1, this term equals the explicit tagged LPI
local projection later assembled by Lemma H4.
    The nonzero frequencies ℎ ≠ 0 are handled by CKPX10M. This master input includes the G3a
reduction to DFI/Kloosterman-fraction sums, the CKPD two-variable smooth-weight derivative
check, the G4a/X10 central DFI saving, and the X10ER routing of high-frequency, small-conductor,
large-𝑔, and boundary layers. Therefore the total nonzero-frequency contribution is 𝑜(𝑁 ).
    Hence

                                                 203
                                      CKP
                                     𝑅ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                       CKP
                                                           (𝑁 ) + 𝑜(𝑁 ).

    Summing over the finite tagged CKP family gives the theorem. The number of tags is polyloga-
rithmic and all error estimates have sufficient savings to survive this summation. The theorem is
proved.
    —


                               Remark F.15 (G8a.9. Output).
          Every tagged CKP atom equals its LPI canonical local projection plus 𝑜(𝑁 ).
    The LPI-admissible statement is:

                                   𝑀CKP,ℬ,𝜏 (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏
                                                         CKP
                                                             (𝑁 ) + 𝑜(𝑁 ).

   The AP expansion is written in weighted fibre form; finite-convolution coefficients are retained;
zero frequency is identified with the canonical tagged local projection; nonzero frequencies are
separated from Edge boundary ranges; and the dependence on DFI remains explicit through
CKPX10M.
   —

G8a.10. Logical Dependencies External dependency: X10 / DFI through CKPX10M.
  Internal dependencies: G1a, G2a, CKPX10M, C1A, C1, B1LD, and LPI.
  Children served: H4, H4M, I1, and the CKP branch closure.


G     TC1, BRS, TTH, and X16 Package
The external X9 and X16 theorem statements used here are stated once in Appendix B.

G.1     TNGTTHM master TC1 no-rogue-short-interval theorem
G.1.1     TNGTTHM. Master TC1 No-Rogue-Short-Interval Theorem
TNGTTHM.0. Statement and Role Lemma TNGTTHM is the reader-facing master
theorem for the TC1 part of the GoodAWACK branch. It packages the component route

TGT-MF → TGT → TTH-SC → MRT/TTD → ROC/BRS/X16BRS/X16C → TTH → X9L-GT

    into one autonomous proof unit.
    The theorem proves the following point.

Every actual B1-origin TC1 coarea test is either near-global and X9L-admissible, or is routed away before X9L is invoked.

   Thus the TC1 route does not require, and does not assert, pointwise Liouville cancellation on
arbitrary shifted short intervals. The theorem is a theorem about the coarea tests actually produced
by the B1/B3/F3/F4 terminal interface, not about all possible intervals inside [𝑋, 2𝑋].
   This lemma is independent of E10L. E10L consumes TNGTTHM as the TC1 branch input;
E10L is not used in its proof.
   —



                                                          204
TNGTTHM.1. Setup: Released TC1 Test Records Fix a terminal TC1-GoodAWACK
macro-template 𝜅 arising from an actual B1/B3/F3/F4 descendant after:

  1. the F3P intrinsic terminal predicate catalogue;

  2. the F3/F4 routing interface, equivalently the F3F4M partition theorem;

  3. C1 boundary removal;

  4. fixed macro-template normalization;

  5. controlled CRT, divisor, affine, scale, modulus, and smooth-weight normalizations.

   Let 𝐿𝑚 be the marked Liouville affine form. The measured Fourier/coarea transfer TGT-MF
constructs a finite measured family

                                             (𝒫𝜅 (𝑁 ), 𝜈𝜅 )

   of tests
                          1 ∑︁
               ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),      𝑝 ∈ 𝒫𝜅 (𝑁 ),         (TNGTTHM-test)
                          𝐻𝑝 𝑛∈𝐼
                                𝑝

   where:

  1. 𝐼𝑝 is a marked B1-origin coarea image interval or AP image piece;

  2. 𝐻𝑝 = |𝐼𝑝 |;

  3. 𝑋𝑝 is the height of the marked image;

  4. the AP modulus, content, and smooth-weight complexity are bounded by a fixed power of
     log 𝑁 depending only on 𝜅;

  5. the probability measure 𝜈𝜅 is inherited from the B1/B3/F3/F4 cell volume and the TGT-MF
     Fourier/coarea normalization.

   A released test is an atom of this TGT-MF coarea family whose cell has not already been
routed to Edge, LongAP/Local, CKP, LocalDiag, empty support, or impossible support by the F3/
F4/C1 boundary layer before the TC1 testing measure is formed. Equivalently, released means
membership in the structural TGT-MF testing family before the later MRT/TTD/ROC/BRS/TTH
decision is applied.
   This is a mathematical definition. It is not a convention about how the proof is read.
   —

TNGTTHM.2. The Possible Rogue Object The only object that would obstruct the TC1
route is a released test 𝑝 ∈ 𝒫𝜅 (𝑁 ) satisfying all of the following conditions.

  1. 𝑝 is still in the TC1-GoodAWACK branch.

  2. 𝑝 is not Edge, LongAP/Local, CKP, LocalDiag, empty, or impossible.



                                                 205
  3. 𝑝 is not in the near-global range for the final exponent chosen after the TTH-SC and BRS/
     X16 losses:

                                           𝐻𝑝 < 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 .

  4. 𝑝 is nevertheless passed to X9L-GT as an independent test.
    Such a test will be called a rogue short-interval test. The theorem below proves that this
class is empty for actual B1-origin TC1 coarea tests.
    There are two ways a short interval can appear syntactically during the proof. TTH-SC separates
them:
  1. a non-structural analytic subdivision of an already released parent test;
  2. a genuine structural short-image child of the B1-origin coarea algebra.
   The first is not an independent released test and must be reaggregated into the parent functional.
The second is exported to the singular routing chain TTD/ROC/BRS/X16BRS/X16C before X9L-
GT is invoked.
   —

TNGTTHM.3. Finite Decision Table For a fixed 𝜅, every structural coarea test produced by
TGT-MF lies in one of the following mutually exclusive cases after applying MRT and TTH-SC.

   Case                    Structural status        Decision before X9L-        Output
                                                    GT
   R1                     regular MRT-admissible PACK holds; TTH                near-global X9L-GT
                          start distribution        proves 𝐻𝑝 ≥                 input
                                                    𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅
   R2                     controlled structural     loses only (log 𝑁 )𝑂𝜅 (1)   still near-global after
                          refinement of R1          in length/height            enlarging 𝐵𝜅
   S1                     non-structural post-      not an element of           reaggregated into the
                          release subdivision       𝒫𝜅 (𝑁 )                     parent functional
   S2                     genuine structural        exported by TTH-SC          routed before X9L-GT
                          short-image child         to TTD/ROC/BRS
   S3                     singular start concentra- TTD identifies the          ROC/BRS route it
                          tion                      singular-origin certifi-
                                                    cate
   S4                     short B1 marked image BRS applies X16BRS/             strict C1P Edge unless
                          with hidden transverse X16C                           tagged
                          mass
   S5                     tagged quotient, local, the F3/F4 tag is al-          routed terminal class
                          CKP, LocalDiag, Edge, ready present
                          or impossible origin



    The first two rows are the only rows allowed to reach X9L-GT. All other rows are removed from
the TC1 testing family before any Liouville/AP input is used.
    This table is finite because the B1 depth, B3 grouping data, F3/F4 grammar, TGT-MF coarea
algebra, and coefficient complexity are fixed for 𝜅.
    —

                                                  206
TNGTTHM.4. Master Theorem

Theorem G.1 (Theorem TNGTTHM. TC1 no-rogue-short-interval theorem). For every fixed
actual B1-origin terminal TC1-GoodAWACK macro-template 𝜅, the released testing family 𝒫𝜅 (𝑁 )
admits a disjoint decomposition

                        𝒫𝜅 (𝑁 ) = 𝒫𝜅ng (𝑁 ) ⊔ 𝒫𝜅rt (𝑁 ),                      (TNGTTHM-partition)
   with the following properties.

  1. Near-global branch. For every 𝑝 ∈ 𝒫𝜅ng (𝑁 ), the testing family is MRT-admissible, the start-
     pushforward satisfies PACK, and

                                               𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅                 (TNGTTHM-NG)

     for the height 𝑋𝑝 of the coarea image. Hence the near-global Davenport/AP form of X9L-GT
     applies to the same measured family.

  1. Routed branch. Every test in 𝒫𝜅rt (𝑁 ) is removed before X9L-GT is invoked. Its cell carries
     one of the intrinsic routing outputs

        Edge,    LongAP/Local,         CKP,     LocalDiag,     empty/impossible. (TNGTTHM-routed)

    There is no third class consisting of an arbitrary shifted short interval, an unclassified short AP
fibre, or a post hoc subdivision of an already released test.
    Consequently

                         𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).                           (TNGTTHM-output)
   —

TNGTTHM.5. Proof: Construction of the Testing Family Assume that a fixed TC1 macro-
template 𝜅 has non-negligible terminal GoodAWACK contribution. The global TC1 generalized von
Neumann step in TGT, combined with the measured Fourier/coarea transfer TGT-MF, produces
the same finite measured testing family (𝒫𝜅 , 𝜈𝜅 ) and a fixed lower bound
                          ∫︁
                                     |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≫𝜅 1.                      (TNGTTHM-lower)
                           𝒫𝜅 (𝑁 )

    This is the only place where the TC1 obstruction is converted into Liouville tests. The tests are
structural B1-origin coarea tests as in TNGTTHM.1.
    No arbitrary interval is inserted at this step. The interval or AP piece 𝐼𝑝 is the image of the
marked B1-origin affine form on a cell released by the F3/F4/TGT-MF structural algebra.
    —

TNGTTHM.6. Proof: No Rogue Refinement Can Be Released Let 𝑝 ∈ 𝒫𝜅 (𝑁 ) be released.
Consider any short interval or short AP subpiece that appears after 𝑝 has been formed.
    If the subpiece is not generated by the finite TGT-MF coarea algebra, then TTH-SC classifies it
as a non-structural analytic subdivision. It is not an element of 𝒫𝜅 (𝑁 ), carries no independent 𝜈𝜅
mass, and is reaggregated into the parent functional ℒ𝑝 before X9L-GT is invoked.


                                                     207
  If the subpiece is generated by the coarea algebra and the length loss is only polylogarithmic,
TTH-SC gives

                               𝐻𝑝′ ≥ 𝐻𝑝 (log 𝑁 )−𝐶𝜅 .                             (TNGTTHM-refine)
    Thus a near-global parent remains near-global after increasing the logarithmic exponent in TTH.
    If the subpiece is structural but violates this controlled lower bound, it is not a refinement of an
already accepted X9L test. It is a genuine B1-origin short-image certificate. TTH-SC exports this
certificate to the singular routing chain, where TTD/ROC/BRS, using X16BRS/X16C, routes it to
strict C1P Edge unless it already has a LongAP/Local, CKP, LocalDiag, Edge, empty, or impossible
tag.
    Therefore no short interval satisfying the rogue conditions of TNGTTHM.2 can survive as a
released TC1 input to X9L-GT.
    —

TNGTTHM.7. Proof: Regular Branch Lemma MRT separates the testing family into regular
and singular alternatives.
    In the regular alternative, MRT gives PACK for the same measure 𝜈𝜅 . By TNGTTHM.6, every
released test that remains in the regular branch is a structural B1-origin coarea test, up to controlled
polylogarithmic refinement.
    Lemma TTH, using BRS together with the X16BRS/X16C carrier-slice estimate, supplies

                                              𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅

   for every remaining released B1-origin test. Thus the family is in the near-global X9L-GT range.
X9L-GT gives
                          ∫︁
                                       |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) = 𝑜𝜅 (1).                   (TNGTTHM-X9)
                           𝒫𝜅ng (𝑁 )

   This contradicts the lower bound (TNGTTHM-lower) on any non-negligible regular TC1 contri-
bution.
   —

TNGTTHM.8. Proof: Singular and Short-Image Branches In the singular alternative,
MRT exports a singular-origin certificate to TTD. TTD identifies the singular geometry. Direct B1
dyadic-coordinate origins and their controlled CRT/divisor/full-rank transports satisfy the ROC
range-origin comparability lemma. Tagged failures already carry one of the terminal routing labels
in (TNGTTHM-routed).
    The complementary, solved-affine, quotient, and short-image carrier cases are handled by BRS.
BRS uses X16BRS and X16C to show that a genuinely short marked B1 image cannot hide large
transverse mass: it is strict C1P Edge unless it already carries a LongAP/Local, CKP, LocalDiag,
Edge, empty, or nonterminal routing tag.
    Therefore the singular branch never reaches X9L-GT. It is exported to the routed alternatives
in (TNGTTHM-routed) before any Liouville/AP theorem is invoked.
    The regular and singular alternatives exhaust the released tests by MRT. The no-third-class
assertion follows from TTH-SC together with BRS: a short piece either is not a released structural
test, in which case it is reaggregated, or it is structural, in which case it is routed through TTD/
ROC/BRS/X16BRS/X16C before X9L-GT.


                                                       208
   There are only boundedly many TC1 structural macro-templates, depending on the fixed Heath-
Brown depth. Summing the 𝑜(𝑁 ) bounds over them gives 𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ). The
theorem is proved.
   —

Parameter check G.2 (TNGTTHM.9. Parameter Check). The theorem uses the following order
of constants.

  1. The B1 depth, B3 grouping complexity, and F3/F4 routing grammar are fixed.

  2. The TGT-MF Fourier/coarea complexity is fixed for the macro-template 𝜅.

  3. TTH-SC exports only a polylogarithmic refinement loss.

  4. X16BRS/X16C fix 𝐵16 , 𝐶16 , and 𝜌16 .

  5. TTH chooses 𝐵𝜅 larger than the TTH-SC loss, the BRS/X16 losses, and the height/content
     distortion losses.

  6. X9L-GT is invoked with a Davenport logarithmic saving exponent larger than the PACK, AP-
     modulus, smooth-weight, and (log 𝑋)𝐵𝜅 losses.

   Hence (TNGTTHM-NG) is a near-global statement

                                       𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 ,

                                             1/3+𝜀
    which is stronger than any fixed 𝐻𝑝 ≥ 𝑋𝑝    𝜅
                                                  for sufficiently large 𝑋𝑝 . The proof does not use
a low-𝜃 polylog-modulus input for arbitrary short shifted intervals.
    —

TNGTTHM.10. Interface Corollary

Corollary G.3 (Corollary TNGTTHM.1. TC1 input for E10L). E10L may import the TC1 branch
through the single statement

                    TNGTTHM + X9L-GT =⇒ 𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

    This import has no logical dependence on E10L itself. The routed alternatives are terminal
routing outputs already defined by the F3/F4 and C1P/LPI/CKP interfaces, while the near-global
branch is closed by X9L-GT after MRT, TTH-SC, BRS, X16BRS, X16C, and TTH.
    —

TNGTTHM.11. Logical Dependencies External dependency: X9L-GT in the near-global
Davenport/AP form.
    Internal dependencies: TGD, F3F4M, TGT-MF, TGT, TTH-SC, TNG, MRT, TTD, ROC,
BRS, X16BRS, X16C, TTH, C1P, C1A, C1, E5, and the parameter register. The routed terminal
outputs are later estimated or assembled by C1, D1/H4M, G8a, H4M, or zero, but those downstream
estimates are not part of the definition of the active TC1 testing family.
    Children served: E10L, GEB, I1, and the full proof assembly.



                                                209
G.2     TC1 GoodAWACK dichotomy
G.2.1    TGD. Terminal GoodAWACK True-Complexity Split
TGD.0. Statement and Role Lemma TGD records a non-recursive refinement of the terminal
GoodAWACK class:

                  GoodAWACK = TC1-GoodAWACK ⊔ HighTC-GoodAWACK.

    The purpose is not to prove the HighTC contribution is small. The purpose is to make the split
finite and structural, so that the remaining HighTC class is a certified algebraic obstruction rather
than an indefinitely recurring tail.
    The guiding principle is:

                     TC1 is decided by a quadratic tensor independence test.

   If the test fails, the failure itself is the HighTC certificate.
   Logical dependencies are the F3/F4 terminal GoodAWACK interface, E5 content stability, BGS
normal-form data, and bounded tensor-linear algebra. TGD is used by TGT, TTD, TNG, HGO2R,
E10M, E10K, and E10L.
   —

TGD.1. Setup: Terminal GoodAWACK Data Let (𝒜, 𝜏 ) be a tagged terminal GoodAWACK
atom produced by

                                         𝐵1 → 𝐵3 → 𝐹 3/𝐹 4.

  By the F3/F4 terminal GoodAWACK interface and E5 content stabilization, the atom has a
model form
                                                             𝑡
                                𝒜=         𝑊 (𝑧)𝜆(𝐿0 (𝑧))         𝑓𝑖 (𝐿𝑖 (𝑧)),
                                     ∑︁                     ∏︁

                                     𝑧∈Ω                    𝑖=1

   where:

  1. Ω is a smooth box-like domain in a fixed-rank parameter lattice;

  2. 𝑊 is a smooth tagged weight of polylogarithmic complexity;

  3. 𝐿0 , 𝐿1 , . . . , 𝐿𝑡 are affine forms of bounded affine and Cauchy-Schwarz complexity;

  4. at least one active affine form carries a Liouville-type oscillatory factor;

  5. all active forms have controlled content;

  6. no terminal Edge, CKP, LongAP/Local, or LocalDiag predicate applies.

   For the true-complexity test, write

                                                   𝐿˙ 𝑖

   for the homogeneous linear part of 𝐿𝑖 . Constants are irrelevant for the tensor test.
   Let

                                                  210
                                    𝑄𝑖 := 𝐿˙ 𝑖 ⊙ 𝐿˙ 𝑖 ∈ Sym2 (𝑉Q* )

   be the quadratic tensor attached to 𝐿𝑖 on the active parameter space 𝑉 .
   Let

                                               ℳ(𝒜)

   be the finite set of marked Liouville-type affine forms in the atom. In E10 notation this set
contains the chosen marked form 𝐿0 , but the refined split allows one to choose any marked form
that passes the TC1 test.
   —

TGD.2. Statement: Definition of TC1-GoodAWACK                         A terminal GoodAWACK atom
(𝒜, 𝜏 ) is called

                                        TC1-GoodAWACK

   if there exists a marked form 𝐿𝑚 , 𝑚 ∈ ℳ(𝒜), such that

                               / spanQ {𝑄𝑖 : 𝑖 ̸= 𝑚, 𝐿𝑖 active in 𝒜}.
                            𝑄𝑚 ∈                                                             (TC1)
   Equivalently, the active affine system is true-complexity one relative to at least one marked
Liouville form.
   This is a deliberately relative condition. It is stronger than merely saying the forms are not
equal or proportional, and weaker than requiring all tensors 𝑄𝑖 to be linearly independent.
   The intended analytic consequence is the following replacement for the high-order E10 generalized
von Neumann step:

  TC1-GoodAWACK non-small =⇒ ‖𝜆(𝐿𝑚 )‖𝑈 2 ≫ the corresponding normalized lower bound.

  In this proof this analytic consequence is supplied by the global testing chain recorded in Lemma
TNG:

                     TGT + MRT + TTD + ROC + BRS + TTH + X9L-GT.

   The present document only proves the structural split.
   —

TGD.3. Statement: Definition of HighTC-GoodAWACK                       A terminal GoodAWACK atom
(𝒜, 𝜏 ) is called

                                      HighTC-GoodAWACK

   if it is terminal GoodAWACK and no marked Liouville form satisfies (TC1).
   Equivalently, for every marked 𝑚 ∈ ℳ(𝒜),

                            𝑄𝑚 ∈ spanQ {𝑄𝑖 : 𝑖 ̸= 𝑚, 𝐿𝑖 active in 𝒜}.                     (HighTC)
   Thus each marked form has a quadratic dependence certificate. After clearing denominators, for
every marked 𝑚 there are integers 𝑐𝑖 , not all zero, with 𝑐𝑚 ̸= 0, such that

                                                 211
                                                      𝑐𝑖 𝑄𝑖 = 0.                        (HighTC-cert)
                                                 ∑︁

                                                  𝑖

   The relation (HighTC-cert) is the terminal HighTC obstruction. It is not a new unresolved
routing instruction.
   Examples of this kind include the four-term progression pattern

                                    𝑥,       𝑥 + 𝑟,    𝑥 + 2𝑟,     𝑥 + 3𝑟,

   for which

                                      𝐿20 − 3𝐿21 + 3𝐿22 − 𝐿23 = 0.

   This pattern is not a mere equality/proportionality collision. It is a higher true-complexity affine
configuration.
   —

TGD.4. Proof: Finite TC1/HighTC Dichotomy

Lemma G.4 (Lemma TGD.1). Every tagged terminal GoodAWACK atom belongs to exactly one of

                         TC1-GoodAWACK,                    HighTC-GoodAWACK.

    Moreover, if it belongs to HighTC-GoodAWACK, then it carries the explicit finite algebraic
certificate (HighTC-cert) for every marked Liouville-type form.

Proof. Fix a tagged terminal GoodAWACK atom (𝒜, 𝜏 ).
   By the GoodAWACK terminal predicate, the set of active forms is finite and has bounded
cardinality depending only on 𝐽0 . The set of marked Liouville-type forms is also finite and nonempty.
   For each marked 𝑚 ∈ ℳ(𝒜), form the quadratic tensor

                                               𝑄𝑚 = 𝐿˙ 𝑚 ⊙ 𝐿˙ 𝑚

   in the finite-dimensional rational vector space

                                                  Sym2 (𝑉Q* ).

   There are two possibilities.
   First, for at least one marked 𝑚,

                                         / spanQ {𝑄𝑖 : 𝑖 ̸= 𝑚}.
                                      𝑄𝑚 ∈

   Then (𝒜, 𝜏 ) is TC1-GoodAWACK by definition.
   Second, for every marked 𝑚,

                                      𝑄𝑚 ∈ spanQ {𝑄𝑖 : 𝑖 ̸= 𝑚}.

    Then (𝒜, 𝜏 ) is HighTC-GoodAWACK by definition. Since the vector space and the active set
are finite-dimensional and rational, each span membership gives a rational linear relation among
the 𝑄𝑖 . Clearing denominators gives an integer relation

                                              𝑐𝑖 𝑄𝑖 = 0,      𝑐𝑚 ̸= 0,
                                         ∑︁

                                         𝑖


                                                       212
     which is exactly (HighTC-cert).
     The two alternatives are mutually exclusive by the law of excluded middle applied to the finite
list of marked tensors. They are exhaustive because every marked tensor either is or is not in the
rational span of the remaining active tensors.
     Therefore the dichotomy is finite, disjoint and non-recursive. Lemma proved.
     —


Parameter check G.5 (TGD.5. Parameter Check: No Infinite Tail). The class HighTC-
GoodAWACK is not defined by saying "whatever remains after another analytic decomposition." It
is defined by the explicit algebraic condition (HighTC).
    Thus a HighTC atom is terminal at the level of this split. Future work has only three legitimate
options:

  1. prove an analytic estimate for all atoms satisfying (HighTC-cert);

  2. prove that some certified HighTC patterns are actually CKP, Edge, or genuine LocalDiag
     under additional already-terminal criteria;

  3. refine the terminal predicate by a new finite invariant that strictly decreases.

   What is not allowed is an unmeasured iteration

                      HighTC → smaller HighTC → smaller HighTC → · · · .

   Such an iteration would need a separate well-founded complexity measure. The present split
avoids that problem by making HighTC a certified finite obstruction class.
   —

TGD.6. Compatibility with LocalDiag The HighTC certificate must not automatically be
routed to LocalDiag.
   Lemma F3 defines LocalDiag as forced equality, proportionality, gcd-local dependence, or
unavoidable collision that makes the contribution a canonical local term. A quadratic tensor relation
such as

                                     𝐿20 − 3𝐿21 + 3𝐿22 − 𝐿23 = 0

   does not by itself produce a canonical local main term.
   Therefore:

                               HighTC-GoodAWACK ̸⇒ LocalDiag.

   Only those HighTC atoms whose certificate also forces a genuine local/main degeneracy may be
passed to H4. Otherwise they remain in the HighTC-GoodAWACK branch.
   This resolves the ambiguity between the broad B3 phrase "affine dependence among active forms"
and the narrower F3/H4 terminal meaning of LocalDiag.
   —

TGD.7. Output for E10         After this split, E10 should be treated as two sub-branches:

               𝑅GoodAWACK (𝑁 ) = 𝑅TC1-GoodAWACK (𝑁 ) + 𝑅HighTC-GoodAWACK (𝑁 ).

                                                213
TC1 branch             The TC1 branch is handled by the global-testing route:

                            TC1 =⇒ 𝑈 2 -generalized von Neumann =⇒ TNG =⇒ 𝑜(𝑁 ).

    This replaces X8 on the TC1 sub-branch. The orthogonality input is X9L-GT.

HighTC branch                The HighTC branch is the explicit algebraic obstruction:

                                                       HighTC-GoodAWACK

    It is closed structurally: origin-degenerate HighTC is rerouted by HGO2R, and the free-affine
residual is excluded by E10M plus E10K.
    The important gain is conceptual: HighTC is an explicit finite algebraic obstruction, not an
open-ended residual tail, and it is discharged by HGO2R and the E10YMX finite-grammar theorem,
whose origin-tag interface is supplied by E10M/E10K.
    —
   Remark G.6 (TGD.8. Output). TC1/HighTC dichotomy proved as a finite structural split.
    TGD does not by itself close E10. It supplies the stable interface:

                         GoodAWACK = TC1-GoodAWACK ⊔ HighTC-GoodAWACK.

   The TC1 branch is handled by TNG. The HighTC branch is handled by HGO2R/E10YMX and
then by E10L.

TGD.9. Logical Dependencies Internal dependencies: the F3/F4 terminal GoodAWACK
interface, E5, BGS, and bounded tensor-linear algebra.
    Children served: TGT, TTD, TNG, HGO2R, E10M, E10K, and E10L.

G.3       TC1 global testing
G.3.1       TGT. Aggregated Testing Route for TC1-GoodAWACK
TGT.0. Statement and Role Lemma TGT records the aggregation and regular-branch testing
replacement for the pointwise short-interval TC1 route.
    The statement is:

after aggregation over a fixed TC1 macro-template, an MRT-admissible near-global testing family is closed by the averaged Liouville input X9L-GT.


   Equivalently, one first aggregates all TC1 atoms with the same structural macro-template and
only then tests Liouville against the induced measured family of intervals or arithmetic progressions.
This avoids selecting a single bad fibre before the averaging structure has been exposed.

         TGT supplies the measured testing family and the regular MRT-admissible closure.

    The complementary singular or short-image alternatives are not inputs to TGT. They are routed
later by the TTD/ROC/BRS/TTH part of the TNG package. Logical dependencies are TGD, TGT-
MF, MRT, TTH, E5, X9L-GT, and the parameter register. TGT is used by TNG and E10L; E10L
is a downstream consumer of the TC1 testing route, not an input to it.
    —


                                                                     214
TGT.1. Setup: Macro-Template Aggregation Fix a structural TC1 macro-template 𝜅. The
template fixes:

  1. the B1 typed parent pattern;

  2. the B3 grouping skeleton;

  3. the F3/F4 routing grammar;

  4. the marked Liouville origin;

  5. the affine coefficient transport type;

  6. the TC1 tensor certificate.

   It does not select a single dyadic atom. Instead, it contains all dyadic, CRT, divisor, and
smoothing cells compatible with the same structural template.
   Write the corresponding terminal TC1 atoms as

                                              𝒜𝑗 ,        𝑗 ∈ 𝐽𝜅 (𝑁 ),

   with effective volumes 𝑉𝑗 , domains Ω𝑗 , marked forms 𝐿𝑚,𝑗 , and normalized contributions

                                  1 ∑︁
                          𝑎𝑗 :=          𝑊𝑗 (𝑧)𝜆(𝐿𝑚,𝑗 (𝑧))      𝑓𝑖,𝑗 (𝐿𝑖,𝑗 (𝑧)).
                                                            ∏︁
                                  𝑉𝑗 𝑧∈Ω                   𝑖̸=𝑚
                                          𝑗


   The aggregated contribution is

                                         𝑅𝜅 (𝑁 ) =
                                                            ∑︁
                                                                      𝑉𝑗 𝑎𝑗 .
                                                          𝑗∈𝐽𝜅 (𝑁 )

   Since the number of structural macro-templates is bounded in terms of 𝐽0 , if the total TC1
contribution is not 𝑜(𝑁 ), then along an infinite sequence there is a fixed 𝜅 and 𝜀 > 0 such that

                                                |𝑅𝜅 (𝑁 )| ≥ 𝜀𝑁.                                (1)
   No dyadic polylogarithmic pigeonhole is used at this stage.
   —
Remark G.7 (TGT.2. Proof: Global TC1 Generalized von Neumann Output). For each atom 𝑗, the
TC1 weighted generalized von Neumann step gives

                                   |𝑎𝑗 | ≤ 𝐶𝜅 ‖𝜆(𝐿𝑚,𝑗 )‖𝑐𝑈𝜅2 (Ω′ ) + 𝑜𝜅 (1),                   (2)
                                                                      𝑗

   after the usual C1 boundary removals and content normalizations.
   Multiply by 𝑉𝑗 , sum over 𝑗, and use (1). Since
                                                 ∑︁
                                                         𝑉𝑗 ≪𝜅 𝑁
                                                     𝑗




                                                         215
   for a fixed macro-template, (1)–(2) imply
                                  1 ∑︁
                                             𝑉𝑗 ‖𝜆(𝐿𝑚,𝑗 )‖𝑐𝑈𝜅2 (Ω′ ) ≫𝜅,𝜀 1.                               (3)
                                  𝑁 𝑗∈𝐽 (𝑁 )                     𝑗
                                           𝜅

   After replacing 𝑐𝜅 by a harmless bounded power and using 0 ≤ ‖ · ‖𝑈 2 ≤ 1, this gives the fixed-
threshold global obstruction

                                     E𝑗∼𝑉𝑗 ‖𝜆(𝐿𝑚,𝑗 )‖4𝑈 2 (Ω′ ) ≫𝜅,𝜀 1.                               (GT-U2)
                                                                     𝑗


   This proves the internal aggregation step.
   —

TGT.3. Proof: Measured Fourier Transfer Apply Lemma TGT-MF to the normalized box/
coset models and the obstruction (GT-U2). The lemma uses the Fourier normalization

                                               ‖𝐹𝑗 ‖4𝑈 2 =        |𝐹̂︀𝑗 (𝜉)|4
                                                             ∑︁

                                                             𝜉

   and the finite coarea normal form of the marked affine form 𝐿𝑚,𝑗 . It constructs a finite probability
measure 𝜈𝜅 on tests
                              1 ∑︁
                   ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),                  𝑝 = (𝑗, 𝜉, coarea piece),         (4)
                              𝐻𝑝 𝑛∈𝐼
                                    𝑝

   where 𝐼𝑝 is a shifted interval or AP image, 𝐻𝑝 = |𝐼𝑝 |, the AP modulus/content and the weight
complexity are polylogarithmically controlled, and C1 boundary pieces have already been discarded.
TGT-MF gives the fixed testing lower bound
                                     ∫︁
                                                  |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≫𝜅,𝜀 1,                          (GT-Test)
                                        𝒫𝜅 (𝑁 )

   for the induced probability measure 𝜈𝜅 .
   This is the global replacement for a pointwise shifted short-interval input:

               not one bad interval, but a whole measured family of Liouville tests.

   —

Parameter check G.8 (TGT.4. Parameter Check: MRT-Admissible Testing Families). The
averaged Liouville input can only apply if the testing measure genuinely averages over starts/scales.
Define a testing family (𝒫𝜅 , 𝜈𝜅 ) to be MRT-admissible if, after partitioning into 𝑂𝜅 ((log 𝑁 )𝐶 )
scale/modulus/weight-complexity classes, the pushforward of 𝜈𝜅 to interval starts is dominated by a
polylogarithmic multiple of normalized counting/Lebesgue measure:

                                                                                𝑑𝑥
                                          (start)# 𝜈𝜅 ≪𝜅 (log 𝑁 )𝐶                                    (PACK)
                                                                                𝑋
   on each dyadic 𝑥 ≍ 𝑋, with

                                                     𝐻𝑝 ≥ 𝑋𝑝𝜃𝜅


                                                         216
   outside C1-negligible boundary pieces.
   This condition is the common form of:

  1. E7 pushforward regularity;

  2. coarea image regularity when the marked image sweeps many starts;

  3. absence of rank-one/point-mass short-image concentration.

    PACK is not supplied by TTH alone. The verification of PACK and the routing of PACK failures
are recorded in MRT. TGT invokes X9L-GT only on the branch selected there as MRT-admissible.
    Assume the external averaged Liouville theorem in the qualitative form:
                                     ∫︁
                                                |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) = 𝑜𝜅 (1)                    (X9L-GT)
                                      𝒫𝜅 (𝑁 )

   for every MRT-admissible TC1 testing family.
   Then (GT-Test) contradicts (X9L-GT).

Lemma G.9 (Lemma TGT.1. Admissible global testing closure). For a fixed structural macro-
template 𝜅, if the induced TC1 testing family is MRT-admissible in the sense of MRT and X9L-GT
holds, then

                                                𝑅𝜅 (𝑁 ) = 𝑜(𝑁 ).

Proof. Assume not. Then (1) holds for some 𝜀 > 0. TGT.2 and TGT-MF give the fixed lower
bound (GT-Test). MRT-admissibility allows the averaged Liouville input X9L-GT, giving 𝑜(1) for
the same left side. This is a contradiction. Lemma proved.
   —


TGT.5. Singular testing measures The route does not close arbitrary TC1 testing families. If
the test measure is concentrated on one or very few interval starts, then averaged Liouville theorems
say nothing.
    The model obstruction is exactly the earlier SAI/rank-one model:

                    Ω = [𝑋, 𝑋 + 𝑌 ] × [1, 𝑀 ],           𝐿𝑚 (𝑢, 𝑣) = 𝑢,      𝑌 𝑀 ≍ 𝑁.               (5)
   The coarea image is the single interval [𝑋, 𝑋 + 𝑌 ]. Averaging over 𝑣 increases the weight of the
same interval; it does not create an average over starts. The pushforward in (PACK) is a point
mass, so the family is not MRT-admissible.
   Thus global aggregation does not require pointwise X9L-SI. Instead, it isolates the exact structural
branch:

          singular testing measure        ⇐⇒       rank-one / short affine-image concentration.

   This is the structural obstruction handled by the singular branch of TTD.
   —




                                                       217
TGT.6. Output Form and Downstream Structural Closure The output supplied by TGT
itself is the regular testing closure:

                MRT-admissible near-global TC1 testing families contribute 𝑜(𝑁 ).

   The full structural dichotomy is supplied downstream in the consolidated form TNG-A. It says
that for every actual B1/B3/F3/F4 terminal TC1 macro-template 𝜅, after C1 boundary removal,
exactly one of the following holds:

  1. the induced global testing family is MRT-admissible, so Lemma TGT.1 closes it using averaged
     Liouville cancellation;

  2. the non-admissible/singular part has an origin tag forcing strict C1P Edge;

  3. it is a genuine LPI/H4M-admissible LongAP/Local main term;

  4. it exposes a CKP grouping handled by G8a;

  5. it exposes LocalDiag;

  6. it is empty/impossible by parent B1 scale or congruence constraints.

   Internally TNG-A uses TTD, TTH-SC, ROC, BRS, X16BRS/X16C, and TTH. It replaces:

  1. pointwise X9L-SI;

  2. atomwise E7-REG-CARRIER;

  3. TC1-SAI-ROUTE;

  4. ad hoc coarea short-image routing.

    It asks for regularity or origin-routing of the global testing measure, not of each presentation
of the same local tail.
    —

TGT.7. Compatibility with Auxiliary Reductions The E7 averaged-fibre argument proves
the averaged slicing part for one coordinate presentation. In the present language, it constructs
part of 𝒫𝜅 .
    The E7 regular-pushforward check concerns condition (PACK) for E7 fibres and finds that rank-
one carriers are exactly the non-admissible case.
    The TC1 coarea Fourier step constructs the coarea tests (4). Theorem TNG-A says that near-
global images are closed by X9L-GT, while genuinely short or singular images are routed by TTD/
ROC/BRS using X16BRS/X16C before X9L-GT is invoked.
    The TC1-SAI route shows that short image alone is not enough to route an atom by the terminal
predicates. In the present language, it says that non-admissible testing measure is not automatically
C1/D1/G8a/LocalDiag.
    X9L-GT is the external averaged input. The global testing formulation explains why a qualitative
𝑜(1) theorem may suffice: after macro-template aggregation, the lower bound in (GT-Test) is a
fixed ≫𝜀 1, not a polylogarithmic threshold.
    —

                                                218
Remark G.10 (TGT.8. Output). The global testing route is a genuine conceptual improvement.
  Together with Theorem TNG-A and X9L-GT, it gives the TC1 closure:

                               TC1 macro-templates contribute 𝑜(𝑁 ).

   After TNG-A, the singular structural branch is not a residual. Lemma TTH supplies the near-
global length information in the near-global alternative,
                                               𝐻 ≥ 𝑋(log 𝑋)−𝐵
for B1-origin coarea tests. Therefore the only analytic 𝑋9𝐿 input required by the TC1 branch is
the near-global Davenport/AP form X9L-GT.
    The single-source statement of this chain is Lemma TNG. TGT supplies the aggregation and
testing lower bound; Lemma TNG verifies that the unrouted tests seen by X9L-GT are exactly the
MRT-admissible, near-global B1-origin coarea tests.
    —

TGT.9. External Input Check X9L-GT records the external input: Davenport closes the near-
global AP-fibre range

                                               𝐻 ≥ 𝑋(log 𝑋)−𝐵 ,

   and this is the only X9L input used by this proof.
   The proof does not invoke a normalized AP-fibre estimate for arbitrary shifted intervals through-
out the range 𝐻 ≥ 𝑋 𝜃 , 0 < 𝜃 < 1/3. The only AP-fibre estimate required after the TNG reduction
to B1-origin coarea tests is the near-global Davenport/AP estimate stated above.

TGT.10. Logical Dependencies External dependency: X9L-GT in the near-global Davenport/
AP range.
    Internal dependencies: TGD, TGT-MF, MRT, TTH, E5, and the parameter register.
    Children served: TNG, E10L, and the TC1-GoodAWACK closure.
    Direction note: TGT.2 and TGT-MF construct the measured testing family, while TGT.4 closes
only the MRT-admissible regular branch. The full TC1 closure uses the later TNG-A interface
to dispose of singular or short-image tests. Thus references from TTD/TTH/TNG back to the
TGT construction refer only to this construction and regular-branch output, not to a theorem that
already assumes TNG-A.

G.4     TGT-MF measured Fourier transfer
G.4.1    TGT-MF. Measured Fourier Transfer for TC1 Global Testing
TGT-MF.0. Statement and Role Lemma TGT-MF is the measure-theoretic and Fourier
normalization step used inside TGT. It turns the global 𝑈 2 -obstruction produced by TC1 aggregation
into a finite measured family of Liouville tests.
    The statement is:

                                       E𝑗∼𝑉𝑗 ‖𝐹𝑗 ‖4𝑈 2 (Ω′ ) ≥ 𝑐
                                                          𝑗

                                                   =⇒                                       (TGT-MF)
                                        ∫︁
          ∃ (𝒫𝜅 (𝑁 ), 𝜈𝜅 ) such that               |ℒ𝑝 (𝜆)| 𝑑𝜈𝜅 (𝑝) ≥ 𝑐/𝐶MF (𝜅) − 𝑜𝜅 (1).
                                                          2
                                         𝒫𝜅 (𝑁 )


                                                         219
   Here 𝐹𝑗 (𝑧) = 𝜆(𝐿𝑚,𝑗 (𝑧)) on the normalized box/coset model Ω′𝑗 , after the C1 boundary and
content-normalization removals used in TGT.2. The constant 𝐶MF (𝜅) depends only on the fixed
TC1 macro-template 𝜅, the bounded dimension of its boxes, and the fixed coarea complexity of the
template. It is independent of 𝑁 .
   Logical dependencies are the TGT.1–TGT.2 setup, the C1 boundary removal interface, E5
content/affine transport control, and the finite F3/F4 coarea normal form. The lemma is used by
TGT, TTD, TNG, and TTH.
   —

TGT-MF.1. Setup: Normalized Fourier Models For every atom 𝑗 ∈ 𝐽𝜅 (𝑁 ), let Ω′𝑗 be the
finite box/coset model remaining after C1-negligible boundary pieces and controlled content factors
have been removed. It is endowed with normalized counting measure
                                                                  1 ∑︁
                                                 EΩ′𝑗 𝑓 :=                𝑓 (𝑧).
                                                                 |Ω′𝑗 | ′𝑧∈Ω𝑗


   Let 𝐺𝑗 be the finite abelian group obtained by completing the box/coset model with the same
                 ̂︀ 𝑗 be its character group. Fourier coefficients are normalized by
periods, and let 𝐺

                                              𝐹̂︀𝑗 (𝜉) := E𝑧∈𝐺𝑗 𝐹𝑗 (𝑧)𝜉(𝑧).                                      (1)
   With this normalization,

                     ‖𝐹𝑗 ‖4𝑈 2 (𝐺𝑗 ) =           |𝐹̂︀𝑗 (𝜉)|4 ,                   |𝐹̂︀𝑗 (𝜉)|2 = E𝐺𝑗 |𝐹𝑗 |2 ≤ 1.   (2)
                                         ∑︁                              ∑︁

                                         𝜉∈𝐺
                                           ̂︀𝑗                           𝜉∈𝐺
                                                                           ̂︀𝑗

    Replacing the box by its completed coset model changes the 𝑈 2 -quantity only by the 𝑜𝜅 (1)
boundary term already assigned to C1. Thus the TGT lower bound may be read with 𝐺𝑗 in place
of Ω′𝑗 .
    Set

                                                                         𝑉𝑗
                                                     𝑤𝑗 = ∑︀                       .                             (3)
                                                                   𝑖∈𝐽𝜅 (𝑁 ) 𝑉𝑖

   The global obstruction entering this lemma is

                                                                         |𝐹̂︀𝑗 (𝜉)|4 ≥ 𝑐.                        (4)
                                                 ∑︁              ∑︁
                                                         𝑤𝑗
                                            𝑗∈𝐽𝜅 (𝑁 )            𝜉∈𝐺
                                                                   ̂︀𝑗
   —

TGT-MF.2. Setup: Coarea Normal Form for One Fourier Coefficient For each pair (𝑗, 𝜉),
the marked form 𝐿𝑚,𝑗 and the finite F3/F4 coarea normal form decompose the Fourier coefficient as

                                 𝐹̂︀𝑗 (𝜉) =                 𝛽𝑗,𝜉,𝑟 ℒ𝑗,𝜉,𝑟 (𝜆)𝑂𝜅 (𝑁 −100 ),                       (5)
                                                   ∑︁

                                                 𝑟∈ℛ(𝑗,𝜉)

   where:

  1. #ℛ(𝑗, 𝜉) ≤ 𝐵co (𝜅);

  2.    𝑟 |𝛽𝑗,𝜉,𝑟 | ≤ 𝐵co (𝜅);
       ∑︀


                                                                  220
  3. every ℒ𝑗,𝜉,𝑟 has the normalized form

                                                  1
                              ℒ𝑗,𝜉,𝑟 (𝜆) =                           𝜆(𝑛)𝜌𝑗,𝜉,𝑟 (𝑛)𝑒(𝛼𝑗,𝜉,𝑟 𝑛);      (6)
                                                           ∑︁
                                              𝐻𝑗,𝜉,𝑟 𝑛∈𝐼
                                                             𝑗,𝜉,𝑟



  4. 𝐼𝑗,𝜉,𝑟 is a shifted interval or arithmetic-progression image of the marked form;

  5. the AP modulus/content and the derivative complexity of 𝜌𝑗,𝜉,𝑟 are bounded by fixed powers
     of log 𝑁 determined by 𝜅;

  6. the discarded coarea boundary pieces have total contribution 𝑜𝜅 (1) after the 𝑗-average and
     are already C1-admitted.

   Equation (5) is a finite identity on the normalized box/coset model. It is not a pigeonhole over
dyadic atoms. The constants in (5) depend on the fixed dimension and routing grammar of 𝜅, not
on the number of dyadic cells inside the macro-template.
   —

TGT-MF.3. Construction of the Testing Measure                                Let

                               𝑆𝜅 :=                      |𝐹̂︀𝑗 (𝜉)|2                                (7)
                                       ∑︁         ∑︁                      ∑︁
                                             𝑤𝑗                                    |𝛽𝑗,𝜉,𝑟 |.
                                       𝑗          𝜉∈𝐺
                                                    ̂︀𝑗                 𝑟∈ℛ(𝑗,𝜉)

   By (2) and the coarea bound in TGT-MF.2,

                                              0 < 𝑆𝜅 ≤ 𝐵co (𝜅).                                      (8)
   The strict positivity follows from (4). Define the finite parameter set

                                                         ̂︀ 𝑗 , 𝑟 ∈ ℛ(𝑗, 𝜉), |𝛽𝑗,𝜉,𝑟 | > 0}.
                𝒫𝜅 (𝑁 ) := {(𝑗, 𝜉, 𝑟) : 𝑗 ∈ 𝐽𝜅 (𝑁 ), 𝜉 ∈ 𝐺                                           (9)
   Define the probability measure 𝜈𝜅 by

                                                          𝑤𝑗 |𝐹̂︀𝑗 (𝜉)|2 |𝛽𝑗,𝜉,𝑟 |
                                    𝜈𝜅 (𝑗, 𝜉, 𝑟) =                                 .                (10)
                                                                    𝑆𝜅
   This is an ordinary finite probability measure. Hence all measurability assertions are literal:
every subset of 𝒫𝜅 (𝑁 ) is measurable.
   For 𝑝 = (𝑗, 𝜉, 𝑟), set

                        ℒ𝑝 (𝜆) := ℒ𝑗,𝜉,𝑟 (𝜆),             𝐼𝑝 = 𝐼𝑗,𝜉,𝑟 ,            𝐻𝑝 = 𝐻𝑗,𝜉,𝑟 .    (11)
   —

TGT-MF.4. Proof of the Lower Bound                        From (5) and Cauchy’s inequality,

                    |𝐹̂︀𝑗 (𝜉)|2 ≤ 2𝐵co (𝜅)                |𝛽𝑗,𝜉,𝑟 | |ℒ𝑗,𝜉,𝑟 (𝜆)|2 + 𝑂𝜅 (𝑁 −100 ).   (12)
                                               ∑︁

                                             𝑟∈ℛ(𝑗,𝜉)

   Multiplying (12) by 𝑤𝑗 |𝐹̂︀𝑗 (𝜉)|2 and summing over 𝑗, 𝜉 gives



                                                           221
                          |𝐹̂︀𝑗 (𝜉)|4 ≤ 2𝐵co (𝜅)                    |𝐹̂︀𝑗 (𝜉)|2              |𝛽𝑗,𝜉,𝑟 | |ℒ𝑗,𝜉,𝑟 (𝜆)|2 + 𝑜𝜅 (1).   (13)
           ∑︁        ∑︁                              ∑︁        ∑︁                   ∑︁
                𝑤𝑗                                        𝑤𝑗
           𝑗         𝜉                                𝑗        𝜉                  𝑟∈ℛ(𝑗,𝜉)

   Using (4), (7), and (10), (13) implies

                                                                                   𝑐 − 𝑜𝜅 (1)
                                          ∫︁
                                                     |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≥                      .                                  (14)
                                           𝒫𝜅 (𝑁 )                                2𝐵co (𝜅)𝑆𝜅
   By (8),

                                                                              𝑐
                                     ∫︁
                                                 |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≥                − 𝑜𝜅 (1)                                     (15)
                                      𝒫𝜅 (𝑁 )                               𝐶MF (𝜅)
   with

                                                      𝐶MF (𝜅) = 2𝐵co (𝜅)2 .                                                      (16)
   This proves the measured Fourier transfer.
   —

Parameter check G.11 (TGT-MF.5. Parameter Check: Complexity and Normalizations). The
construction preserves the exact normalization needed later by MRT and X9L-GT:

  1. 𝜈𝜅 is a probability measure by (10).

  2. Each test is normalized by 𝐻𝑝−1 .

  3. The AP modulus/content is polylogarithmic because the F3/F4/E5 transport operations have
     controlled content and the macro-template 𝜅 is fixed.

  4. The weight 𝜌𝑝 has polylogarithmic derivative complexity inherited from the original smooth
     dyadic and CRT cutoffs.

  5. Boundary components are not part of 𝒫𝜅 ; they are routed to C1 before this lemma is invoked.

  6. No single dyadic fibre, interval start, or Fourier frequency is selected as the obstruction. The
     obstruction is carried by the finite probability measure 𝜈𝜅 .

   The start-pushforward regularity of 𝜈𝜅 is not asserted here. It is the separate PACK/MRT
question handled by MRT, TTD, ROC, BRS, and TTH.
   —

TGT-MF.6. Output Form                          For use in TGT, TTD, TNG, and TTH, the output is:

                                                      GT-U2 =⇒ GT-Test

   where
                                                          ∫︁
                                   GT-Test :                         |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≫𝜅,𝑐 1.
                                                           𝒫𝜅 (𝑁 )

    The implicit constant is 𝑐/𝐶MF (𝜅) up to the C1-negligible 𝑜𝜅 (1) boundary term. This is the
closed measure-theoretic/Fourier bridge required by the TC1 global testing route.

                                                                   222
G.5     TTH-SC structural coarea closure
G.5.1    TTH-SC. Structural Coarea Closure and No Artificial Short-Interval Refinement
TTH-SC.0. Statement and Role Lemma TTH-SC is the formal closure principle used in
the TC1 near-global route. It proves that a released near-global structural coarea image cannot be
replaced by arbitrary short shifted intervals inside the active TC1 testing family.
    Fix a TC1 macro-template 𝜅 after the B1/B3/F3/F4 routing interface, C1 boundary removal,
and the TGT.2/TGT-MF coarea construction. Let 𝒫𝜅 (𝑁 ) be the finite family of structural coarea
tests released by TGT-MF, with probability measure 𝜈𝜅 . For 𝑝 ∈ 𝒫𝜅 (𝑁 ), write

                                           1 ∑︁
                                ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),
                                           𝐻𝑝 𝑛∈𝐼
                                                 𝑝


   where 𝐼𝑝 is the marked B1-origin coarea image piece and 𝐻𝑝 = |𝐼𝑝 |.
   Then every refinement of a released test that can occur inside the proof is classified by exactly
one of the following alternatives.

  1. Controlled structural refinement. The refinement is generated by the finite TGT-MF
     coarea algebra: dyadic scale subdivision, AP/modulus normalization, smooth-weight partition,
     controlled CRT restriction, fixed divisor quotienting, full-rank affine transport, or primitive
     slicing. It produces at most (log 𝑁 )𝐶𝜅 child tests, and each child has length

                                           𝐻𝑝′ ≥ 𝐻𝑝 (log 𝑁 )−𝐶𝜅 .                             (SC1)

      Therefore a near-global parent remains near-global after enlarging the logarithmic exponent.

  1. Non-structural analytic subdivision. The subdivision is a partition of 𝐼𝑝 chosen after the
     structural coarea test has already been released and is not one of the generators in the TGT-
     MF coarea algebra. Such pieces are not elements of 𝒫𝜅 (𝑁 ), carry no independent testing mass,
     and are reassembled into the parent functional before the X9L-GT input is invoked.

  1. Genuine structural short-image alternative. The refinement is structural and produces
     a child image shorter than the controlled lower bound (SC1). Then this is not an artificial
     subdivision of an already released near-global test. It is a genuine short-image B1-origin
     certificate exported to the singular routing package before X9L-GT is applied.

   Consequently no arbitrary shifted short interval can survive as an active unrouted TC1 input to
X9L-GT.
   Logical dependencies are TGT-MF, TGD, F3/F4, C1P/C1A/C1, E5, and the parameter register.
The lemma does not use TTH, TTD, ROC, BRS, or the full TNG closure theorem.
   —

TTH-SC.1. Setup: The Structural Coarea Algebra For a fixed macro-template 𝜅, let 𝒜𝜅
be the finite coarea algebra generated by the operations that are already present in the TGT-MF
construction and the preceding F3/F4 routing interface:

  1. fixing/projection of bounded coordinates;

  2. CRT restriction by polylogarithmic moduli;


                                                 223
   3. fixed divisor quotienting by controlled divisors;

   4. full-rank affine transport with controlled content;

   5. dyadic scale and AP/modulus normalization;

   6. smooth-weight partition of bounded differentiability complexity;

   7. primitive slicing;

   8. post-terminal Fourier/cube subdivisions that preserve the marked Liouville origin.

   The number of atoms produced by this algebra inside a fixed 𝜅-cell is bounded by (log 𝑁 )𝐶𝜅 .
This follows from the fixed macro-template complexity, the polylogarithmic modulus bounds, and
the parameter register.
   A released TC1 coarea test is an atom of 𝒜𝜅 which has not been routed to Edge, LongAP/
Local, CKP, LocalDiag, empty support, or an impossible support class before the TGT-MF testing
measure is formed.
   Thus 𝒫𝜅 (𝑁 ) is supported only on released atoms of 𝒜𝜅 .
   —

TTH-SC.2. Proof: Controlled Structural Refinements Let 𝑝 ∈ 𝒫𝜅 (𝑁 ) be released and
suppose that a child 𝑝′ is obtained by applying further generators of 𝒜𝜅 which are allowed after
release only for scale, modulus, smoothness, or bounded primitive normalization.
    Each such generator has one of two effects.
    First, it may restrict to one of finitely many residue or smooth-weight classes. The number of
classes is at most (log 𝑁 )𝐶𝜅 , and empty or boundary classes are routed to C1P/C1A/C1.
    Second, it may change the AP modulus or the smooth weight while preserving the marked
image 𝐿𝑚 (Ω* ) up to a polylogarithmic partition. Again the number of nonempty pieces is at most
(log 𝑁 )𝐶𝜅 .
    Therefore every non-routed child satisfies

                                        𝐻𝑝′ ≥ 𝐻𝑝 (log 𝑁 )−𝐶𝜅 .

   If the parent satisfies

                                        𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 ,

    then, after absorbing the fixed polylogarithmic losses and the height distortion 𝑋𝑝′ =
𝑋𝑝 (log 𝑋𝑝 )𝑂𝜅 (1) , the child satisfies
                                                             ′
                                       𝐻𝑝′ ≥ 𝑋𝑝′ (log 𝑋𝑝′ )−𝐵𝜅

   for a larger exponent 𝐵𝜅′ . Thus controlled structural refinement does not create a low-theta
short-interval input.
   —




                                                 224
TTH-SC.3. Proof: Non-Structural Analytic Subdivisions Suppose that 𝐼𝑝 is partitioned
into subintervals or AP subpieces
                                                         𝐼𝑝 =
                                                                  ⨆︁
                                                                       𝐼𝑝,𝜔
                                                                𝜔∈Ω𝑝

   after 𝑝 has already been released, and assume that this partition is not generated by 𝒜𝜅 .
   Then the subpieces 𝐼𝑝,𝜔 are not elements of 𝒫𝜅 (𝑁 ). In particular, TGT-MF assigns no indepen-
dent testing mass to them, and the global lower bound supplied by TGT-MF is not a statement
about these subpieces. The only functional exported by TGT-MF at this location is the parent
functional ℒ𝑝 .
   Algebraically, after splitting the sum one has
                                                         ∑︁ 𝐻𝑝,𝜔
                                            ℒ𝑝 (𝜆) =                   ℒ𝑝,𝜔 (𝜆; 𝜌𝑝 , 𝛼𝑝 )
                                                        𝜔∈Ω𝑝
                                                                𝐻𝑝

    up to the harmless smoothing errors already included in the C1 boundary accounting. This
identity is used only for internal estimates if needed; it does not create a new released testing family.
Before invoking X9L-GT the pieces are reassembled into ℒ𝑝 .
    Thus an arbitrary shifted short interval obtained in this way is not an active TC1 test.
    —

TTH-SC.4. Proof: Genuine Structural Short Images Are Routed It remains to consider
a structural child 𝑝′ whose image is genuinely shorter than the controlled bound (SC1). Since 𝑝′ is
structural, the shortness is not an analytic refinement chosen after release. It is a property of the
marked B1-origin image on a routed subcell.
    Thus TTH-SC does not estimate this child and does not invoke the singular routing theorems. It
records that the child is a genuine B1-origin short-image certificate and removes it from the active
X9L-GT testing family. The later TTD/ROC/BRS chain consumes exactly this certificate.
    —
Parameter check G.12 (TTH-SC.5. Parameter Check). The only loss exported by TTH-SC is
polylogarithmic. If 𝐵𝜅 is the near-global exponent before structural refinement, choose 𝐵𝜅′ so that
                                            𝐵𝜅′ ≥ 𝐵𝜅 + 𝐶𝜅 + 𝐶height (𝜅) + 10.
    The parameter register chooses the TTH exponent after the TGT-MF coarea complexity, the
BRS/X16 constants, and the smooth-weight decomposition constants. Hence this enlargement is
already absorbed in the final exponent used by TTH.
    No power-saving estimate is weakened by TTH-SC: alternatives 2 and 3 are not estimated
by X9L-GT, while alternative 1 remains inside the same near-global Davenport/AP input after
enlarging 𝐵𝜅 .
    —

TTH-SC.6. Output Form For use in TTH and TNG, the lemma exports the following closed
barrier:


Every short subtest of a released TC1 coarea test is either non-structural and reaggregated, or structural and exported away from X9L.

    Equivalently, every test that is actually passed to X9L-GT is a structural TGT-MF coarea
test, up to controlled polylogarithmic subdivision, and satisfies the near-global length lower bound
supplied by TTH.

                                                                225
G.6     TC1 near-global chain
G.6.1    TNG. B1-Origin TC1 Near-Global-or-Routed Theorem
TNG.0. Statement and Role Lemma TNG is the bridge lemma for the TC1 branch of
GoodAWACK. It packages the route

        𝐵1-origin coarea → 𝑇 𝑇 𝐻-𝑆𝐶 → 𝑀 𝑅𝑇 /𝑇 𝑇 𝐷 → 𝑅𝑂𝐶 + 𝐵𝑅𝑆 → 𝑇 𝑇 𝐻 → 𝑋9𝐿-𝐺𝑇

   into a single checkable source statement.
   The reader-facing master theorem for the combined TNG/TTH route is Lemma TNGTTHM. The
present file supplies the component proof of Theorem TNG-A and the TC1 cancellation statement
consumed there.
   It introduces no new analytic estimate. Its role is to make explicit that the TC1 branch never
invokes a pointwise shifted short-interval theorem for 𝜆. The only X9L input used in the proof is
the near-global Davenport/AP form

                                        𝐻 ≥ 𝑋(log 𝑋)−𝐵

    after the structural B1-origin reductions have been applied.
    Here an active or unrouted coarea test means a structural TGT.2/TGT-MF test whose cell
has not already been sent to Edge, LongAP/Local, CKP, LocalDiag, or empty support. Logical
dependencies are the TGT.2/TGT-MF global-testing construction, TTH-SC, MRT, TTD, ROC,
BRS, TTH, C1P/C1A/C1, D1/H4M, G8a, X16BRS, X16C, E5, TGD, X9L-GT, and the parameter
register. TNG is used by E10L.
    —

TNG.1. Setup: Active B1-Origin Coarea Tests Fix a terminal TC1-GoodAWACK macro-
template 𝜅. It consists of:

  1. a B1 typed parent block;

  2. a B3 grouping record;

  3. the F3/F4 routing history;

  4. a marked Liouville affine form 𝐿𝑚 ;

  5. the TC1 tensor certificate;

  6. the C1-clean smooth box/coset cell Ω* on which the TC1 Fourier/coarea argument is performed.

   An active B1-origin coarea test is a test produced from this data by the coarea decomposition

                                     𝑛 = 𝐿𝑚 (𝑧),    𝑧 ∈ Ω* ,

   after only the following normalizations:

  1. polylogarithmically many scale, modulus, and smooth-weight subdivisions;

  2. controlled CRT restrictions;


                                              226
  3. fixed divisor quotienting with controlled divisor;

  4. full-rank affine transports with controlled content;

  5. removal of C1 boundary pieces.

   Thus a test has the form
                                            1 ∑︁
                                 ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),                   (TNG-test)
                                            𝐻𝑝 𝑛∈𝐼
                                                   𝑝

   where:

        𝑔𝑝 ≤ (log 𝑋𝑝 )𝐶𝜅 ,    𝐻𝑝 = |𝐼𝑝 |,     𝜌𝑝 has polylogarithmic smoothness complexity.

   The word active excludes cells already routed to Edge, LongAP/Local, CKP, LocalDiag, or
empty support. Those cells are handled by C1P/C1A/C1, D1/H4M, G8a, H4M, or contribute zero.
   —

TNG.2. Structural Coarea Closure The coarea interval 𝐼𝑝 is a structural image piece of
the terminal marked B1-origin carrier 𝐿𝑚 (Ω* ). The formal barrier against rogue short-interval
refinements is Lemma TTH-SC.
    More precisely, TTH-SC classifies every refinement of a released coarea test. Controlled scale,
AP/modulus, and smooth-weight subdivisions remain structural and lose only a fixed power of
log 𝑋. A subdivision chosen after release which is not generated by the structural coarea algebra is
not an element of the TGT-MF testing family and is reaggregated into its parent functional before
X9L-GT is invoked. A genuinely structural short-image child is routed through TTD/ROC/BRS/
X16BRS/X16C and C1P/C1A/C1 before any Liouville/AP input is applied.
    Thus arbitrary shifted short intervals are not active TC1 tests, and this is a closure lemma
rather than a convention of exposition.
    —

TNG.3. Proof: Regular Branch Assume that the TC1 testing family for 𝜅 is MRT-admissible.
Then MRT supplies the start-pushforward bound

                                                            𝑑𝑥
                                    (start)# 𝜈𝜅 ≪𝜅 (log 𝑁 )𝐶𝜅  .                           (PACK)
                                                            𝑋
   For every active B1-origin coarea test in this family, TTH supplies

                                        𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 .                                (TTH)
   Together with the polylogarithmic modulus and smoothness bounds in TNG.1, this is exactly
the hypothesis set of the near-global X9L-GT theorem:

              PACK + {𝑔𝑝 ≤ (log 𝑋𝑝 )𝐶𝜅 } + {𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 } =⇒ X9L-GT-NG.

   Indeed X9L-GT uses Davenport’s estimate in AP form. The loss in passing from global prefixes
to AP fibres is bounded by

                                        (log 𝑋𝑝 )2𝐶𝜅 +𝐵𝜅 +𝑂𝜅 (1) .


                                                  227
    Choosing the Davenport logarithmic saving exponent larger than this loss and the required final
saving gives
                                    ∫︁
                                         |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) = 𝑜𝜅 (1).                     (X9L-NG)

    By the TGT.2/TGT-MF global-testing construction, a non-small TC1 macro-template would
force a fixed lower bound for the same left side. Therefore the MRT-admissible branch contributes
𝑜(𝑁 ).
    —

TNG.4. Proof: Singular Branch Routes Before X9L If MRT-admissibility fails, the testing
measure has singular start concentration. TTD identifies the only possible unrouted singular
geometry: the marked form moves through a short additive image while transverse B1-origin
variables carry the volume.
    The route is then structural, not analytic.
    First, ROC proves range comparability for direct dyadic-coordinate origins and controlled full-
rank transports. It also routes tagged failures to the already existing terminal classes.
    Second, the complementary solved-affine or quotient-origin case is handled by BRS. BRS applies
the B1 carrier-slice estimate, supplied by X16BRS and X16C, and proves the dichotomy

                             short marked image =⇒ strict C1P Edge

   unless the failure already carries a LongAP/Local, CKP, LocalDiag, Edge, or empty routing tag.
   Thus a singular TC1 testing family is never sent to X9L-GT. It is routed to:

                  𝐶1𝑃/𝐶1𝐴/𝐶1,            𝐷1/𝐻4𝑀,          𝐺8𝑎,         𝐻4𝑀,   or 0.

   —

TNG.5. Output Theorem: TC1 Near-Global-or-Routed The TTH/BRS/X16 part of the
TC1 proof is used through the following single theorem-interface. It is intentionally stronger as an
interface than the individual component lemmas: it classifies the actual tests that reach the TC1
global-testing stage.

Theorem G.13 (Theorem TNG-A. TC1 tests are near-global or routed away). Fix a B1/B3/
F3/F4 terminal TC1-GoodAWACK macro-template 𝜅 whose cell has not already been routed away,
after C1 boundary removal, fixed macro-template normalization, and polylogarithmic scale/modulus/
smooth-weight decomposition. Let ℒ𝑝 (𝜆) be any unrouted coarea test produced by TGT from the
marked B1-origin form.
    Then exactly one of the following alternatives holds.

  1. Near-global testing alternative. The test belongs to the regular MRT-admissible branch.
     The start-pushforward satisfies PACK, the AP modulus and smoothness complexity are polylog-
     arithmic, and TTH gives

                                            𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 .

     Hence this test is an allowed input to the near-global Davenport/AP theorem X9L-GT.


                                                   228
  2. Routed alternative. The test is not sent to X9L-GT. Before any Liouville/AP input is in-
     voked, TTD, ROC, BRS, and the X16BRS/X16C carrier-slice estimate route the corresponding
     cell to one of

                     𝐶1𝑃/𝐶1𝐴/𝐶1,            𝐷1/𝐻4𝑀,      𝐺8𝑎,      𝐻4𝑀,         or 0.

   In particular, there is no third case consisting of an arbitrary shifted short interval or an
unclassified short AP fibre. The exclusion of that third case is supplied by TTH-SC.

Proof. Start with the coarea tests constructed in TGT from the fixed macro-template 𝜅. MRT first
separates the regular branch from the singular start-concentration branch.
    In the regular branch, MRT supplies PACK for the same testing family. The coarea test still has
B1-origin in the sense of TTH.2, because the only normalizations are controlled CRT restrictions,
fixed-divisor quotients, full-rank transports, and post-terminal analytic subdivisions that do not
replace the terminal marked carrier. TTH-SC prevents the released test from being replaced by a
new arbitrary short-interval family. TTH then gives the near-global length lower bound for every
remaining coarea image piece. The modulus and smoothness complexity bounds are those recorded
in TNG.1. Thus the test is exactly an X9L-GT input.
    In the singular branch, TTD identifies a singular-origin mechanism. Direct dyadic-coordinate and
tagged full-rank transport cases are handled by ROC. The complementary solved-affine, quotient,
and carrier-slice cases are handled by BRS. In BRS, a genuinely short marked B1 image cannot
carry uncontrolled transverse mass: X16BRS reduces all BRS carrier types to X16-Core, and X16C
proves X16-Core. Therefore a short B1 image is a strict C1P Edge contribution unless it already
carries a LongAP/Local, CKP, LocalDiag, Edge, empty, or nonterminal routing tag. These are
precisely the routed alternatives listed above.
    Finally, TTH-SC gives the closure barrier for refinements of an already released near-global
structural image. Non-structural short pieces are aggregated back to the parent piece, while genuine
structural short-image children are routed before X9L-GT. Hence no pointwise shifted short-interval
escape case remains. The theorem follows.
    For publication checking, the component bridge behind the theorem is

                                   BRS/X16 =⇒ TTH =⇒ X9L

   is the following finite decision table on an unrouted TC1 coarea test.


  Test status after TGT Structural source         BRS/X16 action            Result before X9L
  coarea
  Direct B1 product car- B1/B3/F3/F4 marked       BRS range comparabil- 𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 ;
  rier, full-rank transport, carrier              ity holds              X9L-GT may be in-
  no short image                                                         voked.
  Direct B1 product car- same                     X16BRS/X16C carrier- strict C1P Edge via
  rier with short marked                          slice estimate bounds  C1A E6; no X9L invo-
  image                                           the short-image mass   cation.
  Complementary carrier F4/BRS solved-affine      Replace by product     near-global or strict
  𝑁 −𝑃                       origin               carrier 𝑃 and apply    Edge.
                                                  X16BRS/X16C
  Quotient carrier 𝑠 in   F4 quotient tag         Transfer 𝑠 ∈ 𝐼 to 𝐿 ∈ near-global or strict
  𝐿 = 𝑑𝑠 with tagged 𝑑                            𝑑𝐼; controlled divisor Edge.
                                                  sum is absorbed


                                                229
   Untagged quotient/        unresolved F4 ordinary F4 does not release the routed to Edge,
   divisor relation          divisor predicate      cell to TC1 testing     LocalDiag, CKP,
                                                                            GoodAWACK with
                                                                            tag, or nonterminal de-
                                                                            crease.
   Singular start measure TTD singular branch       ROC handles direct/     routed before X9L.
   from non-direct origin                           tagged origins; BRS
                                                    handles solved-affine
                                                    complement
   Artificial subdivision of TTH-SC non-structural Aggregated back to the no pointwise shifted
   an already near-global case                      structural image piece  short-interval input is
   image                                                                    created.
   Genuine structural        TTH-SC structural      TTD/ROC/BRS/            no unclassified short
   short-image refinement short-image case          X16BRS/X16C and         AP fibre remains.
                                                    C1P/C1A/C1 route it



    Thus the only tests actually passed to X9L-GT are the first row: unrouted structural coarea
image pieces whose length is near-global after BRS/TTH. The second row is the critical use of X16.
It says that a genuinely short marked B1 image cannot hide a large transverse mass: the B1 carrier-
slice estimate converts it into a strict C1P Edge contribution.
    This formulation also fixes the quantifiers. TTH is not a theorem about arbitrary E7 directional
fibres or arbitrary shifted subintervals. It is a theorem about the unrouted B1-origin coarea tests
selected by TGT after F3/F4 routing, C1 boundary removal, and TTD/MRT normalization.
    —


TNG.6. Output: TC1 Cancellation Theorem

Theorem G.14 (Theorem TNG). For every unrouted B1/B3/F3/F4 terminal TC1-GoodAWACK
macro-template 𝜅, after C1 boundary removal and fixed macro-template normalization, Theorem
TNG-A applies to every TC1 coarea test. Consequently:

  1. every test sent to X9L-GT is near-global and MRT-admissible;

  2. every non-near-global or singular test is routed to an already handled terminal class before
     X9L-GT is invoked.

   Consequently

                                   𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

Proof. Aggregate terminal TC1 atoms by the fixed macro-template 𝜅 as in TGT. Apply Theorem
TNG-A to the unrouted coarea tests.
   On the near-global alternative, MRT supplies PACK and TTH supplies the near-global length
lower bound for the same B1-origin coarea tests. Hence the near-global X9L-GT theorem applies
with the parameters listed in TNG.3. This contradicts the fixed TGT testing lower bound unless
the 𝜅-contribution is 𝑜(𝑁 ).


                                                 230
   On the routed alternative, the cell is sent to Edge, LongAP/Local, CKP, LocalDiag, or empty
support by TTD/ROC/BRS using X16BRS/X16C where the carrier-slice estimate is needed. These
outputs are outside terminal TC1-GoodAWACK and are handled by C1P/C1A/C1, D1/H4M, G8a,
H4M, or zero. Therefore the routed branch contributes no terminal TC1-GoodAWACK mass.
   There are only boundedly many structural TC1 macro-templates, depending on the fixed
parameter 𝐽0 . Summing over them gives the displayed 𝑜(𝑁 ) estimate. Theorem proved.
   —


     Remark G.15 (TNG.7. Output). TNG-A + X9L-GT =⇒ 𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).
    Here TNG-A is the single TC1 structural theorem packaging TGT/TTH-SC/MRT/TTD/ROC/
BRS/X16BRS/X16C/TTH. The chain uses X9L-GT only in the near-global Davenport/AP form.
It does not use:

  1. X8 inverse-Gowers input;

  2. pointwise shifted short-interval Liouville cancellation;

  3. a low-𝜃 polylog-modulus theorem for arbitrary short AP fibres.

TNG.8. Logical Dependencies External dependency: X9L-GT in the near-global Davenport/
AP form.
  Internal dependencies: the TGT.2/TGT-MF global-testing construction, TTH-SC, MRT, TTD,
ROC, BRS, TTH, C1A, C1, D1, H4M, G8a, X16BRS, X16C, E5, TGD, and the parameter register.
  Children served: TNGTTHM, E10L, and the GoodAWACK TC1 branch.

G.7     TC1 testing dichotomy
G.7.1    TTD. TC1 Testing Dichotomy
TTD.0. Statement and Role Lemma TTD is the testing-dichotomy reduction. Regular TC1
testing families close by MRT/TTH and the near-global X9L input, while singular B1-origin cases
are closed by ROC and BRS.
    The target isolated in TGT is:

                                 TC1-TESTING-DICHOTOMY.

   The desired statement is:
   For every actual B1/B3/F3/F4 terminal TC1 macro-template 𝜅, after C1 boundary removal,
the induced global Liouville testing family is either:

  1. averaged-admissible in the sense of TGT; or

  2. its singular part has a B1-origin route to strict C1P Edge, D1/H4M LongAP/Local, G8a CKP,
     LocalDiag, or empty/impossible.

   The theorem supplied by TTD is:


regular families close by X9L-GT after MRT and TTH, and singular families route by ROC/BRS.


                                                231
   The singular-origin component is:

 TC1-SINGULAR-ORIGIN : every singular TC1 testing measure has an existing routing origin.

   This is narrower than the pointwise X9L-SI obstruction. It is a structural B1-origin problem,
not an analytic short-interval estimate.
   Logical dependencies are the TGT.2/TGT-MF global-testing construction, MRT, ROC, BRS,
TTH, C1P/C1A/C1, D1/H4M, G8a, X16BRS, X16C, and X9L-GT. TTD is used by TNG and
E10L.
   —

TTD.1. Setup and Regular Branch                Let 𝜅 be a fixed TC1 macro-template. By TGT, if

                                              |𝑅𝜅 (𝑁 )| ≥ 𝜀𝑁                                     (1)
  along an infinite sequence, then the global TC1 generalized von Neumann step and Lemma
TGT-MF produce a measured testing family

                                                (𝒫𝜅 (𝑁 ), 𝜈𝜅 )

   whose tests have the form
                                              1 ∑︁
                                ℒ𝑝 (𝜆) =             𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),                          (2)
                                              𝐻𝑝 𝑛∈𝐼
                                                     𝑝

   and satisfy the fixed lower bound
                                  ∫︁
                                              |ℒ𝑝 (𝜆)|2 𝑑𝜈𝜅 (𝑝) ≫𝜅,𝜀 1.                          (3)
                                    𝒫𝜅 (𝑁 )

   If (𝒫𝜅 , 𝜈𝜅 ) is MRT-admissible, then the averaged Liouville input gives the opposite bound 𝑜(1).
Hence:

Lemma G.16 (Lemma TTD.1. Regular testing branch). Assume the averaged Liouville input X9L-
GT for MRT-admissible testing families. If the TC1 testing family induced by 𝜅 is MRT-admissible,
then

                                              𝑅𝜅 (𝑁 ) = 𝑜(𝑁 ).

Proof. This is the regular-branch part of the TGT/TGT-MF construction, with MRT-admissibility
checked through MRT. The point is that the macro-template aggregation gives a fixed lower bound
(3), so qualitative averaged cancellation suffices. Lemma proved.
    Thus the only remaining case is non-admissible, or singular, testing measure.
   —




                                                    232
TTD.2. Setup: Geometry of Singular Testing Measure After the standard polylogarithmic
decomposition into dyadic scale, AP modulus, and smooth weight-complexity classes, the failure of
MRT-admissibility means that the start pushforward

                                             (start)# 𝜈𝜅

   is not dominated by a polylogarithmic multiple of normalized measure on its dyadic start range.
   At the affine-geometric level this can happen only if a positive fraction of the testing mass is
carried by pieces where the marked affine image has too few independent start directions. The
model is

                   Ω = [𝑋0 , 𝑋0 + 𝐻] × [1, 𝑀 ],      𝐿𝑚 (𝑢, 𝑣) = 𝑢,    𝐻𝑀 ≍ 𝑁,                    (4)
   with

                                         𝐻 < 𝑋0 (log 𝑋0 )−𝐵 .                                     (5)
    The transverse coordinate 𝑣 supplies volume, but it does not move the Liouville interval. The
testing measure is concentrated on essentially one short interval near 𝑋0 .
    This is the common geometric content of:

   1. the rank-one/nonregular E7 carrier;

   2. the short affine-image residual in the TC1 coarea/Fourier decomposition;

   3. the singular affine-image model;

   4. the singular testing measure of TGT.

   —

TTD.3. Proof: Tagged Singular Origins Route Away The singular geometry is harmless if
it comes from an already tagged origin.

Lemma G.17 (Lemma TTD.2. Tagged singular origin routes away). Suppose a singular TC1
testing subfamily arises because the marked affine image has lost start directions through one of the
following tagged operations:

  1. fixing/projection with short residual volume;

  2. fixed divisor quotient with a short quotient range;

  3. variable quotient residual whose quotient range is short;

  4. local/diagonal forcing;

  5. CKP balanced grouping;

  6. strict C1P Edge origin;

  7. impossible or inconsistent fibre;

  8. post-terminal primitive slicing that does not create a new terminal GoodAWACK skeleton.


                                                  233
   Then the subfamily contributes only to C1, D1/H4M, G8a, LocalDiag, or zero, and does not
remain in terminal TC1-GoodAWACK.

Proof. This follows directly from the routing interface fixed by B3, F3, F4, E5, and the terminal-
operation rule stated in TGD.
    Items 1, 2, 3, and 6 are C1P-certified or F4 short-volume/Type-I cases. F4.6 routes short divisor
or short quotient cases to Edge, and C1 counts only those Edge cases with an explicit summable
budget.
    Item 4 is F4.7/F3 LocalDiag detection.
    Item 5 is the CKP route handled by G8a.
    Item 7 is empty.
    Item 8 is terminal-interface clean: post-terminal primitive slicing, Cauchy/cube operations,
and Fourier expansion do not generate new terminal GoodAWACK skeletons. The terminal TC1/
HighTC test uses the pre-slicing affine vectors.
    Thus every tagged singular origin is routed away from terminal TC1-GoodAWACK. Lemma
proved.
   —


TTD.4. Statement: Singular-Origin Criterion The only possible obstruction to the regular
branch is a singular testing subfamily whose marked Liouville form moves through a short additive
image while transverse B1-origin variables carry the volume. In model form this geometry is

                                Ω = [𝑋0 , 𝑋0 + 𝐻] × [1, 𝑀 ],                             𝐿𝑚 (𝑢, 𝑣) = 𝑢,                   𝐻𝑀 ≍ 𝑁,                                       (6)
     where:

    1. 𝐻 ≥ 𝑁 𝜃 , so no short-direction C1P predicate is automatic;

    2. 𝐻 < 𝑋0 (log 𝑋0 )−𝐵 , so the Liouville image is a shifted short interval;

    3. the full effective volume is 𝐻𝑀 ≍ 𝑁 , so C1 short-volume Edge is not automatic;

    4. 𝐿𝑚 has controlled content;

    5. no LocalDiag or CKP relation is forced at the terminal interface;

    6. the marked 𝜆(𝐿𝑚 ) factor remains a nonlocal oscillatory coefficient, so LongAP/Local does not
       apply.

     The singular-origin assertion is:

TC1-SINGULAR-ORIGIN : model (6), and every equivalent singular testing measure, cannot arise from an actual B1-origin terminal TC1 macro-template unless Lemma TTD.2 applies.



     —




                                                                                   234
TTD.5. Proof: Range-Origin Comparability and BRS Closure

Lemma G.18 (Lemma ROC). For every actual terminal GoodAWACK marked Liouville form 𝐿𝑚 ,
after C1 boundary removal and after passing to a fixed TC1 macro-template, either:

  1. the affine image satisfies near-global range comparability


             |𝐿𝑚 (Ω)| ≥ 𝑋𝑚 (log 𝑋𝑚 )−𝐶 ,      𝑋𝑚 ≍ max(2, dist(𝐿𝑚 (Ω), 0) + |𝐿𝑚 (Ω)|);        (ROC)

  1. or the failure of (ROC) is caused by a tagged origin from Lemma TTD.2.

   If ROC holds, the marked image is closed by the near-global coarea argument and the near-global
X9L-GT input. If ROC fails, Lemma TTD.2 routes it away.
   Therefore ROC proves the direct-origin part of the TC1 testing dichotomy.

Proof of Lemma ROC. B1 begins with dyadically localized product variables, whose value and
additive range are comparable. Controlled CRT restrictions and fixed divisor quotients only lose
polylogarithmic factors. Variable quotient residuals are routed by F4 if the quotient is short, local,
or CKP; otherwise the quotient is central-long. E10M and E10K forbid untagged rank-dropping
affine changes in a terminal GoodAWACK skeleton.


BRS closure of the complementary singular case Lemma ROC proves range-origin compa-
rability for direct dyadic-coordinate origins and their controlled CRT/divisor/full-rank transports.
It also confirms that tagged failures route by Lemma TTD.2. The part not covered by direct
comparability is the complementary affine-origin case, where a short marked image carries hidden
transverse B1 multiplicity.
    Lemma BRS closes exactly that case using X16BRS/X16C. It proves:

                              B1-RANGE-SKELETON/ROC-SLICE.

   Thus TC1-SINGULAR-ORIGIN is supplied by ROC plus BRS.
   —

TTD.6. Output Theorem           Use:

  1. X9L-GT: averaged Liouville cancellation for MRT-admissible testing families;

  2. ROC and BRS, which prove TC1-SINGULAR-ORIGIN;

  3. TTH, which puts unrouted tests in the cited X9L-GT range.

   Then

                                   𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).




                                                 235
Proof. Aggregate by TC1 macro-template. Lemma MRT first selects the regular MRT-admissible
branch or a singular-origin branch. If a template is MRT-admissible, Lemma TTD.1 closes it. If it
is singular, ROC plus BRS gives either ROC, Edge by slice mass, or one of the tagged origins in
Lemma TTD.2. Hence the singular part is closed by the near-global coarea argument or routes to
C1, D1/H4M, G8a, LocalDiag, or zero. Summing over the bounded number of macro-templates
gives the claim. Theorem proved using the BRS/X16-Core input supplied by Lemmas X16BRS and
X16C.
    —


                                  Remark G.19 (TTD.7. Output).
TC1-TESTING-DICHOTOMY is proved, with MRT selection explicit and BRS/X16-Core supplied by X16BRS and X16C.
   What is proved here:


MRT-admissible testing families close by X9L-GT after TTH, and singular origins route away by ROC/BRS.

    The low-theta alternative is not used. Lemma TTH supplies the near-global bound 𝐻 ≥
𝑋(log 𝑋)−𝐵 , and X9L-GT supplies the averaged Liouville input for that range. The singular
structural branch is not a residual.

TTD.8. Logical Dependencies External dependency: X9L-GT after TTH supplies the near-
global range.
   Internal dependencies: the TGT.2/TGT-MF global-testing construction, MRT, ROC, BRS,
TTH, C1P/C1A/C1, D1/H4M, G8a, X16BRS, and X16C.
   Children served: TNG and E10L.

G.8     MRT admissibility
G.8.1    MRT. PACK Interface for TC1 Global Testing
MRT.0. Statement and Role Lemma MRT verifies the PACK interface required before TGT
applies the averaged Liouville input.
   The important distinction is:

  1. TTH supplies a near-global length lower bound 𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 .

  2. MRT-admissibility also needs a start-pushforward bound:
                                                                   𝑑𝑥
                                         (start)# 𝜈𝜅 ≪ (log 𝑁 )𝐶      .                          (PACK)
                                                                   𝑋

    Length alone does not imply PACK. The exact interface is: regular TC1 macro-templates satisfy
PACK by finite B1/B3/F3/F4 multiplicity; failure of PACK is a singular-origin event exported to
the TTD/ROC/BRS routing package before X9L-GT is invoked.
    Logical dependencies are B1, B3, F3/F4, E5, TGD, TGT-MF, and the parameter register. MRT
is used by TGT, TTD, TNG, and E10L.
    —




                                                   236
MRT.1. Setup: Testing Family            Fix a TC1 macro-template 𝜅. It consists of:

  1. a parent B1 block;

  2. a B3 grouping candidate;

  3. the F3/F4 routing history;

  4. a marked affine Liouville form 𝐿𝑚 ;

  5. the measured coarea/Fourier decomposition supplied by TGT-MF.

   Each test in the family has the form

                          1 ∑︁
               ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),      𝐼𝑝 = [𝑥𝑝 , 𝑥𝑝 + 𝐻𝑝 ] ∩ 𝐿𝑚 (Ω𝑝 ).
                          𝐻𝑝 𝑛∈𝐼
                                𝑝


    The measure 𝜈𝜅 is the normalized volume weight inherited from the tagged B1/B3/F3/F4 cell.
    For Lemma MRT, a regular TC1 macro-template means a B1-origin TC1 macro-template
for which the marked coarea map 𝐿𝑚 has rank one on an active full-rank direction of the current
parameter lattice and the induced start map is not collapsed onto a lower-dimensional/rank-deficient
image. Rank-deficient, point-mass, or short-image failures are by definition sent to the singular
branch in MRT.3.
    —

MRT.2. Proof: Regular Multiplicity Condition For a fixed macro-template 𝜅, say that the
start map is regular if for every interval 𝐽 ⊂ [𝑋, 2𝑋]

                                                                   |𝐽|
                                    𝜈𝜅 {𝑝 : 𝑥𝑝 ∈ 𝐽} ≪ (log 𝑁 )𝐶𝜅       .                    (REG-start)
                                                                    𝑋
    This is exactly PACK, with 𝐶 = 𝐶𝜅 .
    In the B1-origin setting, REG-start follows from finite routing multiplicity when the start
coordinate is the image of a full-rank dyadic coordinate map. The source of the full-rank condition
is the regular branch just defined: rank-deficient coarea maps are not regular MRT inputs but
singular-origin inputs exported to TTD/ROC/BRS.
    Full-rank affine transports distort length and lattice index only by polylogarithmic factors by
E5. Specifically, E5.2 controls CRT restrictions, E5.4 controls primitive slicing by writing the image
as 𝑔𝑢 + 𝑏 with controlled 𝑔, and E5.5 states that full-rank affine changes preserve content up to
controlled factors. The parent B1 variables are dyadically localized, and the routing grammar has at
most (log 𝑁 )𝐶0 cells. Hence a subinterval of relative length |𝐽|/𝑋 captures at most a polylogarithmic
multiple of that relative volume.
    This proves MRT-admissibility for the regular full-rank B1-origin TC1 testing families.
    —

MRT.3. Proof: Failure of PACK is Singular If REG-start fails, the TC1 tests concentrate
too much mass on too few starts. For a B1-origin macro-template, this can only happen through
one of the structural singular mechanisms already named in TTD/ROC:

  1. rank-one/nonregular E7 carrier;


                                                  237
  2. short image of the marked B1 carrier;

  3. fixed or variable quotient range collapse;

  4. forced local dependence or diagonal collision;

  5. impossible/empty support.

  These are not sent to X9L-GT. MRT records them as singular-origin certificates; TTD/ROC/
BRS consume those certificates and route them to C1P Edge, LongAP/Local, CKP, LocalDiag, or
empty support.
  Thus X9L-GT is invoked only on tests satisfying PACK.
  —

MRT.4. Statement: Interface Lemma Lemma MRT. Let 𝜅 be a B1-origin TC1 macro-
template after F3/F4 routing and C1 boundary removal.
   Then exactly one of the following holds:

  1. the induced testing family (𝒫𝜅 , 𝜈𝜅 ) is MRT-admissible in the sense of TGT.4;

  2. 𝜅 has a singular-origin certificate exported to the TTD/TNG-A structural branch before X9L-
     GT is invoked.

Proof. If the start map satisfies REG-start, PACK follows by MRT.2. This regular case uses only
the E5/TGD full-rank transport control and the finite B1/B3/F3/F4 routing multiplicity; it does
not use X16-Core.
   If REG-start fails, the failure is a concentration/rank-collapse event in the B1-origin coarea map.
The finite B1/B3/F3/F4 grammar leaves only the five mechanisms listed in MRT.3. MRT therefore
emits a singular-origin certificate for the later TTD/TNG-A branch; it does not itself invoke BRS,
TTH, or X16C. Lemma proved.
   —


                                 Remark G.20 (MRT.5. Output).
              X9L-GT is applied only after Lemma MRT selects the MRT branch.
   Downstream, the singular branch is handled by TTD/ROC/BRS using X16BRS and X16C
where the B1 carrier-slice estimate is needed, not by a pointwise short-interval Liouville theorem.

MRT.6. Logical Dependencies Internal dependencies: B1, B3, F3/F4, E5, TGD, TGT-MF,
and the parameter register.
   Children served: TGT, TTD, TNG, and E10L.

G.9     ROC singular-origin routing
G.9.1    ROC. Range-Origin Lemma for Singular TC1 Testing
ROC.0. Statement and Role Lemma ROC is the singular-origin reduction feeding the BRS/
TTH route. Direct B1-origin short-image cases are closed by BRS and X16BRS, while remaining
tagged origins route to Edge, CKP, LocalDiag, LongAP/Local, or Impossible.
   The lemma supplies the singular-origin block consumed by the TC1 dichotomy:


                                                  238
                                         TC1-SINGULAR-ORIGIN/ROC.

   The desired range-origin comparability statement is:
   For every actual B1-origin terminal TC1-GoodAWACK marked form 𝐿𝑚 , after C1 boundary
removal and fixed macro-template normalization, either

                |𝐿𝑚 (Ω)| ≥ 𝑋𝑚 (log 𝑋𝑚 )−𝐶 ,               𝑋𝑚 ≍ max(2, dist(𝐿𝑚 (Ω), 0) + |𝐿𝑚 (Ω)|),                     (ROC)
    or the failure of (ROC) has an existing C1/D1/G8a/LocalDiag/empty origin tag.
    The direct-origin part proved in ROC is:


direct dyadic-coordinate origins and their controlled CRT/divisor/full-rank transports satisfy ROC.

   The complementary/solved affine origins are supplied by the subsequent BRS carrier-slice
theorem:


B1-RANGE-SKELETON/ROC-SLICE : enrich terminal GoodAWACK skeletons with additive image length and coarea slice-multiplicity data.


    Logical dependencies are B1, BGS, BRS.1, X16BRS, X16C, and the E10Y/E10M/E10K terminal-
affine grammar interface. The dependency on BRS is noncircular: BRS uses only ROC.1 and ROC.2,
while the full ROC closure is obtained after BRS is invoked. ROC is consumed by TTD, TNG,
TNGTTHM, and E10L. It is not an input to TTH-SC; TTH-SC only exports the structural short-
image certificates that the later TTD/ROC/BRS chain consumes.
    —

ROC.1. Proof: Clean Dyadic-Coordinate Origins Satisfy ROC Suppose 𝐿𝑚 is a surviving
parent/grouped coordinate origin in the sense of Lemma BGS, Type A, and that no Goldbach
complement or quotient-solving step is used to define it.
   At the B1/B3 level, the corresponding grouped variable 𝑢 is dyadically localized:

                                                           𝑢 ≍ 𝑈.

    If 𝑢 is terminal GoodAWACK and not C1-routed, it is long:

                                                          𝑈 ≥ 𝑁 𝜃.

    Its additive image on the dyadic cell has length

                                                        |𝑢(Ω)| ≍ 𝑈.

    Also

                                                          𝑋𝑚 ≍ 𝑈.

    Therefore

                                                     |𝐿𝑚 (Ω)| ≍ 𝑋𝑚 ,

    which is stronger than (ROC).

                                                             239
Stability under controlled transports The same conclusion survives the following operations:

  1. controlled CRT restriction, losing at most a polylogarithmic index;

  2. fixed divisor quotient 𝐿 ↦→ 𝐿/𝑑 with 𝑑 ≤ (log 𝑁 )𝐶 ;

  3. full-rank affine coordinate changes with polylogarithmically bounded minors and inverse
     minors, as normalized by the E10Y/E10M/E10K terminal-affine grammar interface;

  4. removal of C1 boundary pieces.

    Indeed, each operation changes additive image length and height by at most a polylogarithmic
factor unless it is rank-dropping. Rank-dropping operations are tagged by the terminal-affine
grammar interface and are exported to the tagged-origin routing alternatives consumed later by
TTD.
    Hence:

Lemma G.21 (Lemma ROC.1. Direct-origin range comparability). Direct dyadic-coordinate marked
forms and their controlled full-rank CRT/divisor transports satisfy (ROC), after C1 boundary
removal.

Proof. Dyadic localization gives value and additive range comparable to the same scale. Controlled
CRT/divisor/full-rank transports distort both by only a polylogarithmic factor. If the transport
loses the direction responsible for the image length, it is a tagged rank drop, not a direct-origin
case. Lemma proved.
   —


ROC.2. Proof: Tagged Singular Origins Route Away If (ROC) fails because one of the
following tagged origins is present:

  1. short residual volume;

  2. Type I error budget;

  3. short fixed divisor or short quotient;

  4. forced local dependence;

  5. CKP balanced multiplicative origin;

  6. impossible/inconsistent support;

  7. post-terminal primitive slicing that does not create a new terminal GoodAWACK skeleton;

   then the singular testing family already carries an intrinsic terminal routing tag. The TC1
dichotomy consumes these tags, but the tags themselves are supplied by the B1/B3/F3/F4 and
E10Y/E10M/E10K grammar interfaces.
   Thus the only possible obstruction to ROC is an **untagged range-defective origin**.
   —




                                               240
ROC.3. Setup: Complementary Affine-Origin Problem The dangerous case is not a direct
dyadic coordinate. It is a marked affine form obtained from the Goldbach relation or from solving a
grouped equation.
   The schematic source is:

                                         𝑃𝐴 (𝑎) + 𝑃𝐵 (𝑏) = 𝑁.

   After grouping and partial solving, a surviving marked form may look like

                                          𝐿𝑚 = 𝑁 − 𝐿other ,                                     (1)
   or, in a two-group presentation,

                                                     𝑁 − 𝑢𝑣
                              𝑢𝑣 + 𝑢′ 𝑣 ′ = 𝑁,        𝐿𝑚 = 𝑢′ =.                          (2)
                                                          𝑣′
   Here the absolute height of 𝐿𝑚 can be 𝑋𝑚 ≍ 𝑁 or 𝑁/𝑣 ′ , while its additive image length is
controlled by the variation of the other side:

                                        |𝐿𝑚 (Ω)| ≍ |𝐿other (Ω)|.

   Thus it is possible at the interface level to have

                                      |𝐿𝑚 (Ω)| ≪ 𝑋𝑚 (log 𝑋𝑚 )−𝐵

   without any immediate contradiction.
   The model is:

                  Ω = [𝑇, 𝑇 + 𝐻] × [1, 𝑀 ],          𝐿𝑚 (𝑡, 𝑟) = 𝑁 − 𝑡,   𝐻𝑀 ≍ 𝑁,               (3)
   with

                                       𝑁 𝜃 ≤ 𝐻 < 𝑁 (log 𝑁 )−𝐵 .                                 (4)
   The 𝑡-range is long, so C1 short-direction Edge is not automatic. The full abstract box volume
can be ≍ 𝑁 because of the transverse 𝑟-range. The marked image is a shifted short interval near 𝑁 .
   This is exactly the singular testing model consumed later by TTD.
   —

ROC.4. Setup: Why Actual B1 Saves This Case Although model (3) is allowed by the
abstract terminal interface, it may be impossible as an actual B1 descendant.
   The reason is that the transverse variable 𝑟 cannot be an arbitrary free volume direction if it
only records factorizations of a fixed integer 𝐿𝑚 = 𝑛. In the true B1 finite-convolution expansion,
once the marked integer 𝑛 and the complementary product are fixed, the remaining factorization
multiplicity is divisor-type, not a free interval of length 𝑀 ≍ 𝑁/𝐻.
   If one could prove a uniform coarea slice bound of the form

                                                 |𝑊 (𝑧)| ≪ (log 𝑁 )𝐶                        (Slice)
                                        ∑︁

                                        𝑧∈Ω
                                      𝐿𝑚 (𝑧)=𝑛

   or an averaged version strong enough after summing over 𝑛 ∈ 𝐿𝑚 (Ω), then every short-image
complementary case would be Edge:

                                                    241
                                             |𝑊 (𝑧)| ≪ |𝐿𝑚 (Ω)|(log 𝑁 )𝐶 = 𝑜(𝑁 )
                            ∑︁      ∑︁

                         𝑛∈𝐿𝑚 (Ω) 𝐿𝑚 (𝑧)=𝑛

   whenever

                                         |𝐿𝑚 (Ω)| ≤ 𝑁 (log 𝑁 )−𝐵

    with 𝐵 chosen large.
    This is the bridge supplied below by the BRS component between the actual B1 factorization
origin and the terminal affine/coarea interface.
    —

ROC.5. Interface Passed to BRS Lemma B1 records that elementary coefficients are polylog-
arithmically bounded per tuple. It does not state a coarea slice-multiplicity theorem for terminal
affine forms 𝐿𝑚 .
    Lemma BGS records:

  1. parent block;

  2. grouping;

  3. routing history;

  4. current lattice/domain;

  5. current affine forms and their origin types.

   It does not record, for each marked form:

  1. additive height 𝑋𝑚 ;

  2. additive image length 𝑌𝑚 = |𝐿𝑚 (Ω)|;

  3. coarea slice mass

                                             𝑀𝑚 (𝑛) =              |𝑊 (𝑧)|;
                                                          ∑︁

                                                        𝐿𝑚 (𝑧)=𝑛


  4. whether 𝑀𝑚 (𝑛) is divisor-bounded by the parent product origin or can genuinely have free
     transverse volume.

   The E10Y/E10M/E10K terminal-affine grammar interface forbids untagged rank-dropping affine
maps, but model (3) need not be a rank-drop of the terminal affine span. It is a range/slice-
multiplicity defect.
   Therefore full ROC is supplied in the TC1 route through the BRS augmentation of the terminal
range/slice data.
   —




                                                    242
ROC.6. Statement: B1 Range Skeleton Lemma                   The needed strengthening is:

                                     B1-RANGE-SKELETON.

   Every terminal GoodAWACK skeleton is augmented in BRS with, for every marked form 𝐿𝑚 :

  1. a height scale 𝑋𝑚 ;

  2. an image-length scale 𝑌𝑚 ;

  3. a coarea slice-mass majorant 𝑆𝑚 ;

  4. an origin tag explaining whether 𝑆𝑚 is divisor-bounded or a genuine free-volume direction.

   The BRS theorem asserts:

Lemma G.22 (Lemma BRS). For every actual B1/F3/F4 terminal TC1-GoodAWACK macro-
template and marked form 𝐿𝑚 , after C1 boundary removal, one of the following holds:

  1. 𝑌𝑚 ≥ 𝑋𝑚 (log 𝑋𝑚 )−𝐶 , so ROC holds;

  2. 𝑌𝑚 𝑆𝑚 = 𝑜(𝑁 ), so the short image is strict C1P Edge;

  3. the free-volume slice origin exposes LongAP/Local, CKP, LocalDiag, or impossible support;

  4. the singular slice is a tagged quotient/divisor/rank-drop already classified by the terminal-affine
     tagged-origin grammar.

    This lemma proves the BRS component of TC1-SINGULAR-ORIGIN/ROC. It uses the direct-
origin comparability sublemma ROC.1 but not the full ROC closure statement.
    —

ROC.7. Proof: Closure from BRS             Assume Theorem BRS.1. Then:

  1. direct dyadic-coordinate origins satisfy ROC by Lemma ROC.1;

  2. tagged singular origins route away by Section ROC.2;

  3. complementary/solved affine origins are controlled by BRS: either near-global, Edge by slice
     mass, or routed to D1/G8a/LocalDiag/empty.

    Therefore every singular TC1 testing measure is routed or impossible. When this singular-origin
closure is imported into TTD/TNG together with the regular TGT output, one obtains

                                    𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

   —
                                  Remark G.23 (ROC.8. Output).
ROC reduces the remaining singular-origin case to B1-RANGE-SKELETON/ROC-SLICE, which is supplied by BRS.
   What is proved:

        direct dyadic-coordinate origins and controlled full-rank transports satisfy ROC.

                                                  243
    Block supplied by BRS:

                                            B1-RANGE-SKELETON/ROC-SLICE.

    This is a smaller and more concrete target than pointwise X9L-SI: prove that a terminal marked
affine form with short image cannot carry large hidden transverse B1 multiplicity unless the origin
is already Edge, LongAP/Local, CKP, LocalDiag, or impossible.
    —

ROC.9. Output for the Proof Tree Lemma BRS proves the BRS block stated above, using
X16BRS/X16C. Therefore the combined status is:

                            TC1-SINGULAR-ORIGIN/ROC is supplied by ROC + BRS.

    The proof tree uses ROC and BRS together.

ROC.10. Logical Dependencies Internal dependencies: B1, BGS, Theorem BRS.1, X16BRS,
X16C, and the E10Y/E10M/E10K terminal-affine grammar interface. The dependency is noncircular:
BRS uses only the direct-origin sublemma ROC.1 and the tagged-origin routing in ROC.2, while
the full ROC closure is obtained in Sections ROC.7–ROC.9 after Theorem BRS.1 has been invoked.
    Children served: sublemma ROC.1 serves BRS; the full ROC+BRS closure serves TTD, TNG,
TNGTTHM, and E10L.

G.10        BRS range/slice closure
G.10.1         BRS. B1 Range/Slice Closure for Singular TC1 Testing
BRS.0. Statement and Role                              Lemma BRS proves the structural block isolated in Lemma
ROC:

                                            B1-RANGE-SKELETON/ROC-SLICE.

    The point is to rule out the artificial model

                              Ω = [𝑋, 𝑋 + 𝑌 ] × [1, 𝑀 ],                     𝐿𝑚 (𝑢, 𝑣) = 𝑢,               𝑌 𝑀 ≍ 𝑁,

   when 𝐿𝑚 is an actual B1-origin terminal marked form. In the genuine B1 descendant, the
transverse variable is not arbitrary free mass. It is tied to boundedly many finite-convolution
product variables. Restricting the marked carrier to a short additive image therefore cuts the B1
tuple mass by the same relative factor, up to the standard divisor-sum losses already recorded as
X16 in the ledger.
   The result is:


the singular short-image B1-origin residual is Edge unless it already has a LongAP/Local, CKP, LocalDiag, impossible, or tagged quotient/divisor origin.




                                                                         244
    Thus BRS closes the structural singular-origin branch using the BRS-specific divisor-sum estimate
X16BRS. Lemma X16BRS reduces the four BRS carrier types to the fixed-depth divisor-correlation
input X16-Core, and Lemma X16C proves X16-Core.
    Equivalently, BRS supplies the routed alternative in Theorem TNG-A: a TC1 coarea test with a
genuinely short B1-origin marked image is routed to strict Edge or to an already handled tagged
class before X9L-GT is invoked.
    Logical dependencies are B1, C1P, C1A, C1, F3P, F3, F4, ROC.1, X16BRS, and X16C. BRS is
used by ROC, TTD, TTH, TNG, TNGTTHM, and E10L. It is not an input to TTH-SC; rather,
TTH-SC exports genuine structural short-image certificates which are later consumed by the TTD/
ROC/BRS routing chain.
    —

BRS.1. Statement: X16-B1 Dyadic Carrier Estimate Let ℬ be a fixed B1 typed dyadic
block. Its parent variables are

                               𝑥1 , . . . , 𝑥𝑟 , 𝑦1 , . . . , 𝑦𝑠 ,          𝑟, 𝑠 ≤ 2𝐽0 ,

   with dyadic supports and parent equation

                                            𝑃𝐴 (𝑥) + 𝑃𝐵 (𝑦) = 𝑁.

    Let 𝐶(𝑥, 𝑦) be a B1 carrier attached to a terminal marked form. It is one of the following,
after a bounded number of controlled CRT restrictions, fixed-divisor quotients, and full-rank affine
coordinate changes:

  1. a grouped product carrier;

  2. a Goldbach complementary carrier 𝑁 − 𝑃 ;

  3. a quotient carrier 𝑠 occurring in a recorded equation 𝐿 = 𝑑𝑠;

  4. a controlled divisor quotient of one of the previous carriers.

    For quotient carriers, the divisor in 𝐿 = 𝑑𝑠 is always tagged before BRS is invoked. This is the
quotient-tag completeness statement of F4.9/F4.11: an untagged variable divisor would still be an
unresolved ordinary divisor predicate and could not pass the F3.13 terminal GoodAWACK labelling
step.
    Let 𝑋𝐶 be its dyadic height and let 𝐼 be an additive interval. Put

                               𝑌16 := max{|𝐼 ∩ Z|, 𝑋𝐶 (log 𝑁 )−𝐵16 },

   where 𝐵16 is the X16 slice-floor exponent fixed in the parameter register. Then the total absolute
B1 tuple mass on the subcell

                                                        𝐶∈𝐼

   satisfies
                                                                      𝑌16
                                                                 (︂         )︂
                      Massℬ (𝐶 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )            𝐶1
                                                                                 + 𝑁 1−𝜌 (log 𝑁 )𝐶1 ,
                                                                      𝑋𝐶


                                                          245
   more precisely,

                                                                  𝑌16
                        Massℬ (𝐶 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶1                  + 𝑁 1−𝜌 (log 𝑁 )𝐶1 ,   (BRS-slice)
                                                                  𝑋𝐶
   for constants 𝐶1 , 𝜌 > 0 depending only on 𝐽0 and the fixed routing architecture.

Proof. This is the B1 form of the divisor-sum input X16. The exact statement is X16BRS; its carrier
reductions are recorded in Lemma X16BRS, while the analytic core is proved in Lemma X16C.
    For a grouped product carrier, fixing 𝐶 = 𝑐 does not leave a one-variable divisor average. The
remaining variables still contain the same-side complementary product 𝑢, and the opposite B1 side
is forced to have product 𝑁 − 𝑐𝑢. Thus the required bound is the fixed-depth correlation

                                    𝜏𝑂(𝐽0 ) (𝑐)         𝜏𝑂(𝐽0 ) (𝑢)𝜏𝑂(𝐽0 ) (𝑁 − 𝑐𝑢),
                              ∑︁                  ∑︁

                              𝑐∈𝐼                 𝑢≍𝑈

    with positive support 𝑁 − 𝑐𝑢 > 0, not merely 𝑐∈𝐼 𝜏𝑂(𝐽0 ) (𝑐). This is exactly X16-Core, proved
                                                             ∑︀

in Lemma X16C by Shiu AP divisor averages. It gives the main term proportional to 𝑌16 /𝑋𝐶 , plus
a power-saving boundary error. The smooth dyadic weights are handled by partial summation, and
the elementary B1 coefficient types 𝜇, 1, log cost only (log 𝑁 )𝐶1 .
    For a complementary carrier 𝐶 = 𝑁 − 𝑃 , the condition 𝐶 ∈ 𝐼 is equivalent to 𝑃 ∈ 𝑁 − 𝐼, so
the same estimate applies to the product carrier 𝑃 .
    For a quotient carrier 𝐿 = 𝑑𝑠, put 𝐶 = 𝑠, 𝑠 ≍ 𝑋𝐶 , and 𝑑 ≍ 𝐷. The restriction 𝑠 ∈ 𝐼 restricts
the product 𝑑𝑠 to a set whose X16 length has the same ratio 𝑌16 /𝑋𝐶 inside its dyadic carrier scale
𝐷𝑋𝐶 . Applying the grouped-product estimate to 𝑑𝑠 gives
                                                    𝑌16
                                    𝑁 (log 𝑁 )𝐶1        + 𝑁 1−𝜌 (log 𝑁 )𝐶1 ,
                                                    𝑋𝐶
    which is (BRS-slice). If 𝑑 is controlled/fixed, this is just fixed-divisor absorption. If 𝑑 varies
over a tagged dyadic divisor family, the divisor boundedness of the B1/F4 coefficient contributes
only an additional (log 𝑁 )𝑂𝐽0 (1) factor, as recorded in Lemma X16BRS. If the variable quotient
equation instead forces local dependence, balanced multiplicative structure, short residual volume,
or an impossible support, F4 routes the atom to LocalDiag, CKP, Edge, or empty before it reaches
terminal GoodAWACK.
    Controlled CRT restrictions and full-rank affine coordinate changes alter the lattice index,
carrier height, and interval length by at most polylogarithmic factors. These losses are absorbed in
(log 𝑁 )𝐶1 . C1 boundary pieces are discarded before the estimate is applied. Lemma proved.
    —


BRS.2. Proof: Short Image Implies Strict Edge Let 𝐿𝑚 be a terminal TC1-GoodAWACK
marked form with B1 carrier 𝐶𝑚 . Let

                                     𝑋𝑚 ≍ 𝑋𝐶𝑚 ,              𝑌𝑚 = |𝐿𝑚 (Ω)|.

   Assume the marked image is singular:

                                           𝑌𝑚 < 𝑋𝑚 (log 𝑋𝑚 )−𝐵 .                                  (SAI)
   Choose


                                                          246
                               𝐵𝜅 > 𝐵16 + 𝐶0 + 𝐶1 + 𝐶16 + 𝜌−1
                                                           16 + 20,

    where 𝐶0 is the C1 saving budget, 𝐶1 is the internal C1/B1 coefficient loss, 𝐵16 is the X16 slice-
floor exponent, and 𝐶16 , 𝜌16 come from X16-BRS as registered in the parameter register. Then

                    Mass(𝐿𝑚 (Ω)) ≪ 𝑁 (log 𝑁 )−𝐶0 −10 + 𝑁 1−𝜌 (log 𝑁 )𝐶1 = 𝑜(𝑁 ).

   Therefore the singular short-image subcell satisfies the strict C1P short residual volume predicate
E6 and is registered in the C1A admission ledger.

Proof. Apply (BRS-slice) with 𝐶 = 𝐶𝑚 and 𝐼 = 𝐿𝑚 (Ω). The singular image condition gives
|𝐼 ∩ Z|/𝑋𝑚 ≤ (log 𝑁 )−𝐵𝜅 , after harmless polylogarithmic renormalization of 𝑋𝑚 . The X16 floor
contributes only (log 𝑁 )−𝐵16 . The displayed choice of 𝐵𝜅 and 𝐵16 makes the first term logarithmically
saved, while the second term has power saving. This is exactly the strict C1P E6 budget. Lemma
proved.
   —


Parameter check G.24 (BRS.3. Parameter Check: Compatibility with Routing Tags). The
previous lemma applies only to terminal GoodAWACK descendants that actually reach the B1
carrier estimate. If the short image is caused by any of the following, the atom does not need BRS:

  1. short residual volume or Type I error budget;

  2. short fixed divisor or short quotient;

  3. forced local dependence or proportionality;

  4. CKP-balanced multiplicative structure;

  5. impossible congruence or support;

  6. tagged rank drop or quotient/divisor origin already present in the routing record.

   These cases are exactly the tagged alternatives of the F3/F4 tagged-origin decision tree.
   Thus BRS only handles the previously untagged complementary/solved affine case. In that case
the carrier remains a genuine B1 product or quotient carrier, so BRS.1 applies.
   —

BRS.4. Output Theorem

Theorem G.25 (Theorem BRS.1. B1 range/slice dichotomy). For every actual B1/B3/F3/F4
terminal TC1-GoodAWACK macro-template and every marked form 𝐿𝑚 , after C1 boundary removal
and fixed macro-template normalization, one of the following holds:

  1. 𝐿𝑚 satisfies range-origin comparability

                                        |𝐿𝑚 (Ω)| ≥ 𝑋𝑚 (log 𝑋𝑚 )−𝐵 ;

  2. the short-image subcell is strict C1P Edge by BRS.2;



                                                 247
  3. the origin is tagged and routes to LongAP/Local, CKP, LocalDiag, Edge, or empty by the F3/
     F4 tagged-origin decision tree.

Proof. If 𝐿𝑚 is a direct dyadic-coordinate origin or a controlled full-rank transport of one, the
direct-origin comparability sublemma ROC.1 gives case 1 unless the image is restricted to a smaller
subcell. In that subcell, BRS.2 gives case 2.
    If 𝐿𝑚 is a fixed divisor quotient, the carrier scale is changed by a polylogarithmic factor only, so
BRS.1 and BRS.2 apply.
    If 𝐿𝑚 is a variable quotient residual or complementary solved affine origin, F4 first removes all
short quotient, forced local, balanced CKP, and impossible cases. If any such tag is present, we are
in case 3. Otherwise the quotient/complement carrier remains an actual B1 product carrier with
controlled content. BRS.1 applies, and a failure of range comparability gives case 2.
    Finally, the E10Y/E10M/E10K terminal-affine grammar interface excludes arbitrary untagged
rank-dropping affine regrouping. Full-rank affine transports preserve BRS up to polylogarithmic
loss; rank drops carry one of the tags already covered by case 3. These cases exhaust the B1-to-
GoodAWACK skeleton. Theorem proved.
   —


BRS.5. Output for Singular TC1 Testing Combining Theorem BRS.1 with the direct-origin
and tagged-origin parts of ROC gives

                                  TC1-SINGULAR-ORIGIN/ROC.

   Indeed, a singular testing measure is precisely a concentration on marked forms whose image fails
near-global range comparability. By BRS.1, such a failure is either strict C1P Edge or an existing
routing tag. Hence it cannot remain as an untagged terminal TC1-GoodAWACK contribution.
   When imported into Lemma TTD, this closes the BRS part of the singular branch of the TC1
global-testing route. The MRT-admissible branch is still the branch handled by TGT using the
averaged Liouville input X9L-GT.
   —
                                 Remark G.26 (BRS.6. Output).
            B1-RANGE-SKELETON/ROC-SLICE is proved using X16-BRS/X16-Core.
    This does not prove a pointwise shifted short-interval theorem for 𝜆. It shows that the only
TC1 situations where such a pointwise theorem appeared to be needed are not genuine terminal B1-
origin GoodAWACK mass: short image mass is Edge after the B1 carrier slice estimate, and all
non-Edge failures carry existing routing tags.
    The structural reduction in BRS is separate from the analytic carrier-slice input. The analytic
input is discharged by Lemma X16C.

BRS.7. Logical Dependencies Internal dependencies: B1, C1P, C1A, C1, F3P, F3, F4, the
direct-origin sublemma ROC.1, X16BRS, and X16C. BRS does not depend on TTD or on the full
ROC closure theorem; full ROC is obtained only after BRS is combined with ROC.1 and the tagged-
origin routing part of ROC.
    Children served: ROC, TTD, TTH, TNG, TNGTTHM, and E10L.




                                                 248
G.11     X16BRS carrier-slice reduction
G.11.1    X16BRS. Carrier-Slice Divisor Estimate for BRS
X16BRS.0. Statement and Role Lemma X16BRS is the carrier-slice estimate used by BRS.
It reduces the four BRS carrier types to one fixed-depth divisor-correlation estimate, called X16-
Core below. The core estimate is proved by Lemma X16C using Shiu’s arithmetic-progression
Brun–Titchmarsh theorem for divisor-bounded multiplicative functions.
    Its role in the TC1 proof is local and structural: it supplies the short-image carrier-slice bound
used by BRS in the routed alternative of Theorem TNG-A. It is not a Liouville short-interval input.
    Logical dependencies are B1, C1, BRS, X16C, and the parameter register. X16BRS is used by
BRS, TTH, TNG, and X16.
    —

X16BRS.1. Statement: B1 Carrier-Slice Estimate Let ℬ be a typed B1 dyadic block of
fixed depth 𝐽0 . Let Massℬ (𝐶 ∈ 𝐼) denote the sum of absolute values of the B1 coefficient weights
over tuples in ℬ for which the carrier 𝐶 lies in an additive interval 𝐼 of length 𝑌 . The carrier height
is 𝑋𝐶 .
    Fix the X16 slice-floor exponent 𝐵16 from the parameter register, and put

                                𝑌16 := max{|𝐼 ∩ Z|, 𝑋𝐶 (log 𝑁 )−𝐵16 }.

   The BRS carrier estimate needed by TTH is

                                                        𝑌16
                       Massℬ (𝐶 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶16        + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 ,         (X16-BRS)
                                                        𝑋𝐶
   where 𝐶16 = 𝐶16 (𝐽0 ) and 𝜌16 = 𝜌16 (𝐽0 ) > 0.
   The allowed BRS carriers are:

  1. grouped product carriers;

  2. Goldbach complementary carriers 𝑁 − 𝑃 ;

  3. quotient carriers 𝑠 from a recorded equation 𝐿 = 𝑑𝑠;

  4. controlled divisor quotients 𝐿/𝑑0 , with 𝑑0 ≤ (log 𝑁 )𝐶 .

   —

X16BRS.2. Setup: Core Divisor-Correlation Input The one analytic input required for the
reductions below is:
    X16-Core. For every fixed-depth B1 finite-convolution support and every grouped product
carrier 𝑃 of height 𝑋𝑃 , with 𝑌16 = max{|𝐼 ∩ Z|, 𝑋𝑃 (log 𝑁 )−𝐵16 },

                                                        𝑌16
                       Massℬ (𝑃 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶16        + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 .         (X16-Core)
                                                        𝑋𝑃
   This is the fixed-depth divisor-correlation estimate proved in Lemma X16C. The proof keeps the
genuine 𝑁 − 𝑝𝑢 divisor correlation and controls it by Shiu-type averages in arithmetic progressions,
switching between the carrier and complementary variables according to the dyadic range.



                                                  249
X16BRS.2a. Excluded Shortcut: One-Variable Divisor Averaging The estimate cannot
be proved by fixing 𝑃 = 𝑛 ∈ 𝐼 and bounding only

                                                             𝜏𝑘 (𝑛)
                                                       ∑︁

                                                       𝑛∈𝐼

   by a standard average divisor estimate. After fixing 𝑃 = 𝑛, the remaining variables still satisfy
a Goldbach-complementary equation of the form

                                                   𝑛 𝑣 + 𝑤 = 𝑁.

  Thus the relevant majorant is not merely 𝜏𝑘 (𝑛); it is a fixed-depth divisor correlation along the
moving complementary values 𝑁 − 𝑛𝑣, for example schematically

                                         𝜏𝑘1 (𝑛)          𝜏𝑘2 (𝑣)𝜏𝑘3 (𝑁 − 𝑛𝑣).
                                   ∑︁              ∑︁

                                   𝑛∈𝐼             𝑣≍𝑉𝑛

   A bound for 𝑛∈𝐼 𝜏𝑘 (𝑛) alone does not control the correlation with 𝑁 − 𝑛𝑣, especially when the
                 ∑︀

modulus/step 𝑛 is large.
   The sufficient input is the fixed-depth divisor-correlation statement X16-Core above. It is
supplied by Lemma X16C, not by the rejected one-variable divisor average.
   —

X16BRS.3. Proof: Product and Complementary Carriers For a grouped product carrier
𝐶 = 𝑃 , (X16-BRS) is exactly X16-Core.
   For a complementary carrier 𝐶 = 𝑁 − 𝑃 , the condition 𝐶 ∈ 𝐼 is equivalent to 𝑃 ∈ 𝑁 − 𝐼. Since
|𝑁 − 𝐼| = |𝐼| and the dyadic height is unchanged up to fixed constants, X16-Core gives (X16-BRS).
   —

X16BRS.4. Proof: Quotient Carriers                     Let 𝐶 = 𝑠 occur through a recorded quotient equation

                                   𝐿 = 𝑑𝑠,             𝑑 ≍ 𝐷,          𝑠 ≍ 𝑋𝐶 .

   If 𝑑 is fixed or controlled by a dyadic divisor tag, then 𝑠 ∈ 𝐼 implies
                      𝐿 ∈ 𝑑𝐼,       |𝑑𝐼 ∩ Z| ≪ 𝑑|𝐼 ∩ Z| + 𝑂(𝑑),                    𝑋𝐿 ≍ 𝑑𝑋𝐶 .
Applying X16-Core to the carrier 𝐿 gives
                                                   𝑌16
                                 𝑁 (log 𝑁 )𝐶16         + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 ,
                                                   𝑋𝐶
which is (X16-BRS).
   If the divisor 𝑑 is not fixed by a routing tag and summing over 𝑑 would introduce uncontrolled
cross-correlations, the term is not an X16-BRS carrier. That case must be routed by F4 as local
dependence, CKP balance, strict Edge, or a tagged quotient residual before BRS is invoked.
   For a variable but tagged divisor family, B1 coefficient bounds give |𝛼(𝑑)| ≪ 𝜏𝑂𝐽0 (1) (𝑑)(log 𝑁 )𝑂(1) .
On a dyadic 𝑑-block,
                                    ∑︁ 𝜏𝑂𝐽 (1) (𝑑)
                                               0
                                                             ≪ (log 𝑁 )𝑂𝐽0 (1) .
                                    𝑑≍𝐷
                                                   𝑑
Thus the controlled sum over tagged 𝑑-layers preserves (X16-BRS), after enlarging 𝐶16 by a constant
depending only on 𝐽0 .
   —

                                                          250
X16BRS.5. Proof: Controlled Divisor Quotients                Let 𝐶 = 𝐿/𝑑0 , where 𝑑0 ≤ (log 𝑁 )𝐶 is
fixed or controlled. Then

                                    𝐶∈𝐼      ⇐⇒        𝐿 ∈ 𝑑0 𝐼.

The carrier height changes from 𝑋𝐶 to 𝑑0 𝑋𝐶 , while the interval length changes from 𝑌 to 𝑑0 𝑌 .
Their ratio is unchanged, and the polylogarithmic factor 𝑑0 is absorbed into 𝐶16 . Hence X16-Core
again gives (X16-BRS).
   —
Remark G.27 (X16BRS.6. Output). By X16-Core, all four BRS carrier types satisfy X16-BRS.
Therefore BRS may use the estimate with constants 𝐶16 (𝐽0 ), 𝜌16 (𝐽0 ) > 0.

          X16BRS is proved from X16-Core, and X16-Core is proved by Lemma X16C.

    The remaining external-theorem check is the standard verification of the Shiu invocation and
local-factor averaging, both made explicit in Lemma X16C.

X16BRS.7. Logical Dependencies Internal dependencies: B1, C1, BRS, X16C, and the
parameter register.
   Children served: BRS and TTH.

G.12     X16-Core Shiu/AP proof
G.12.1    X16C. Proof of the BRS Carrier-Slice Estimate
X16C.0. Statement and Role Lemma X16C proves the analytic core isolated in Lemma
X16BRS.
   The proof does not use the insufficient one-variable estimate

                                                  𝜏𝑘 (𝑛).
                                            ∑︁

                                            𝑛∈𝐼

   Instead it uses the arithmetic-progression form of Shiu’s Brun–Titchmarsh theorem for non-
negative multiplicative functions, applied to the moving complementary values 𝑁 − 𝑐𝑢. This is the
point where the carrier-complement correlation is controlled.
   The conclusion is:

                        X16-Core is proved for the BRS carrier interface.

    The only external input used here is Shiu’s theorem in the standard divisor-function corollary
stated below.
    Reference:
    P. Shiu, "A Brun-Titchmarsh theorem for multiplicative functions", J. Reine Angew. Math. 313
(1980), 161–170.
    Logical dependencies are X16BRS, BRS, F4, CKPD, the parameter register, and Shiu’s arithmetic-
progression Brun–Titchmarsh theorem for multiplicative functions. X16C is used by X16BRS, BRS,
TTH, and TNG.
    —




                                               251
X16C.1. External Input: Shiu in Divisor-Function Form We use the following standard
consequence of Shiu’s theorem.
Lemma G.28 (Lemma X16-SH. Divisor functions in AP intervals). Fix 𝐾, 𝐴 ≥ 1 and 0 < 𝛿 < 1/10.
Let

                                                𝑓 (𝑛) = 𝜏𝐾 (𝑛)𝐴 .

    Let 𝐽 ⊂ [1, 𝑁 ] be an interval of length 𝐻, where 𝑁 𝛿 ≤ 𝐻 ≤ 𝑁 , and let 𝑞 ≤ 𝐻 1−𝛿 . Then for
every residue class 𝑎 mod 𝑞,

                                                           𝐻
                                                      (︂         )︂
                                     𝑓 (𝑛) ≪𝐾,𝐴,𝛿            + 1 (log 𝑁 )𝐶SH (𝐾,𝐴,𝛿) ℰ𝑞,𝑎 ,      (SH)
                          ∑︁

                         𝑛∈𝐽
                                                           𝑞
                      𝑛≡𝑎 (mod 𝑞)

    where the possible non-coprime local factor is supported on primes dividing (𝑎, 𝑞). The local
factors which occur in the applications below are controlled by Lemma X16-LFA. This is the only
point in the X16C proof where non-coprime AP classes enter.
Lemma G.29 (Lemma X16-LFA. Local factor averaging). Fix 𝐾0 , 𝐾, 𝐴 ≥ 1. Let ℰ𝑐,𝑁 denote any
local factor produced by applying X16-SH to the residue class 𝑁 mod 𝑐, after extracting the common
                                                       #                     #
divisor (𝑐, 𝑁 ). Then, for every X16 carrier interval 𝐼16 ⊂ [𝑋/2, 3𝑋] with |𝐼16 | ≫ 𝑋(log 𝑁 )−𝐵16 ,
                                                1/2            #
                                    𝜏𝐾0 (𝑐)𝐴 ℰ𝑐,𝑁 ≪𝐾0 ,𝐾,𝐴,𝛿 |𝐼16 |(log 𝑁 )𝐶loc .             (SH-loc)
                             ∑︁

                               #
                            𝑐∈𝐼16

    The same bound holds for a full dyadic interval 𝑐 ≍ 𝑋. Consequently it also applies in the
interchanged orientation of Case 2, where the averaging variable is the same-side complement 𝑢 ≍ 𝑈 .
Proof of X16-LFA. Shiu’s theorem is stated for coprime residue classes. For a non-coprime class
write 𝑔 = (𝑎, 𝑞), 𝑎 = 𝑔𝑎1 , 𝑞 = 𝑔𝑞1 , with (𝑎1 , 𝑞1 ) = 1. The summand is 𝑓 (𝑔𝑛1 ) on a coprime class
modulo 𝑞1 . Since 𝑓 = 𝜏𝐾𝐴 is submultiplicative up to constants depending only on 𝐾, 𝐴,


                                         𝑓 (𝑔𝑛1 ) ≪𝐾,𝐴 𝑓 (𝑔)𝑓 (𝑛1 ).

   The local cost is therefore bounded by a fixed divisor power of 𝑔 = (𝑎, 𝑞), together with the
harmless Euler factor 𝑝|𝑞 (1 + 𝑂𝐾,𝐴 (1/𝑝)). In our application 𝑞 = 𝑐 and 𝑎 = 𝑁 , so 𝑔 = (𝑐, 𝑁 ).
                      ∏︀
                               #
   Average this cost over 𝑐 ∈ 𝐼16 . Equivalently, apply Shiu’s ordinary interval theorem to the
multiplicative function

                                      𝑔𝑁 (𝑐) = 𝜏𝐾0 (𝑐)𝐴 𝜏𝑀 ((𝑐, 𝑁 ))𝐵 ,

   where 𝐵 is fixed large enough to dominate the local factor. This is a non-negative multiplicative
                                                                                              #
function of 𝑐, uniformly divisor-bounded for fixed 𝐾0 , 𝐴, 𝑀, 𝐵. If 𝑋 > (log 𝑁 )2𝐵16 , then |𝐼16 |≫
𝑋(log 𝑁 )−𝐵16 ≫ 𝑋 1/2 , and Shiu gives
                                                         #
                                              𝑔𝑁 (𝑐) ≪ |𝐼16 |(log 𝑁 )𝑂(1) .
                                       ∑︁

                                         #
                                      𝑐∈𝐼16

    If 𝑋 ≤ (log 𝑁 )2𝐵16 , the same bound is trivial after increasing the logarithmic exponent, because
every 𝑐 ∈ [𝑋/2, 3𝑋] is polylogarithmic. The full-dyadic-interval case is the ordinary mean-value
estimate for the same fixed divisor-bounded multiplicative function. The proof for the interchanged
variable is identical. Lemma proved.


                                                       252
Lemma G.30 (Lemma X16-SH-class. Squared divisor functions are admissible). For every fixed
𝐾 ≥ 1, the function

                                                𝑓 (𝑛) = 𝜏𝐾 (𝑛)2

   belongs to the divisor-bounded multiplicative class to which X16-SH applies, with constants
depending only on 𝐾. Indeed, for prime powers,
                                                         )︃2
                                         ℓ+𝐾 −1
                                        (︃
                          𝜏𝐾 (𝑝ℓ )2 =                          ≪𝐾 (1 + ℓ)2𝐾−2 ≤ 𝐴ℓ𝐾 ,
                                          𝐾 −1

   after increasing 𝐴𝐾 . Also

                                         𝜏𝐾 (𝑛)2 ≤ 𝜏𝐾 2 (𝑛) ≪𝐾,𝜀 𝑛𝜀

   for every 𝜀 > 0. Hence the applications of X16-SH below with 𝑓 = 𝜏𝐾
                                                                     2 are legitimate.
                                                                      3
   —

X16C.2. Statement: X16-Core                  Let ℬ be a B1 typed dyadic block of depth at most 𝐽0 . Its
parent equation is
                                   𝑟            𝑠
                                        𝑎𝑖 +         𝑏𝑗 = 𝑁,         𝑟, 𝑠 ≤ 2𝐽0 .                 (B1)
                                  ∏︁           ∏︁

                                  𝑖=1          𝑗=1

   Fix a slice-floor exponent 𝐵16 , chosen in the parameter register after 𝐶16 and before 𝐵𝜅 .
   Let 𝑃 be a one-side grouped product carrier. Thus, after possibly interchanging the two sides of
(B1),
                                                     𝑟
                                                         𝑎𝑖 = 𝑃 𝑈,
                                                    ∏︁

                                                 𝑖=1

   where 𝑈 is the complementary product of the remaining variables on that side. Let 𝑋𝑃 be the
dyadic height of 𝑃 , and let

                                𝐼 # = 𝐼 ∩ Z,              𝑌 # = max(1, |𝐼 # |).

   Define

                                  𝑌16 := max{𝑌 # , 𝑋𝑃 (log 𝑁 )−𝐵16 }.                            (Y16)
    If 𝑌 # < 𝑋𝑃 (log 𝑁 )−𝐵16 , enlarge 𝐼 to an interval 𝐼16 ⊂ [𝑋𝑃 /2, 3𝑋𝑃 ] with |𝐼16 ∩ Z| ≍ 𝑌16 . This
only enlarges the mass. Hence the proof below is carried out for 𝐼16 ; if 𝑌 # ≥ 𝑋𝑃 (log 𝑁 )−𝐵16 , take
𝐼16 = 𝐼.
    The BRS form of X16-Core is

                                                          𝑌16
                Massℬ (𝑃 ∈ 𝐼) ≪𝐽0 𝑁 (log 𝑁 )𝐶16               + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 .       (X16-Core)
                                                          𝑋𝑃
   One may take, after harmless enlargement,
                                                                     1
                       𝐶16 = 100𝐽02 + 100,               𝜌16 =              .           (X16-constants)
                                                                  106 𝐽04

                                                          253
   The 𝑌 # convention is the usual integer-lattice correction. The floor in 𝑌16 is essential: a single
highly composite carrier value may carry a local divisor factor larger than any fixed power of log 𝑁 .
BRS does not need such a one-point estimate. If the actual marked image is shorter than the floor,
the monotone enlargement to 𝐼16 still gives a strict C1P saving once 𝐵16 is chosen large enough.
   This last point is not a circular appeal to TTH. BRS uses X16-Core before TTH: the floor term
contributes at most

                                          𝑋𝑃 (log 𝑁 )−𝐵16
                          𝑁 (log 𝑁 )𝐶16                   = 𝑁 (log 𝑁 )𝐶16 −𝐵16 ,
                                                𝑋𝑃

    up to the fixed C1/B1 coefficient losses. The parameter condition recorded in the parameter
register,

                                      𝐵16 > 𝐶0 + 𝐶1 + 𝐶16 + 20,

  makes this a strict C1P Edge contribution. Thus replacing a shorter image by the X16 floor is a
monotone upper-bound device whose extra mass remains within the C1 budget.
  —

X16C.3. Setup: Reduction to a Bilinear Divisor Correlation The elementary B1 coefficients
are of type 𝜇 · 1≤𝑦 , 1, and log. Hence, after dyadic localization and taking absolute values, each
coefficient product is bounded by

                                               (log 𝑁 )𝑂𝐽0 (1) .

    If 𝑃 = 𝑝, the number of factorizations of 𝑝 by the carrier variables is ≪ 𝜏𝐾1 (𝑝), with 𝐾1 ≤ 2𝐽0 . If
the complementary product on the same side is 𝑈 = 𝑢, the number of its factorizations is ≪ 𝜏𝐾2 (𝑢),
with 𝐾2 ≤ 2𝐽0 . The opposite side is then forced to have product

                                               𝑄 = 𝑁 − 𝑝𝑢,

    and, on the positive support 𝑁 − 𝑝𝑢 > 0, the number of its factorizations is ≪ 𝜏𝐾3 (𝑁 − 𝑝𝑢),
with 𝐾3 ≤ 2𝐽0 .
    Discarding dyadic restrictions on 𝑄 only enlarges the count. The support condition 𝑁 − 𝑝𝑢 > 0
is retained; terms with 𝑁 − 𝑝𝑢 ≤ 0 contribute nothing. Therefore

            Massℬ (𝑃 ∈ 𝐼) ≪ (log 𝑁 )𝑂𝐽0 (1)            𝜏𝐾1 (𝑝)         𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ,   (1)
                                               ∑︁                ∑︁

                                                  #
                                               𝑝∈𝐼16             𝑢≍𝑈

    where 𝑋𝑃 𝑈 ≍ 𝑁 unless the block has already been routed to a C1 short-volume or impossible
support. This is the true correlation that a one-variable divisor-average shortcut does not capture.
    The parametrization is as follows. Fix the subset 𝑆 of B1 variables whose product is the carrier
𝑃 ; the complementary subset on the same side has product 𝑈 . Every original B1 tuple maps to a
unique pair

                               𝑝 = 𝑃 (𝑎𝑖 : 𝑖 ∈ 𝑆),           𝑢 = 𝑈 (𝑎𝑖 : 𝑖 ∈
                                                                           / 𝑆),




                                                       254
    and then the opposite side is forced to have product 𝑄 = 𝑁 − 𝑝𝑢. Conversely, for fixed 𝑝 and 𝑢,
the number of compatible B1 factorizations is bounded by the displayed divisor factors 𝜏𝐾1 (𝑝)𝜏𝐾2 (𝑢),
and the number of opposite-side factorizations is bounded by 𝜏𝐾3 (𝑁 − 𝑝𝑢). Any dyadic, congruence,
gcd, or routing-tag restriction left inside the original support is either retained by the actual tuple
count or discarded when passing to the upper bound (1). Discarding such restrictions can only
enlarge the mass, and any additional divisor multiplicity is absorbed into the fixed exponent 𝐶16 .
    It remains to prove that the double sum in (1) is

                           ≪𝐽0 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) + 𝑁 1−𝜌16 (log 𝑁 )𝑂𝐽0 (1) .                   (2)
   Since 𝑋𝑃 𝑈 ≍ 𝑁 , (2) is exactly (X16-Core).
   —

Parameter check G.31 (X16C.4. Parameter Check: Shiu/AP Route). The proof of the bilinear
estimate below uses Shiu-type divisor bounds only after the carrier variables and their arithmetic
progressions have been fixed. For reference, the following list records the exact controlled quantity
at each step. The list format is used instead of a compressed table so that the formulae remain
readable in the full manuscript.

  • Carrier fixing. For a fixed product carrier 𝑝 = 𝑃 (𝑎𝑖 : 𝑖 ∈ 𝑆),

                                       #{(𝑎𝑖 ) : 𝑃 (𝑎𝑖 ) = 𝑝} ≤ 𝜏𝐾1 (𝑝).

     This loss is absorbed in 𝐶16 .

  • Same-side complement. For the complementary factor 𝑢 = 𝑈 (𝑎𝑗 : 𝑗 ∈
                                                                     / 𝑆),

                                       #{(𝑎𝑗 ) : 𝑈 (𝑎𝑗 ) = 𝑢} ≤ 𝜏𝐾2 (𝑢).

     This loss is absorbed in 𝐶16 .

  • Opposite side. The remaining Goldbach complement is 𝑁 − 𝑝𝑢. The divisor weight
    𝜏𝐾3 (𝑁 − 𝑝𝑢) remains inside the correlation; it is essential and is not averaged away.

  • **Fixed 𝑝 arithmetic progression.** The expression 𝑁 − 𝑝𝑢, with 𝑢 in a fixed residue class
    and a dyadic interval, is estimated by Shiu’s divisor estimate in an arithmetic progression.
    The modulus is 𝑝 in the non-small-volume range, and the loss is (log 𝑁 )𝑂𝐾 (1) .

  • **Fixed 𝑢 arithmetic progression.** The expression 𝑁 − 𝑢𝑝, with 𝑝 in a fixed residue class
    and a dyadic interval, is estimated by the same Shiu/AP estimate with modulus 𝑢 whenever
    this is the admissible orientation. The loss is again (log 𝑁 )𝑂𝐾 (1) .

  • Cauchy–Schwarz passage. Products of the fixed divisor weights are controlled by Cauchy–
    Schwarz followed by Shiu/AP on the squared divisor weight. The recorded loss is square-
    rooted and polylogarithmic.

  • Divisor second moment. The sums

                                                        𝜏𝐾 (𝑢)2
                                                  ∑︁

                                                  𝑢∼𝑈


                                                   255
       and the analogous restricted sums are bounded by the standard fixed-divisor second moment,
                                                                              2
       for example Tenenbaum, Chapter II.5, Theorem 5. This gives 𝑈 (log 𝑈 )𝐾 −1 .

     • Non-coprime AP class. AP classes with a fixed local gcd are handled by separating the
       local gcd factors before applying Shiu/AP. The contribution is absorbed by SH-loc.

     • CRT and quotient restrictions. Full-rank congruence restrictions and tagged quotients
       are controlled by bounded content, CRT splitting, and the quotient tag from F4. The loss is
       polylogarithmic.

     • Residual small volume. If 𝑌16 𝑈 ≤ 𝑁 1−𝜌 , or if the symmetric analogue holds, the trivial
       divisor bound with 𝜀 ≪ 𝜌 gives a power saving.

   Thus the argument never replaces the carrier-complement correlation by the one-variable average
  𝑝∈𝐼 𝜏𝐾 (𝑝). The complementary variable and the Goldbach expression 𝑁 − 𝑝𝑢 remain present until
∑︀

the Shiu/AP estimate is applied on the correct fixed arithmetic progression.
   —

X16C.5. Proof: The Bilinear Correlation Estimate                                Let

                           𝑆=           𝜏𝐾1 (𝑝)         𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 .
                                ∑︁                ∑︁

                                   #
                                𝑝∈𝐼16             𝑢≍𝑈


     We prove (2).

Case 1. The complementary variable is not too small Assume 𝑋𝑃 ≤ 𝑁 1−𝛿 , with 𝛿 =
                     #
1/(20𝐽02 ). Fix 𝑝 ∈ 𝐼16 . The values 𝑁 − 𝑝𝑢, as 𝑢 ≍ 𝑈 , lie in an arithmetic progression modulo 𝑝, in
an interval of length ≪ 𝑝𝑈 . After intersecting with the positive support 𝑁 − 𝑝𝑢 > 0, this set is
contained in an interval 𝐽𝑝 ⊂ [1, 𝑁 ] of length 𝐻𝑝 = 𝑁 ; this monotone enlargement can only increase
the AP divisor sum. Since 𝑋𝑃 𝑈 ≍ 𝑁 , the expected number of admissible residue-class points is still

                                        𝐻𝑝     𝑁
                                           +1≍   + 1 ≍ 𝑈 + 1 ≍ 𝑈.
                                        𝑝      𝑝
                                                   1−𝛿/2
   The modulus satisfies 𝑝 ≤ 𝑁 1−𝛿 ≤ 𝐻𝑝                            2 , and Cauchy-Schwarz
                                           . Applying (SH) to 𝑓 = 𝜏𝐾3
together with the ordinary second moment bound for 𝜏𝐾2 , gives
                                              (︃                     )︃1/2 (︃                                )︃1/2
            𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ≤               𝜏𝐾2 (𝑢)                      𝜏𝐾3 (𝑁 − 𝑝𝑢) 1𝑁 −𝑝𝑢>0
       ∑︁                                          ∑︁                           ∑︁
                                                                 2                              2
                                                                                                                     .
      𝑢≍𝑈                                         𝑢≍𝑈                       𝑢≍𝑈

     The first factor is

                                             ≪ 𝑈 1/2 (log 𝑁 )𝑂𝐽0 (1)

    by the standard second moment for fixed divisor functions. For instance, the Selberg–Delange
mean-value estimates recorded in Tenenbaum, "Introduction to Analytic and Probabilistic Number
Theory", Graduate Studies in Mathematics 163, American Mathematical Society, 3rd ed. 2015, Ch.
II.5, Theorem 5, give for fixed 𝐾


                                                           256
                                                                         2
                                           𝜏𝐾 (𝑢)2 ≪𝐾 𝑈 (log 2𝑈 )𝐾 −1 .
                                     ∑︁

                                     𝑢≍𝑈

   For the second factor, 𝑁 − 𝑝𝑢 runs through the residue class 𝑁 mod 𝑝 in the enlarged interval
                                     2 gives
𝐽𝑝 ⊂ [1, 𝑁 ]. X16-SH applied to 𝑓 = 𝜏𝐾3


                                𝜏𝐾3 (𝑁 − 𝑝𝑢)2 1𝑁 −𝑝𝑢>0 ≪ 𝑈 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑝,𝑁 .
                          ∑︁

                          𝑢≍𝑈

   Multiplying the two square-root estimates yields
                                                                                   1/2
                             𝜏𝐾2 (𝑢)𝜏𝐾3 (𝑁 − 𝑝𝑢)1𝑁 −𝑝𝑢>0 ≪ 𝑈 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑝,𝑁 .
                       ∑︁

                       𝑢≍𝑈

                     #
   Summing over 𝑝 ∈ 𝐼16 and using X16-LFA yields
                                                  1/2
                                          𝜏𝐾1 (𝑝)ℰ𝑝,𝑁 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) .
                                   ∑︁

                                     #
                                  𝑝∈𝐼16

   Therefore

                                           𝑆 ≪ 𝑈 𝑌16 (log 𝑁 )𝑂𝐽0 (1) .                                (3)
   This is the desired main term 𝑌16 𝑈 .

Case 2. The carrier is very large Assume 𝑋𝑃 > 𝑁 1−𝛿 . Since 𝑋𝑃 𝑈 ≍ 𝑁 , we have 𝑈 ≪ 𝑁 𝛿 .
     If 𝑌16 𝑈 ≤ 𝑁 1−𝜌16 , the trivial divisor bound 𝜏𝐾 (𝑛) ≪𝐾,𝜀 𝑛𝜀 , with 𝜀 chosen much smaller than
𝜌16 , gives the required power-saving term. Explicitly,

                        𝑆 ≪ 𝑁 𝜀 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) ≤ 𝑁 1−𝜌16 +𝜀 (log 𝑁 )𝑂𝐽0 (1) .

   Taking
                                               1           1
                                            𝜀 = 𝜌16 =
                                               2      2 · 106 𝐽04

   and enlarging 𝐶16 absorbs the logarithmic factor. Equivalently, with
                                                        1
                                                  𝜌′16 = 𝜌16 ,
                                                        2
   we have
                                                                   ′
                              𝑁 1−𝜌16 +𝜀 (log 𝑁 )𝑂𝐽0 (1) ≪ 𝑁 1−𝜌16 (log 𝑁 )𝐶16 .

   After this point rename 𝜌′16 as 𝜌16 . This is the harmless initial shrinkage of the displayed positive
constant in (X16-constants).
                                                                            #
   Assume now that 𝑌16 𝑈 > 𝑁 1−𝜌16 . We fix 𝑢 instead of 𝑝. As 𝑝 ∈ 𝐼16        , the values 𝑁 − 𝑢𝑝 lie in
the residue class 𝑁 mod 𝑢, and the positive part is contained in an interval 𝐽𝑢 ⊂ [1, 𝑁 ] of length

                                                  𝐻𝑢 ≍ 𝑢𝑌16 .

   Since 𝑢 ≍ 𝑈 , the non-small-volume assumption gives

                                                        257
                                             𝐻𝑢 ≫ 𝑈 𝑌16 > 𝑁 1−𝜌16 .

   The Shiu modulus condition follows from the explicit parameter inequality

                                             𝛿 < (1 − 𝜌16 )(1 − 𝛿/2).                           (4)
   Indeed, 𝑢 ≍ 𝑈 ≪ 𝑁 𝛿 , while 𝐻𝑢 ≫ 𝑁 1−𝜌16 ; hence, for large 𝑁 ,

                                               𝑢 ≤ 𝑁 𝛿 ≤ 𝐻𝑢1−𝛿/2 .

   For the displayed choices 𝛿 = 1/(20𝐽02 ) and 𝜌16 = 1/(106 𝐽04 ), (4) holds for every 𝐽0 ≥ 1; any
constant loss is absorbed by the harmless initial shrinkage of 𝜌16 .
   For fixed 𝑢, Cauchy-Schwarz gives

                           𝜏𝐾1 (𝑝)𝜏𝐾3 (𝑁 − 𝑢𝑝)1𝑁 −𝑢𝑝>0
                    ∑︁

                      #
                   𝑝∈𝐼16
                              ⎛           ⎞1/2 ⎛                           ⎞1/2

                                 𝜏𝐾1 (𝑝)2 ⎠ ⎝       𝜏𝐾3 (𝑁 − 𝑢𝑝)2 1𝑁 −𝑢𝑝>0 ⎠ .
                              ∑︁                 ∑︁
                           ≤⎝
                            ⎜             ⎟    ⎜                           ⎟
                                      #                     #
                                   𝑝∈𝐼16                 𝑝∈𝐼16


   The first factor is
                                                   1/2
                                              ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) .

   This is the ordinary 𝑞 = 1 divisor-function interval estimate; the X16 floor gives 𝑌16 ≥
𝑋𝑃 (log 𝑁 )−𝐵16 , and in the present case 𝑋𝑃 > 𝑁 1−𝛿 , so the interval is far longer than any fixed
power needed for Shiu’s short-interval corollary.
   For the second factor, 𝑁 − 𝑢𝑝 lies in the residue class 𝑁 mod 𝑢 in the interval 𝐽𝑢 of length 𝐻𝑢 ,
and the modulus condition has just been verified. X16-SH applied to 𝑓 = 𝜏𝐾   2 gives
                                                                               3


                                   𝜏𝐾3 (𝑁 − 𝑢𝑝)2 1𝑁 −𝑢𝑝>0 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑢,𝑁 .
                           ∑︁

                              #
                           𝑝∈𝐼16


   Therefore
                                                                                      1/2
                              𝜏𝐾1 (𝑝)𝜏𝐾3 (𝑁 − 𝑢𝑝)1𝑁 −𝑢𝑝>0 ≪ 𝑌16 (log 𝑁 )𝑂𝐽0 (1) ℰ𝑢,𝑁 .
                         ∑︁

                        #
                     𝑝∈𝐼16


   Summing over 𝑢 ≍ 𝑈 and using the dyadic form of X16-LFA gives

                              𝑆 ≪ 𝑌16 𝑈 (log 𝑁 )𝑂𝐽0 (1) + 𝑁 1−𝜌16 (log 𝑁 )𝑂𝐽0 (1) .

   This proves (2) in the large-carrier case.
   Combining the two cases proves the bilinear correlation estimate.
   —




                                                         258
X16C.6. Proof: Completion of X16-Core Substituting the bilinear estimate (2) into (1), and
absorbing all fixed coefficient and divisor exponents into 𝐶16 = 100𝐽02 + 100, gives

                                                      𝑌16
                      Massℬ (𝑃 ∈ 𝐼) ≪ 𝑁 (log 𝑁 )𝐶16       + 𝑁 1−𝜌16 (log 𝑁 )𝐶16 .
                                                      𝑋𝑃

   This is X16-Core for one-side grouped product carriers.
   Complementary carriers 𝑁 − 𝑃 , quotient carriers 𝑠 with 𝐿 = 𝑑𝑠, and controlled divisor quotients
𝐿/𝑑0 are reduced to this product-carrier case in Lemma X16BRS. The quotient-tag completeness
needed there is recorded in Lemma F4.
   Thus X16-BRS is proved in the BRS interface.
   —

X16C.7. Excluded Shortcut and Correct Routing                 The following shortcut is not used:

                              𝑃 =𝑝∈𝐼       =⇒     only average 𝜏 (𝑝).

   The actual remaining equation is

                                           𝑝𝑢 + 𝑄 = 𝑁.

    The proof keeps the 𝑄 = 𝑁 − 𝑝𝑢 correlation. For fixed 𝑝, the 𝑄-values form an AP modulo
𝑝; for fixed 𝑢, they form an AP modulo 𝑢. Shiu’s theorem gives the required divisor average in
whichever direction has an admissible modulus. The power-saving term covers the residual small-
volume range.
    Thus the proof uses the stated AP-divisor input rather than a one-variable divisor average.
    —
Remark G.32 (X16C.8. Output). Lemma X16C supplies the following input:

  1. X16BRS is proved using X16-Core plus the carrier-type reductions of Lemma X16BRS.

  2. BRS and TTH carry no X16-Core conditionality.

  3. The CKP smooth-weight DFI derivative condition is supplied separately by CKPD.

X16C.9. Logical Dependencies External dependency: Shiu’s Brun–Titchmarsh theorem for
multiplicative functions, used in the divisor-function AP form stated in X16C.1.
   Children served: X16-BRS, BRS, TTH.

G.13     TTH near-global length
G.13.1    TTH. Internal Length Lower Bound for B1-Origin TC1 Coarea Tests
TTH.0. Statement and Role          Lemma TTH proves the internal bypass

                                        TC1-THETA-1/3

   in the form needed by the TGT route.
   The conclusion is:



                                                259
every unrouted B1-origin coarea test in the TC1 testing family has 𝐻 ≥ 𝑋(log 𝑋)−𝐵𝜅 and hence 𝐻 ≥ 𝑋 1/3+𝜀𝜅 .

   Consequently the low-theta external input

                                      X9L-POLYLOG-MOD<1/3

    is not needed for the coarea-normalized TC1 route. The near-global Davenport/AP input X9L-
GT applies.
    TTH is not an independent analytic estimate. It is a structural consequence of BRS. The X16-
BRS/X16-Core input is supplied by Lemmas X16BRS and X16C; the parameter consequences are
recorded in the parameter register.
    In the TC1 proof TTH is used through Theorem TNG-A, the near-global-or-routed theorem:
TTH supplies the near-global alternative, while BRS/X16BRS/X16C route the complementary
short-image alternatives away before X9L-GT is invoked.
    For publication checking, Lemma TNGTTHM packages this TTH output together with TGT-
MF, TGT, TTH-SC, MRT, TTD, ROC, BRS, X16BRS, X16C, and X9L-GT into a single TC1 no-
rogue-short-interval theorem.
    Here "unrouted" means that the cell has not already been sent to Edge, LongAP/Local, CKP,
LocalDiag, or empty support by the preceding routing lemmas. Logical dependencies are the TGT.2/
TGT-MF coarea-test construction, TTH-SC, BRS, E5, X16BRS, X16C, and the parameter register.
TTH is used by TGT, TNG, and E10L; the external input X9L-GT is invoked only downstream,
after TTH has supplied the near-global length lower bound.
    —

TTH.1. Scope Restriction The following stronger statement is outside the scope of Lemma
TTH:

                         every possible E7 directional fibre has 𝐻 ≥ 𝑋 1/3+𝜀 .

     That statement is too strong at the level of the abstract E7 interface. A box may have a short
but long-enough direction 𝑈 = 𝑁 𝜃 and a transverse base sweeping many starts. Such a directional
slicing can be MRT-admissible while still having 𝑈 < 𝑋 1/3 .
     The proof after TGT and the BRS/ROC reductions does not need that stronger E7-fibre
statement. It uses the coarea tests produced from the marked affine image

                                               𝑛 = 𝐿𝑚 (𝑧),

   not an arbitrary directional fibre selected before coarea.
   Thus TC1-THETA-1/3 is proved below for the coarea testing family that is actually fed into
X9L-GT.
   —

TTH.2. Setup: Why B1-Origin Coarea Tests Are the Relevant Tests The quantifier "B1-
origin coarea test" is sufficient for the proof route for the following reason.
    In TGT, a fixed TC1 macro-template 𝜅 fixes:

  1. the B1 typed parent pattern;


                                                   260
  2. the B3 grouping skeleton;

  3. the F3/F4 routing grammar;

  4. the marked Liouville origin;

  5. the affine coefficient transport type;

  6. the TC1 tensor certificate.

   The tests used in the averaged Liouville input are then produced in TGT.3 by Fourier/coarea
decomposition along the same marked form:

                                              𝑛 = 𝐿𝑚,𝑗 (𝑧).

   Thus the Liouville argument in every unrouted test is still the terminal marked B1-origin carrier,
possibly after controlled CRT restriction, fixed divisor quotienting, full-rank transport, and Cauchy/
cube/Fourier post-terminal operations. These post-terminal operations do not create a new non-B1
Liouville carrier:

  1. Lemma E5 preserves controlled content under Cauchy/cube operations;

  2. the TGD/TGT terminal-interface lemma treats Cauchy/cube operations, primitive slicing,
     and Fourier expansion as post-terminal analytic operations, not new terminal origins;

  3. if a post-terminal operation creates a collision, rank drop, local dependence, CKP structure,
     Edge piece, or impossible support, that piece is routed away before entering the TC1 testing
     family to which X9L-GT is applied.

    Therefore every unrouted test to which X9L-GT is applied has a B1 carrier in the sense required
by Lemma BRS.
    The exclusion of arbitrary post hoc short-interval refinements is not a convention. It is the
closure principle TTH-SC:

     a short subtest is either non-structural and reaggregated, or structural and routed away.

   Thus the only tests released to X9L-GT are structural TGT-MF coarea image pieces, after
the controlled polylogarithmic scale/modulus/smoothness decomposition needed to normalize the
weights.
   —

TTH.3. Setup Fix a TC1 macro-template 𝜅 and an actual terminal B1/B3/F3/F4 GoodAWACK
atom in the Branch B route, after:

  1. C1 boundary and strict Edge pieces have been removed;

  2. LongAP/Local pieces have been passed to D1/H4M;

  3. CKP pieces have been passed to G8a;

  4. LocalDiag pieces have been passed to H4M;

  5. impossible or empty routing tags have been discarded.

                                                  261
   Let

                                                 𝐿𝑚 (𝑧)

    be the marked Liouville affine form. Let Ω* denote the C1-clean smooth box/coset cell on which
the coarea test is taken. This may be the original terminal cell or a post-WGVN/Fourier subcell Ω′𝑗 ,
but it is still a subcell of the same B1-origin carrier and has the same marked Liouville origin. Let

                                              𝐼𝑚 = 𝐿𝑚 (Ω* )

   be its marked affine image on this cell.
   Write

                          𝑌𝑚 := |𝐼𝑚 |,       𝑋𝑚 ≍ max(2, dist(𝐼𝑚 , 0) + 𝑌𝑚 )

   for the image length and height.
   The coarea testing step of TGT produces tests

                                            1 ∑︁
                                 ℒ𝑝 (𝜆) =          𝜆(𝑛)𝜌𝑝 (𝑛)𝑒(𝛼𝑝 𝑛),
                                            𝐻𝑝 𝑛∈𝐼
                                                  𝑝


  where 𝐼𝑝 is a coarea image interval or AP image piece of 𝐿𝑚 on Ω* , with polylogarithmic content/
modulus and polylogarithmic smooth partition losses.
  After fixing one scale/modulus/weight-complexity class, TTH-SC gives

                        𝐻𝑝 ≫𝜅 𝑌𝑚 (log 𝑁 )−𝐶𝜅 ,          𝑋𝑝 ≍𝜅 𝑋𝑚 (log 𝑁 )𝑂𝜅 (1) .                  (1)
    This is the controlled-structural-refinement output of TTH-SC. Pieces shorter than this are not
released X9L inputs: if they are non-structural, TTH-SC reaggregates them into the parent coarea
piece; if they are structural, TTH-SC exports them to the singular B1-origin routing package before
X9L-GT is invoked.
    —

TTH.4. Proof: BRS Gives Near-Global Image for Every Unrouted Test Theorem
BRS.1 says that for every actual B1/B3/F3/F4 terminal TC1-GoodAWACK macro-template and
every marked form 𝐿𝑚 , after C1 boundary removal and fixed macro-template normalization, the
B1 carrier slice estimate applies to any surviving carrier subcell. Therefore, applied to the interval
𝐼𝑚 = 𝐿𝑚 (Ω* ), one of the following holds:

  1. range-origin comparability:
                                            𝑌𝑚 ≥ 𝑋𝑚 (log 𝑋𝑚 )−𝐵𝜅 ;                             (ROC)

  2. the short-image subcell is strict C1P Edge;

  3. the origin is tagged and routes to LongAP/Local, CKP, LocalDiag, Edge, or empty.

   In the TC1 coarea testing family to which X9L-GT is applied, cases 2 and 3 have already been
removed by the routing assumptions in TTH.2. Therefore every remaining test comes from a marked
image satisfying (ROC).



                                                  262
    Combining (ROC) with (1), and absorbing all polylogarithmic distortions into a larger exponent
𝐵𝜅′ , gives
                                                                  ′
                                              𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 .                                       (2)
   This is the near-global lower bound needed by the Davenport/AP X9L input. We record it as
the TTH conclusion:
                                                                  ′
                                              𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅                                    (TTH)
    The exponent 𝐵𝜅′ is chosen after the BRS/X16 constants. In the notation of the parameter
register, it must dominate
                                    𝐵16 + 𝐶0 + 𝐶1 + 𝐶16 + 𝜌−1
                                                           16 + 20.

Thus the near-global conclusion uses the X16-Core constants fixed in Lemma X16C.
   It is also stronger than a 1/3-power lower bound.

Interface with TTH-SC and BRS The preceding proof uses BRS only on structural short-
image certificates. Such certificates are exactly the third alternative in TTH-SC: they are generated
by the B1-origin coarea algebra and are exported before X9L-GT is invoked. BRS is not applied to
arbitrary analytic subdivisions of an interval, because those subdivisions are not released TC1 tests
and are reaggregated by TTH-SC.
   Thus the logical order is:


TTH-SC =⇒ {near-global controlled children, or structural short-image certificates} =⇒ BRS/X16BRS/X16C =⇒ TTH.

    This is the barrier used in TNGTTHM to exclude a rogue short-interval input to X9L-GT.
    —
Parameter check G.33 (TTH.5. Parameter Check: No Hidden Short-Fibre Quantifier). The
proof above uses the following exact quantifier structure.


    Object                              Allowed to enter X9L?            Reason
    Structural TGT coarea image         Yes, after TTH                   BRS proves near-global length
    of the marked B1 carrier                                             unless the mass is Edge/tagged.
    Polylogarithmic AP/modulus/         Yes                              It loses only a fixed power of
    smoothness subdivision of that                                       log 𝑋, absorbed into 𝐵𝜅 .
    image
    Artificial subdivision into arbi-   No                              TTH-SC classifies it as non-
    trary shifted short intervals                                       structural and reaggregates it.
    Genuine structural short-image      No                              TTH-SC exports it to the
    child                                                               TTD/ROC/BRS/X16BRS/
                                                                        X16C package before X9L.
    Singular start concentration        No                              MRT/TTD/ROC/BRS routes
                                                                        it before X9L.
    Unresolved quotient/divisor         No                              F4 must tag or route it before
    origin                                                              TTH is invoked.




                                                       263
   Therefore the proof never needs the statement
                                          ⃒             ⃒
                                          ⃒ ∑︁          ⃒
                              sup              𝜆(𝑛)𝑒(𝛼𝑛)⃒ = 𝑜(|𝐼|)   (𝜃 < 1/3),
                                          ⃒             ⃒
                                          ⃒
                        𝐼⊂[𝑋,2𝑋], |𝐼|=𝑋 𝜃 ⃒             ⃒
                                        𝑛∈𝐼

    nor any polylog-modulus analogue for arbitrary short shifted intervals. The only Liouville input
is the near-global averaged AP form after the B1-origin coarea normalization.
    —

TTH.6. Output: The 1/3 Lower Bound                 Choose any fixed

                                                      2
                                              0 < 𝜀𝜅 < .
                                                      3
   For definiteness one may take 𝜀𝜅 = 1/6. Since
                                                     ′
                                     𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 ≥ 𝑋𝑝1/3+𝜀𝜅

   for all sufficiently large 𝑋𝑝 , (TTH) implies

                                            𝐻𝑝 ≥ 𝑋𝑝1/3+𝜀𝜅

    for every unrouted coarea test 𝑝 in the TC1 testing family, outside the already routed C1/tagged
pieces.
    Small bounded 𝑋𝑝 values are harmless and can be absorbed into the finite initial range of the
final sufficiently-large-𝑁 theorem.
    —

TTH.7. Output for X9L-GT X9L-GT supplies the normalized polylog-modulus averaged AP-
fibre/Fourier input for the TC1 range. Its cited form uses the near-global Davenport/AP input
whenever

                                         𝐻 ≥ 𝑋(log 𝑋)−𝐵 .

    By (TTH), every unrouted B1-origin coarea test produced by the TC1 global-testing route lies
in the near-global range. Therefore the low-theta residual

                                     X9L-POLYLOG-MOD<1/3

   is bypassed for the coarea-normalized TC1 branch.
   Combining:

  1. TGT;

  2. TTH-SC;

  3. BRS;

  4. X9L-GT in the near-global Davenport/AP range;

  5. the present length lemma;

                                                   264
    gives

                                    𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

    No pointwise shifted short-interval theorem and no low-theta polylog-modulus external theorem
are used in this route.
    —
                                  Remark G.34 (TTH.8. Output).
           TC1-THETA-1/3 is proved for the unrouted B1-origin coarea testing family.
   The proof is not an independent analytic estimate. It is a structural length consequence of the
BRS/ROC slice theorem: an actual terminal TC1 marked image is either near-global relative to its
B1 carrier height, or it has already left the GoodAWACK branch.

TTH.9. Logical Dependencies Internal dependencies: the TGT.2/TGT-MF coarea-test
construction, TTH-SC, BRS, E5, X16BRS, X16C, and the parameter register.
   External dependency: none for the structural length lower bound. X9L-GT is used only
downstream after the near-global range has been established.
   Children served: TGT, TNG, TNGTTHM, and E10L.


H      GoodAWACK Finite Grammar and Rank-Dropping AFF Clo-
       sure
This appendix uses E5 only through the following clean interface, whose proof is given once in
Appendix D.7. The E5 proof is not repeated in the GoodAWACK appendix.

H.1     E5-Clean Interface Imported from Appendix D.7
For the GoodAWACK finite-grammar arguments, E5 supplies exactly the following interface.

    1. Controlled content is preserved under the F3/F4/E5 routing transports: CRT restriction, fixed-
       divisor absorption, primitive slicing, clean full-rank affine regrouping, Cauchy/cube shifts, and
       local-diagonal extraction.

    2. A full-rank affine transport is E5-clean only when it is full rank on both the active affine-
       difference span and, after the terminal GoodAWACK object is fixed, on the terminal tensor-
       test vector span.

    3. A rank-dropping affine transport is never promoted by E5 into a new terminal GoodAWACK
       generator. It is allowed only when its lost rank has an origin tag already recorded by B1/B3/
       F3/F4 routing data, CKP, Edge, LocalDiag, impossible routing, or post-terminal analytic
       slicing.

    4. Post-terminal analytic slicing estimates a fixed terminal skeleton and does not replace the
       terminal tensor-test vectors.

    Thus Appendix H may cite the phrase ‘E5-clean’ as a content-stability and non-generator
interface, but all proofs of content stability remain in Appendix D.7.



                                                  265
H.2     E10YMX master GoodAWACK finite-grammar theorem
H.2.1    E10YMX. Master GoodAWACK Finite-Grammar Theorem
E10YMX.0. Statement and Role Lemma E10YMX is the reader-facing finite combinatorial
theorem for the HighTC part of the GoodAWACK branch. It proves, in ordinary theorem/proof
form, that the HighTC residual has no live free-affine terminal class.
   The theorem uses the following previously defined structural inputs.

  1. E10Y: the B1/B3/F3/F4/E5-clean routing grammar is complete for actual terminal
     GoodAWACK skeletons.

  2. E10X: the finite GoodAWACK grammar has an induction invariant excluding untagged rank-
     dropping affine operations in reachable states.

  3. E10M: every rank-dropping AFF occurrence in an actual terminal GoodAWACK skeleton is
     tagged.

  4. E10K: the no-untagged theorem implies AFF-origin completeness.

   The output is:

                            𝑅FreeAffineHighTC (𝑁 ) = 0.                          (E10YMX-output)
    No source-file hash, occurrence manifest, search term, or mechanical audit is a premise of this
theorem. Non-logical verification records may be maintained separately as reproducibility aids, but
the proof below uses only the mathematical definitions, lemmas, and finite transition table recorded
here and in E10Y/E10X/E10M/E10K.
    —

E10YMX.1. Definitions

Actual terminal GoodAWACK skeleton An actual terminal GoodAWACK skeleton is
a terminal routing record produced by the independent B1/B3/F3/F4/E5-clean construction. It
consists of:

  1. the B1 typed product block and dyadic cell;

  2. the B3 finite grouping record;

  3. the F3/F4 routing history;

  4. the terminal GoodAWACK label;

  5. the terminal affine forms

                                       ℒS = {𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 };

  6. the origin record for rank-changing operations;

  7. the terminal tensor-test vectors and tensors

                                           ℓ𝜌 ,    𝑄𝜌 = ℓ𝜌 ⊗ ℓ𝜌 .

                                                  266
   Post-terminal TC1 testing, coarea decomposition, Fourier expansion, Cauchy–Schwarz, BRS/
X16 estimates, Davenport/AP estimates, and local projection arguments may estimate this fixed
terminal object. They do not generate a new terminal GoodAWACK skeleton.

Rank-dropping AFF A bounded affine transformation is rank-dropping if its linear part has
a kernel on the active affine-difference span or on the terminal tensor-test span. It is tagged if its
rank drop has one of the following recorded origins:


Fix/Proj,   CRT,       FixedDiv,   VarQuot,   LocalDiag,    CKP,         PostTerminalNonGenerator.
                                                                     Edge,
                                                                                      (E10YMX-tags)
   It is untagged if it appears only as a free affine parametrization or free affine regrouping, with
no origin in the routing record.

FreeAffineHighTC A formal HighTC GoodAWACK certificate belongs to FreeAffineHighTC if
its obstruction requires an untagged rank-dropping affine origin in an actual terminal GoodAWACK
skeleton. Thus the class is live only if an actual terminal skeleton contains an untagged rank-
dropping AFF occurrence.
    —

E10YMX.2. Finite Grammar Let 𝒢GA be the finite GoodAWACK grammar. A state is a tuple

                                         s = (𝑉, ℒ, 𝒞, 𝒬, 𝒯 , 𝒪),

   where 𝑉 is the active variable list, ℒ is the visible affine-form list, 𝒞 is the controlled congruence/
content data, 𝒬 is the quotient/divisor tag data, 𝒯 is the routing tag, and 𝒪 is the origin record for
rank-changing operations.
   The start states are the B1/B3 grouped cells. The allowed transitions are:


   Transition class                  Rank effect                         Required tag or outcome
   fixing/projection                 may lower dimension by fixing Fix/Proj
                                     recorded variables
   controlled CRT restriction        full-rank finite-index restriction, CRT or empty
                                     or incompatible
   fixed-divisor quotient            quotient by a recorded fixed        FixedDiv
                                     divisor
   variable quotient residual        F4 quotient/divisor residual        VarQuot or routed tag
   local/diagonal/gcd dependence     forced equality, proportionality, LocalDiag
                                     repeated form, or local relation
   CKP-balanced relation             balanced Kloosterman-fraction CKP
                                     structure
   strict saving or boundary rela-   C1 Edge, square-divisor, short- Edge
   tion                              volume, high-frequency, small-
                                     conductor, or boundary case
   bounded affine regrouping         full-rank change, or rank drop inherited tag
                                     with recorded upstream origin




                                                   267
   E5 auxiliary inheritance          content transport for an already inherited tag; no new terminal
                                     generated routing record          generator
   primitive/post-terminal slicing   analytic operation after termi- PostTerminalNonGenerator
                                     nal tensor-test vectors are fixed
   terminal labelling                assigns Edge, CKP,                terminal label
                                     GoodAWACK, LocalDiag, or
                                     LongAP/Local




    E10Y proves that this table is complete for actual skeleton-generating pre-terminal operations
in B1/B3/F3/F4/E5-clean descendants. Therefore an operation not represented in the table is not
an actual terminal GoodAWACK skeleton generator.
    —

E10YMX.3. Grammar Invariant               For a state s, define the invariant

      ℐ(s) :   every rank-dropping AFF occurrence visible in s has a tag in (E10YMX-tags).

   The invariant is stated only for reachable states of 𝒢GA .

Lemma H.1 (Lemma E10YMX.1. Invariant preservation). The invariant ℐ holds for every reachable
state of 𝒢GA .

Proof. We induct on the length of a grammar derivation.
    At length zero, the state is a B1/B3 grouped cell. B1 supplies product variables and dyadic
cells, and B3 selects among finitely many product grouping candidates. No rank-dropping affine
transformation has yet been applied.
    Assume ℐ(s) and apply one transition.
    If the transition fixes or projects variables, its origin is recorded as Fix/Proj. If it is a CRT
restriction, it is either full-rank on the active difference lattice or the fibre is empty; the origin
is recorded as CRT. Fixed-divisor and variable-quotient transitions record FixedDiv or VarQuot.
Local, diagonal, gcd, repeated form, or proportionality transitions leave GoodAWACK or record
LocalDiag. CKP-balanced transitions leave GoodAWACK and record CKP. Strict saving, square-
divisor, short-volume, boundary, high-frequency, or small-conductor transitions leave GoodAWACK
and record Edge.
    A bounded affine regrouping is allowed only if it is full-rank on the active affine span and
terminal tensor-test span, or if the rank drop inherits an already recorded upstream origin. E5
auxiliary inheritance transports content or auxiliary variables already present in the routing record
and cannot create a new terminal affine generator. Post-terminal slicing is allowed only after the
terminal tensor-test vectors have been fixed, and is recorded as PostTerminalNonGenerator.
    Thus every transition either preserves rank, records one of the allowed tags, or routes the state
away from terminal GoodAWACK. Hence ℐ is preserved. The induction proves the lemma.
    —




                                                  268
E10YMX.4. Completeness and No-Untagged-AFF

Lemma H.2 (Lemma E10YMX.2. Actual skeletons are reachable). Every actual terminal
GoodAWACK skeleton is a reachable terminal state of 𝒢GA .

Proof. This is exactly the completeness theorem E10Y. B1 and B3 supply the start states. F3/F4
supply all pre-terminal routing operations: CRT absorption, F4 large-divisor and quotient decisions,
square-divisor routing, grouping selection or elimination, LocalDiag detection, Edge detection, and
terminal class labelling. E5 supplies only content stability for transports whose origin tags are
already present. Post-terminal analytic operations estimate a fixed terminal object and do not
generate a new skeleton. Hence the actual skeleton is reachable in the displayed grammar.


Corollary H.3 (Corollary E10YMX.3. No untagged AFF in actual terminal GoodAWACK). No
actual terminal GoodAWACK skeleton contains an untagged rank-dropping AFF occurrence.

Proof. Let S be an actual terminal GoodAWACK skeleton. By Lemma E10YMX.2, it is reachable
in 𝒢GA . By Lemma E10YMX.1, every rank-dropping AFF occurrence in a reachable state carries a
tag in (E10YMX-tags). Therefore no untagged rank-dropping AFF occurrence is present. This is
the E10M theorem in the master-grammar language.
   —


E10YMX.5. Main Theorem

Theorem H.4 (Theorem E10YMX. GoodAWACK finite-grammar closure). Every actual terminal
GoodAWACK skeleton has no untagged rank-dropping AFF occurrence. Consequently the formal
FreeAffineHighTC class has no actual terminal occurrence:

                          𝑅FreeAffineHighTC (𝑁 ) = 0.                        (E10YMX-FreeAffine)

Proof. The first assertion is Corollary E10YMX.3.
    It remains to connect this no-untagged-AFF statement with the formal FreeAffineHighTC
residual isolated in E10G–E10J. The reduction chain E10G, E10H, E10I and E10J isolates exactly
one possible live formal residual: a HighTC GoodAWACK certificate whose survival requires an
actual rank-dropping affine regrouping with no allowed origin tag. E10K translates Corollary
E10YMX.3 into AFF-origin completeness: every rank-dropping affine map in an actual terminal
GoodAWACK skeleton has one of the allowed origins.
    Therefore the live condition defining FreeAffineHighTC is false for actual B1-origin terminal
GoodAWACK skeletons. The formal class is empty in the active proof tree, and
                                     𝑅FreeAffineHighTC (𝑁 ) = 0.
The theorem is proved.
   —




                                                 269
E10YMX.6. Routed HighTC Corollary

Corollary H.5 (Corollary E10YMX.4. HighTC GoodAWACK output for E10L). Every HighTC
GoodAWACK certificate arising from an actual terminal GoodAWACK skeleton is either:

  1. origin-degenerate and routed by HGO2R to CKP, LocalDiag, Edge, impossible, or an already
     handled local class; or

  2. a formal FreeAffineHighTC certificate, which is empty by Theorem E10YMX.

   Thus E10L may import the HighTC branch through the single statement

      E10YMX =⇒ 𝑅HighTC-GoodAWACK (𝑁 ) is routed away or zero as a GoodAWACK class.

    The routed outputs are terminal classes already defined by the F3/F4, C1P, LPI, and CKP
interfaces. Their estimates or local assembly are handled by the corresponding branch lemmas; they
are not additional GoodAWACK assumptions.
    —

Parameter check H.6 (E10YMX.7. Parameter Check). The theorem introduces no analytic
parameter and no new error term. It uses:

  1. the fixed-depth B1 decomposition;

  2. the finite B3 grouping list;

  3. the finite F3/F4 routing interface, packaged by F3F4M;

  4. the E5-clean content-stability interpretation;

  5. the finite transition table in E10X;

  6. the no-untagged rank-drop theorem E10M;

  7. the AFF-origin-completeness consequence E10K.

   All losses are structural or polylogarithmic content losses already registered in E5 and the
parameter register. Since E10YMX proves a zero residual 𝑅FreeAffineHighTC (𝑁 ) = 0, it contributes
no summation loss to the global error budget.
   —

E10YMX.8. Logical Dependencies Internal dependencies: B1, B3, F3P, F3, F3A, F3T, F4,
F3F4M, E5, BGS, HGO2R, BAOC, E10G, E10H, E10I, E10J, E10Y, E10M, E10X, and E10K.
   External dependency: none.
   Children served: E10L, GEB, I1, and the full proof assembly.




                                               270
H.3     E10L Branch B GoodAWACK theorem
H.3.1    E10L. Branch B GoodAWACK Theorem without X8
E10L.0. Statement and Role Lemma E10L is the Branch B / GoodAWACK theorem. It
proves

                                    𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).                        (E10L-output)
   The proof uses two independent GoodAWACK subroutes.
  1. TC1 route. Lemma TNGTTHM proves that every actual B1-origin TC1 coarea test is either
     near-global and X9L-admissible, or routed away before X9L-GT is invoked.
  2. HighTC route. Lemma E10YMX proves the finite GoodAWACK grammar theorem: no
     actual terminal GoodAWACK skeleton contains an untagged rank-dropping AFF occurrence,
     and

                                       𝑅FreeAffineHighTC (𝑁 ) = 0.

   Thus the GoodAWACK branch is closed by

           TGD + TNGTTHM + E10YMX =⇒ 𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).                         (E10L-route)
    The proof uses no X8. The external Liouville input used by the TC1 route is only the near-
global Davenport/AP form X9L-GT, imported through TNGTTHM after TTH supplies the length
lower bound.
    —

E10L.1. Inputs The proof is carried out after the F3/F4 terminal routing interface has fixed a
GoodAWACK terminal atom. The following inputs are used.
  1. TGD. Every terminal GoodAWACK atom belongs to exactly one of

                           TC1-GoodAWACK         or   HighTC-GoodAWACK.

  2. TNGTTHM. The TC1 branch is near-global and X9L-admissible, or routed to Edge, LongAP/
     Local, CKP, LocalDiag, empty, or impossible before X9L-GT.
  3. E10YMX. The HighTC finite grammar has no live FreeAffineHighTC terminal class:

                                       𝑅FreeAffineHighTC (𝑁 ) = 0.

  4. HGO2R. Origin-degenerate HighTC certificates are routed to already handled terminal
     classes.
  5. Terminal branch estimates. Routed outputs are already handled by:

         C1   (Edge),     G8a   (CKP),      H4M       (LongAP/Local, LocalDiag, local reroutes),

      or contribute zero in empty/impossible cases.
   The E5-clean content-stability interface is used inside E10YMX and the routing infrastructure.
E10L itself does not introduce a new affine-regrouping rule.
   —

                                               271
E10L.2. Theorem

Theorem H.7 (Theorem E10L.1. GoodAWACK cancellation). For every tagged terminal
GoodAWACK atom produced by the B1/B3/F3/F4/E5-clean interface, the total GoodAWACK
contribution is 𝑜(𝑁 ). Consequently

                                   𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

Proof. Fix a terminal GoodAWACK macro-template 𝜅. By TGD, the contribution of this template
is the sum of its TC1 contribution and its HighTC contribution.


TC1 branch Apply TNGTTHM to the released B1-origin TC1 coarea tests attached to 𝜅.
TNGTTHM gives a disjoint alternative.
  In the near-global branch, the tests are MRT-admissible, satisfy PACK, and obey

                                     𝐻𝑝 ≥ 𝑋𝑝 (log 𝑋𝑝 )−𝐵𝜅 .

    Therefore the near-global Davenport/AP theorem X9L-GT applies to the same measured family.
It gives the required averaged Liouville cancellation, so the TC1 contribution of 𝜅 is 𝑜(𝑁 ).
    In the routed branch, TNGTTHM removes the test before X9L-GT is invoked. The cell is routed
to Edge, LongAP/Local, CKP, LocalDiag, empty, or impossible. Those outputs are not terminal
TC1-GoodAWACK mass. They are estimated or assembled by C1, G8a, H4M, or zero.
    Thus the TC1 contribution is 𝑜(𝑁 ).

HighTC branch For the HighTC part, HGO2R routes every origin-degenerate HighTC certificate
to CKP, LocalDiag, Edge, impossible, or an already handled local class. These outputs are handled
by G8a, H4M, C1, or zero.
   The only formal HighTC class not discharged by origin-degenerate routing is

                                       FreeAffineHighTC.

    By E10YMX, the finite GoodAWACK grammar is complete for actual terminal GoodAWACK
skeletons, every rank-dropping AFF occurrence in such a skeleton is tagged, and therefore

                                    𝑅FreeAffineHighTC (𝑁 ) = 0.

    Hence the HighTC contribution is routed away or empty as a GoodAWACK class.
    Combining the TC1 and HighTC branches gives an 𝑜(𝑁 ) bound for the 𝜅-contribution. There
are only finitely many macro-templates, depending on the fixed Heath–Brown depth and routing
grammar. Summing over them gives

                                   𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

   The theorem is proved.
   —




                                               272
E10L.3. No-X8 and No-Short-Interval Profile              The proof does not use:

  1. X8 inverse-Gowers input;

  2. a full nilsequence form of X9;

  3. pointwise shifted short-interval cancellation for 𝜆;

  4. low-𝜃 polylog-modulus estimates for arbitrary short AP fibres;

  5. a source-file or occurrence-manifest premise for HighTC closure.

   The TC1 route uses X9L-GT only after TNGTTHM has proved that the relevant tests are near-
global. The HighTC route uses E10YMX as a finite mathematical grammar theorem.
   —
                                  Remark H.8 (E10L.4. Output).
Branch B / GoodAWACK cancellation is proved by TGD, TNGTTHM, E10YMX, and the terminal branch estimates.
   This supplies

                                      𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 )

   for the final assembly.
   —

E10L.5. Logical Dependencies External dependency: none directly; X9L-GT in the near-
global Davenport/AP range is imported through TNGTTHM.
   Internal dependencies: TGD, TNGTTHM, E10YMX, HGO2R, C1, G8a, and H4M. The com-
ponent TC1 route is imported through TNGTTHM, and the component HighTC finite-grammar
route is imported through E10YMX.
   Children served: I1 and the final proof assembly.

H.4     BGS skeleton normal form
H.4.1    BGS. Skeleton Normal Form for Terminal GoodAWACK Descendants
BGS.0. Statement and Role The skeleton record below is an intrinsic B1/B3/F3/F4/E5
normal form. It records B3 product groupings, full-rank coordinate changes, and rank drops with
explicit fixing/projection, quotient/divisor/local, CKP, Edge, impossible, or post-terminal analytic
origin tags. It does not obtain its meaning from E10L. The later E10Y/E10M/E10K/E10X finite-
grammar layer consumes this record to exclude untagged rank-dropping affine regrouping as a
terminal-vector generator.
    Lemma BGS extracts the normal form for terminal GoodAWACK atoms generated by the B1/
B3/F3/F4 routing interface.
    The purpose is to prepare the finite structural theorem HGO.2:

                     HighTC-cert =⇒ CKP ⊔ LocalDiag ⊔ Edge ⊔ Impossible.

   The output is a finite parameterized skeleton normal form that records:

  1. the parent B1 product block;


                                                 273
  2. the B3 grouping choice;

  3. the F3/F4 routing history;

  4. the active affine forms that survive into terminal GoodAWACK;

  5. the origin of each affine form;

  6. the exact tensor test needed for TC1/HighTC.

   This is the object used downstream by the E10YMX/E10L finite-grammar and clean-interface
arguments.
   Logical dependencies: B1, B3, F3, F4, and E5. Outputs served: HGO2R, BAOC, E10Y, E10K,
E10L, and E10X.
   —

BGS.1. B1 parent block          Every terminal GoodAWACK atom has a parent typed B1 block

                                           ℬ = (𝑟, 𝑠, X, Y, t),

   where:

  1. 𝑟, 𝑠 ≤ 2𝐽0 ;

  2. X, Y are dyadic scale vectors;

  3. t records elementary coefficient types

                                        𝜇 · 1≤𝑦 ,          1,       log;

  1. the parent equation is


                                          𝑃𝐴 (𝑎) + 𝑃𝐵 (𝑏) = 𝑁,                           (B1)
   with
                                             𝑟                              𝑠
                                 𝑃𝐴 (𝑎) =                      𝑃𝐵 (𝑏) =
                                            ∏︁                             ∏︁
                                                  𝑎𝑖 ,                           𝑏𝑗 .
                                            𝑖=1                           𝑗=1

   The parent variable set is

                                    𝒳ℬ = {𝑎1 , . . . , 𝑎𝑟 , 𝑏1 , . . . , 𝑏𝑠 }.

   All later descendants keep the parent B1 tag. This is required by H4 and by the clean F3/F4
routing interface.
   —




                                                         274
BGS.2. B3 grouping skeleton A B3 grouping skeleton is a finite partition of selected parent
variables into grouped factors:

                                   𝑢𝜈 =                     𝑣𝜈 =
                                           ∏︁                      ∏︁
                                                 𝑥,                       𝑥,
                                          𝑥∈𝐼𝜈                     𝑥∈𝐽𝜈

   and similarly on both sides of (B1).
   The grouping skeleton records:

  1. which grouped factors are short, long, or central;

  2. whether a balanced bilinear grouping is exposed;

  3. whether a local AP configuration is exposed;

  4. whether a forced local dependence/collision is exposed;

  5. which residual grouped variables remain available for BranchB/GoodAWACK.

   By Lemma B3, the set of possible grouping skeletons is finite:

                                            |𝒢(ℬ)| ≤ 𝐶(𝐽0 ).

    Only grouping skeletons that are not terminal Edge, CKP, LongAP/Local, or LocalDiag can
feed the GoodAWACK skeleton.
    —

BGS.3. F3/F4 routing skeleton Starting from a B1 block and a B3 grouping skeleton, F3/F4
perform only finitely many routing-level operations before terminality.
   For a terminal GoodAWACK descendant, the routing skeleton records the following data:

                                  r = (rCRT , rdiv , rquot , rgrp , rfail ).

   Here:

  1. rCRT records controlled CRT/congruence restrictions;

  2. rdiv records fixed divisor absorptions;

  3. rquot records variable quotient equations 𝐿(𝑧) = 𝑑𝑠 that were resolved without becoming Edge,
     LocalDiag, or CKP;

  4. rgrp records B3 product groupings, full-rank affine changes of variables, and only rank-dropping
     maps carrying one of the E10Y/E10M/E10K tags packaged by the E10YMX finite-grammar
     theorem;

  5. rfail records failed terminal alternatives that were checked and eliminated.

   The GoodAWACK routing condition is:

                    ¬Edge,      ¬CKP,           ¬LongAP/Local,                 ¬LocalDiag,      (G0)
   together with:

                                                      275
  1. no unresolved ordinary large-divisor predicate;
  2. central-long affine/WACLE residual structure;
  3. controlled content;
  4. at least one marked Liouville-type affine form.

   The important limitation is that the F3/F4 routing skeleton does not include the quadratic
tensor test
                                           ?
                                      𝑄𝑚 ∈ spanQ {𝑄𝑖 : 𝑖 ̸= 𝑚}.

   That test is added only at the TC1/HighTC dichotomy stage.
   —

BGS.4. Setup: Active Parameter Lattice After applying the F3/F4 routing skeleton, a
terminal GoodAWACK descendant is supported on a bounded-rank lattice coset

                                      𝑧 ∈ ΩS ⊂ 𝑧* + ΛS ⊂ Z𝑘S ,

   where:

  1. 𝑘S ≤ 𝐾(𝐽0 );
  2. ΩS is a smooth box-like region;
  3. every active long direction has length at least 𝑁 𝜃 , up to C1-routed boundary/short-volume
     exceptions;
  4. the lattice index and all contents are bounded by a power of log 𝑁 .

   We write the active parameter vector as

                                           𝑧 = (𝑧1 , . . . , 𝑧𝑘S ).

   The affine transformations from the parent variables to 𝑧 are recorded by an origin map

                                                    origS .

    This map is part of the skeleton. It is needed to decide whether a later algebraic relation
corresponds to CKP, genuine LocalDiag, Edge, or an impossible parent configuration.
    —

BGS.5. Setup: Active Affine Forms The terminal GoodAWACK descendant has a finite
active affine system

                                       ℒS = {𝐿𝜌 (𝑧) : 𝜌 ∈ ℐS },

   where

                           𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 ,       ℓ𝜌 ∈ Z𝑘S ,    𝑐𝜌 ∈ Z,

   after clearing the controlled lattice denominator.
   Each 𝐿𝜌 has one of the following origins:

                                                     276
Type A. Parent coordinate / grouped factor

                                                     𝐿𝜌 (𝑧)

   is the affine representative of a surviving parent variable or grouped factor after CRT restriction
and affine change of variables.

Type B. Fixed divisor quotient           A controlled divisor relation

                                       𝑑 | 𝐿(𝑧),         𝑑 ≤ (log 𝑁 )𝐶 ,

   has been absorbed on a lattice coset, producing

                                             𝐿𝜌 (𝑧) = 𝐿(𝑧)/𝑑

   as an integer affine form on the restricted lattice.

Type C. Variable quotient residual              A quotient equation

                                                   𝐿(𝑧) = 𝑑𝑠

   has been resolved without short-volume Edge, without forced local dependence, and without
balanced CKP structure. The quotient variable 𝑠 survives as an affine form

                                                𝐿𝜌 (𝑧) = 𝑠(𝑧).

Type D. Primitive slice / fibre form After primitive slicing, a marked form may be written
on a fibre as

                              𝐿𝜌 (𝑧0 + 𝑢𝑣) = 𝑔𝑢 + 𝑏,            𝑔 ≤ (log 𝑁 )𝐶 .

   This is used analytically in E7/E10, but the pre-slicing form remains in ℒS for the tensor test.

Type E. Auxiliary bounded affine factor An auxiliary divisor-bounded or smooth coefficient
factor depends on

                                                     𝐿𝜌 (𝑧)

   and is treated analytically as 𝑓𝜌 (𝐿𝜌 (𝑧)).
   Thus the terminal atom has model form

                         AS =          𝑊S (𝑧)            𝜆𝜌 (𝐿𝜌 (𝑧))          𝑓𝜌 (𝐿𝜌 (𝑧)),     (BGS)
                                ∑︁                 ∏︁                  ∏︁

                                𝑧∈ΩS            𝜌∈ℳS                   𝜌∈𝒰S

   where ℳS ̸= ∅ is the set of marked Liouville-type forms and 𝒰S ⊆ ℐS is the set of auxiliary
bounded/smooth coefficient forms. In the E10 proof one marked form is selected and denoted 𝐿0 .
   All active forms satisfy

                                          cont(𝐿𝜌 ) ≤ (log 𝑁 )𝐶 .

   —

                                                        277
BGS.6. Definition: Skeleton Normal Form
Definition H.9 (Definition. B1-to-GoodAWACK skeleton). A B1-to-GoodAWACK skeleton is a
tuple

                              S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S ),

   where:

  1. ℬ is a B1 parent block;

  2. Γ ∈ 𝒢(ℬ) is a B3 grouping skeleton;

  3. r is the F3/F4 routing skeleton;

  4. ΛS is the active lattice/coset data;

  5. ΩS is the smooth central-long domain;

  6. ℒS is the active affine system;

  7. ℳS ⊆ ℒS is the nonempty marked Liouville-form set;

  8. origS records the B1/F3/F4 origin of each affine form;

  9. 𝒲S records dyadic weights and coefficient types.

   It is admissible if:

                          ¬Edge,   ¬CKP,      ¬LongAP/Local,        ¬LocalDiag,            (Admiss)
   and the terminal GoodAWACK predicate of Lemma F3 holds.
   —

BGS.7. Lemma: terminal GoodAWACK atoms admit skeleton normal form
Lemma H.10 (Lemma BGS.1). Every tagged terminal GoodAWACK atom produced by the chain

                                          𝐵1 → 𝐵3 → 𝐹 3/𝐹 4

   admits an admissible B1-to-GoodAWACK skeleton normal form S, and can be written as (BGS).
Proof. Start with the parent B1 block. By Lemma B1, it has the product equation (B1), finitely
many variables, dyadic weights, and coefficient types 𝜇, 1, and log.
    By Lemma B3, the block receives a finite set of grouping alternatives. Choose the grouping
history that leads to the given terminal descendant. This supplies Γ.
    By Lemma F3, every routing step is one of the allowed finite routing-level operations: con-
trolled CRT absorption, F4 large-divisor decision, square-divisor routing, finite grouping selection/
elimination, LocalDiag detection, or strict Edge detection. Since the atom is terminal GoodAWACK,
all Edge, CKP, LongAP/Local, and LocalDiag outcomes have been checked and eliminated for this
descendant, and no unresolved ordinary large divisor predicate remains.
    By Lemma F4, any ordinary large divisor or quotient equation that survives without becoming
Edge, LocalDiag, or CKP is absorbed into a central-long affine residual with controlled content.
The content quotient lemma ensures that the quotient forms remain controlled on the active lattice.

                                                   278
    By Lemma E5, controlled content is stable under CRT restriction, fixed divisor absorption,
primitive slicing, clean full-rank affine regrouping, and Cauchy/cube operations. Therefore the
active forms that reach E10 are affine forms of controlled content, and every rank-changing operation
appearing in the skeleton record carries one of the explicit routing tags listed above.
    Finally, by the terminal GoodAWACK predicate in the F3/F4 interface, at least one active affine
form carries a Liouville-type oscillatory coefficient, the active directions are long, and the atom has
the model form (BGS).
    Collecting the parent block, grouping, routing history, active lattice, affine system, marked forms,
origin map and weights gives S. Lemma proved.
   —


BGS.8. Setup: Tensor Interface on a Skeleton                 For each active affine form

                                           𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 ,

   define

                                     𝑄𝜌 = ℓ𝜌 ⊙ ℓ𝜌 ∈ Sym2 (Q𝑘S ).

   The TC1/HighTC test on the skeleton is:

                               ∃𝑚 ∈ ℳS           / spanQ {𝑄𝜌 : 𝜌 ̸= 𝑚}
                                              𝑄𝑚 ∈                                            (TC1-Skel)
   for TC1, and

                               ∀𝑚 ∈ ℳS        𝑄𝑚 ∈ spanQ {𝑄𝜌 : 𝜌 ̸= 𝑚}                     (HighTC-Skel)
   for HighTC.
   Equivalently, HighTC supplies for each marked 𝑚 a relation

                                            𝑐𝜌 𝑄𝜌 = 0,      𝑐𝑚 ̸= 0.                            (H-cert)
                                      ∑︁

                                     𝜌∈ℐS

   Because 𝑘S and |ℐS | are bounded in terms of 𝐽0 , this is a bounded-size rational row-reduction
problem for each fixed skeleton.
   —

BGS.9. Output: What This Normal Form Resolves The skeleton normal form resolves
three issues needed before HGO.2:

  1. it separates the parent product equation from the terminal affine system;

  2. it records the origin of each affine form, so a HighTC certificate can be tested for CKP/
     LocalDiag/Edge meaning rather than treated as a bare linear-algebra relation;

  3. it makes clear that HighTC testing is finite and terminal, hence it does not create a new
     routing loop.

   In particular:


                                                   279
                                                    HighTC
   is now a certificate attached to a concrete skeleton
                                                      S,
   not an undefined terminal branch.
   —

BGS.10. Scope Boundary and Structural Closure Lemma BGS proves the skeleton normal
form. It does not by itself prove that all HighTC certificates reroute to CKP, LocalDiag, Edge, or
Impossible. That structural conclusion is supplied by HGO2R, E10M, E10K, and E10L.
     The reason is that the BGS normal form records the finite-parametric grammar of admissible
skeletons, rather than a literal list of all symbolic skeleton instances. It proves that the skeleton set
is finite for fixed 𝐽0 , but it does not by itself list:
  1. all possible affine coefficient vectors ℓ𝜌 ;
  2. all possible quotient-origin maps;
  3. all symbolic dependencies among the ℓ𝜌 ;
  4. all conditions under which a HighTC certificate forces CKP, LocalDiag, Edge, or impossibility.
   Moreover, some skeleton entries may depend on controlled divisor/CRT parameters. These
parameters are polylogarithmically bounded in size, but for structural proof they must be treated
symbolically, not by numerical enumeration.
   Thus the output of BGS is the finite-parametric input for the subsequent finite-grammar theorem.
   —

BGS.11. Structural Use
Theorem H.11 (Theorem BGS/HGO.2). For every admissible B1-to-GoodAWACK skeleton S, if
S has a HighTC certificate (H-cert), then one of the following holds:
  1. the origin map origS exposes an admissible balanced bilinear grouping, so the atom is CKP;
  2. the certificate forces equality, proportionality, fixed gcd-local dependence, or an LPI/H4M-
     admissible canonical local projection, so the atom is LocalDiag;
  3. the certificate forces short residual volume, large content/gcd, or another strict C1P saving
     predicate, so the atom is Edge;
  4. the certificate is incompatible with the parent product equation (B1), the dyadic central-long
     constraints, and the routing history r.
   This theorem is supplied by the HGO2R/E10 closure chain. Once it is combined with the BGS
normal form, one obtains:
                                      𝑅HighTC-GoodAWACK (𝑁 ) = 0
    after rerouting, and Branch B closes using the already proved TC1 Fourier lemma.
    The role of Lemma BGS is to make the statement finite and symbolic; the exclusion of untagged
free-affine HighTC skeletons is handled by E10YMX.
    —

                                                     280
BGS.12. Output for the Proof Tree
                         B1-to-GoodAWACK skeleton normal form proved.
   The normal form is:
                             S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )
   with terminal atom model (BGS).
   This is sufficient to formulate HGO.2 precisely as a finite-parametric skeleton theorem and to
pass the problem to HGO2R/E10YMX/E10L.

BGS.13. Logical dependencies Internal dependencies: B1, B3, F3, F4, and E5.
  Children served: HGO2R, E10M, E10K, E10L.

H.5     HGO2 reduction
H.5.1    HGO2R. Reduction of BGS/HGO.2 to Free-Affine HighTC Exclusion
HGO2R.0. Statement and Role Lemma HGO2R proves the origin-degenerate rerouting
statement HGO2R.1. The free-affine class is treated by the finite-grammar closure theorems E10Y,
E10M, E10X, E10K, and E10L.
    The structural block is:

                    HighTC-cert =⇒ CKP ⊔ LocalDiag ⊔ Edge ⊔ Impossible                    (HGO.2)
   for admissible B1-to-GoodAWACK skeletons in the sense of Lemma BGS.
   The BGS/HGO2R part proves this implication for HighTC certificates whose quadratic depen-
dence is visible through the recorded origin map. The remaining free-affine case is delegated to
E10YMX.
   The reduction statement is:
      HGO.2 reduces to excluding FreeAffine-HighTC skeletons from actual B1 descendants.
   Thus HGO2R is a reduction theorem with an explicit structural-closure dependency.
   Logical dependencies: BGS, C1, the CKP branch, the H4 LocalDiag admission criterion, E10Y,
E10X, E10K, and E10L. Outputs served: E10M, E10K, and E10L.
   —

HGO2R.1. Setup: Starting Point             Let
                             S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )
   be an admissible B1-to-GoodAWACK skeleton.
   Write
                               𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 ,    𝑄𝜌 = ℓ𝜌 ⊙ ℓ𝜌 .
   Assume that S is HighTC. Then for every marked form 𝑚 ∈ ℳS there is an integer relation

                                           𝑐𝜌 𝑄𝜌 = 0,     𝑐𝑚 ̸= 0.                         (H-cert)
                                     ∑︁

                                    𝜌∈ℐS

   The question is whether (H-cert), together with the origin map origS , forces terminal rerouting.
   —

                                                  281
HGO2R.2. Setup: Degenerate-Origin Certificates Call a HighTC certificate origin-
degenerate if at least one of the following is forced by the support of the relation and the origin
data of the participating forms.

                                                       ̸ 𝜎 in the support of (H-cert) such
D1. Repeated or proportional affine origin There are 𝜌 =
that the current lattice forces

                                            𝐿𝜌 = 𝑎𝐿𝜎 + 𝑏

    with fixed rational 𝑎, 𝑏, or the corresponding parent/product factors are repeated after quotient-
ing.

D2. Fixed gcd-local dependence The forms in the support of (H-cert) are tied by a fixed
divisor/gcd relation recorded in rdiv or rquot , so that one active form is determined by another on a
fixed local residue class.

D3. Balanced multiplicative origin The support of (H-cert) splits through origS into two
long grouped multiplicative variables on each side of the parent B1 equation, with the B3 balance
predicates required by Lemmas B3 and F3.

D4. Strict saving origin The certificate forces one of the strict C1P predicates: short residual
volume, large gcd/content budget, square-divisor budget, Type-I error budget, high-frequency
budget, or small-conductor budget with the full ambient normalization.

D5. Parent incompatibility The certificate forces a linear or congruence relation incompatible
with the parent B1 product equation, the dyadic scale cell, or the current CRT lattice. Then the
skeleton has empty support.
    The complementary case is called free-affine:

                                       FreeAffineHighTC(S)

   if (H-cert) holds but none of D1–D5 is forced by the current recorded origins.
   —

HGO2R.3. Lemma: origin-degenerate HGO.2

Lemma H.12 (Lemma HGO2R.1). Let S be an admissible B1-to-GoodAWACK skeleton with a
HighTC certificate. If the certificate is origin-degenerate, then the corresponding terminal atom
reroutes to one of:

                         CKP,       LocalDiag,         Edge,    Impossible.

Proof. We use the cases D1–D5.
    In case D1, the current atom contains a forced equality, proportionality, or repeated factor after
quotienting. This is exactly within the terminal LocalDiag predicate of Lemma F3, provided the
resulting contribution is a canonical local term. The LPI admission condition consumed by H4 is
satisfied because the relation is tagged by the parent B1 block and routing history. Hence the atom
is LocalDiag.


                                                 282
    In case D2, the fixed gcd/divisor data determine one active form from another on the current
lattice. This is the fixed local dependence case of Lemmas F3 and F4. Again the term is admitted
only as a tagged canonical local projection, so it is LocalDiag rather than an arbitrary local-looking
term.
    In case D3, the origin map exposes a balanced finite-convolution bilinear structure. By the B3
CKP candidate criterion and the F3 CKP terminal predicate, this is a CKP atom. The coefficient
and content conditions are preserved by Lemmas F4 and E5. The CKP estimate and canonical zero-
frequency normalization are handled by Lemma G8a.
    In case D4, the certificate forces one of the strict C1P saving predicates. By Lemma C1, ordinary
large-divisor or small-conductor labels alone are not enough; but D4 assumes the full strict budget.
Therefore the atom is terminal Edge and contributes 𝑜(𝑁 ).
    In case D5, the active lattice/domain is empty, or the putative skeleton is incompatible with the
tagged B1 block. The contribution is zero. It can be recorded as Edge-zero or Impossible.
    These cases cover all origin-degenerate certificates. Lemma proved.
   —


HGO2R.4. Scope Boundary The origin-degenerate lemma leaves open the case where the
quadratic tensor relation is a genuine higher-true-complexity relation among distinct primitive affine
forms, with no forced local dependence, no balanced multiplicative origin, no strict saving predicate,
and no parent incompatibility visible from the recorded origin data.
   This is a genuine structural boundary of HGO2R. There is a standard model.
   Let


      𝐿0 (𝑥, 𝑟) = 𝑥,     𝐿1 (𝑥, 𝑟) = 𝑥 + 𝑟,     𝐿2 (𝑥, 𝑟) = 𝑥 + 2𝑟,    𝐿3 (𝑥, 𝑟) = 𝑥 + 3𝑟.       (4AP)

   The homogeneous coefficient vectors are

                                   ℓ𝑖 = (1, 𝑖) ∈ Z2 ,    0 ≤ 𝑖 ≤ 3.

   Their quadratic tensors satisfy

                                      𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.                               (4AP-cert)
   Indeed, this is the system

                 1 − 3 + 3 − 1 = 0,       0 − 3 + 6 − 3 = 0,      0 − 3 + 12 − 9 = 0

   for the (𝑥2 , 𝑥𝑟, 𝑟2 ) tensor coordinates.
   Thus if any 𝐿𝑖 is a marked Liouville-type form, the marked tensor lies in the span of the others.
The skeleton is HighTC.
   But the 4AP pattern has:

  1. no equality or proportionality among the forms;

  2. no fixed gcd-local dependence;

  3. no balanced CKP bilinear multiplicative structure;

  4. no short-volume or large-content saving by itself;

                                                  283
  5. no contradiction with being central-long affine.

   Therefore, under the safe F3/H4 interpretation of LocalDiag as a genuine canonical local term,
the relation (4AP-cert) is not a LocalDiag certificate.
   There is a related linear dependence

                                      𝐿0 − 3𝐿1 + 3𝐿2 − 𝐿3 = 0.

   If the broad B3 phrase "affine dependence among active forms" were read literally as automatic
LocalDiag, this pattern would be routed away. That reading is not part of the proof: Lemma H4
admits LocalDiag only when the term is a tagged canonical local projection, not merely because
an affine identity exists among oscillatory forms. Thus 4AP-like free-affine patterns cannot be
dismissed by the broad B3 phrase unless an additional canonical-local admission proof is supplied.
   This is exactly the interface true-complexity obstruction isolated by the GoodAWACK TC1/
HighTC analysis.
   —

HGO2R.5. Interface Example: Formal Free-Affine Skeleton The BGS normal form is
broad enough to allow the following formal skeleton unless the B1-origin exclusion lemma is used.
   Let the active lattice be a two-dimensional central-long box

                      Ω = {(𝑥, 𝑟) : 𝑥 ≍ 𝑋, 𝑟 ≍ 𝑅, 𝑥 + 3𝑟 ≍ 𝑋, 𝑋, 𝑅 ≥ 𝑁 𝜃 }.

   Let

                           ℒ = {𝑥, 𝑥 + 𝑟, 𝑥 + 2𝑟, 𝑥 + 3𝑟},   ℳ = {𝑥}.

   Let the remaining three forms be auxiliary bounded coefficient forms, and let the weight be
smooth and divisor-bounded. All affine contents are 1.
   This formal skeleton satisfies the explicit GoodAWACK-style features:

  1. central-long affine structure;

  2. bounded affine complexity;

  3. at least one marked Liouville-type form;

  4. controlled content;

  5. no unresolved ordinary large divisor predicate;

  6. no strict Edge predicate;

  7. no CKP-balanced multiplicative form;

  8. no LPI/H4M-admissible LocalDiag relation.

   It is HighTC by (4AP-cert).
   Lemma HGO2R is the origin-degenerate part of the HGO.2 route. The formal skeleton above is
the free-affine class isolated by the interface. That class is routed to E10YMX, which supplies the
actual-origin exclusion needed to complete full HGO.2 for terminal GoodAWACK descendants.
   —

                                                284
HGO2R.6. Free-Affine Exclusion Full HGO.2 is equivalent, over the origin-degenerate lemma
proved here, to the following finite structural exclusion.

Lemma H.13 (Lemma HGO2R.2. No free-affine HighTC skeletons). For every admissible B1-to-
GoodAWACK skeleton S produced by the chain

                                        𝐵1 → 𝐵3 → 𝐹 3/𝐹 4,

   every HighTC certificate is origin-degenerate in the sense of Section HGO2R.2.
   Equivalently:

             FreeAffineHighTC(S)      never occurs for actual B1 descendants.                (NoFAH)
   This exclusion is supplied by E10YMX.
   —

HGO2R.7. Consequence for E10             The E10 decomposition after the TC1 Fourier closure is:

                𝑅GoodAWACK (𝑁 ) = 𝑅TC1-GoodAWACK (𝑁 ) + 𝑅HighTC-GoodAWACK (𝑁 ),

   with

                                    𝑅TC1-GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

   By Lemma HGO2R.1, the origin-degenerate part of HighTC reroutes to already handled branches:

                     𝑅HighTC,deg (𝑁 ) ⊆ CKP ⊔ LocalDiag ⊔ Edge ⊔ Impossible.

   Thus the only obstruction left by HGO2R alone is

                                        𝑅FreeAffineHighTC (𝑁 ).

   This class is empty by E10YMX, so HGO2R supplies the origin-degenerate HighTC rerouting
component of Branch B.
   —

HGO2R.8. Output for the Proof Tree

           Origin-degenerate HighTC reroutes to CKP, LocalDiag, Edge, or Impossible.

   The NoFAH/free-affine class is closed by E10X and E10K.
   What is proved here:

               OriginDegenerateHighTC =⇒ CKP ⊔ LocalDiag ⊔ Edge ⊔ Impossible.

   The surviving free-affine class is not a terminal output of Branch B; it is passed to the downstream
E10YMX/E10L finite-grammar layer.

HGO2R.9. Logical dependencies Internal dependencies: BGS, C1, CKP branch, LocalDiag/
LPI routing, and the F3/F4 terminal routing interface.
   Children served: E10M, E10K, E10L.

                                                 285
H.6     BAOC affine origin catalogue
H.6.1    BAOC. B1 Affine-Origin Catalogue
BAOC.0. Statement and Role Lemma BAOC supplies the B1/B3/F3/F4 transport catalogue
for homogeneous coefficient vectors in terminal GoodAWACK skeletons. It is a provenance grammar,
not the final no-free-affine theorem. The decisive exclusion of untagged rank-dropping AFF is
supplied by E10Y, E10X, E10M, E10K, and E10L.
    The statement is:


every terminal GoodAWACK affine form is generated by a finite B1/B3/F3/F4 transport grammar.

    More precisely, for every terminal GoodAWACK skeleton S and every active affine form
𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 , the vector ℓ𝜌 is obtained from B1/B3 grouped coordinates by finitely many of
the transport rules T1–T7 below.
    BAOC also records the scope boundary of this catalogue: provenance alone does not decide
every true-complexity relation among the ℓ𝜌 . The free-affine class isolated by that boundary is
routed to E10YMX/E10L.
    Logical dependencies: B1, B3, F3, F4, E5, BGS, HGO2R, E10Y, E10X, E10M, E10K, and E10L.
Outputs served: HGO2R, E10Y, E10M, E10K, E10L, and E10X.
    —

BAOC.0a. Setup: What F4 Already Proves Lemma F4 already proves the core ordinary-
divisor part of BAOC.
    In F4.1, every ordinary large-divisor predicate has one of the forms

                        𝑑 | 𝐿(𝑧),    𝐿(𝑧) = 𝑑𝑠,       𝑑 | gcd(𝐿1 (𝑧), 𝐿2 (𝑧)),

   where 𝐿, 𝐿1 , 𝐿2 are affine or product-grouped forms already produced by B1/B3.
   F4.3 records fixed-divisor absorption:

                     Λ𝑑 = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 0 (mod 𝑑)},         𝐿𝑑 (𝑧) = 𝐿(𝑧)/𝑑.

   F4.4 proves the content quotient formula
                                                       𝑔
                                    contΛ𝑑 (𝐿/𝑑) =          ≤ 𝑔.
                                                     (𝑔, 𝑑)
   F4.5–F4.11 handle the variable quotient equation

                                            𝐿(𝑧) = 𝑑𝑠

   and prove the exhaustive alternative:

              OrdinaryLargeDivisor =⇒ Edge ⊔ LocalDiag ⊔ CKP ⊔ GoodAWACK,

     or else the corrected F3 measure strictly decreases.
     Thus the BAOC transport rules T3 and T4 below are not new results. They are F4 translated
into homogeneous-vector bookkeeping.
     What F4 does not do is decide every terminal true-complexity relation among the final active
list of vectors {ℓ𝜌 }. That structural decision is made by E10YMX/E10L.
     —

                                               286
BAOC.1. Setup: Coefficient-Vector Bookkeeping                             Let a terminal GoodAWACK skeleton be

                            S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S ).

  Every active affine form is written on the active parameter lattice as

                                              𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 .

  BAOC must catalogue the homogeneous vectors

                                                     ℓ𝜌 ∈ Z𝑘S

  with enough origin data to decide whether a quadratic tensor relation

                                                    𝑐𝜌 (ℓ𝜌 ⊙ ℓ𝜌 ) = 0                                         (QRel)
                                               ∑︁

                                               𝜌

  is:

  1. CKP-origin;

  2. LPI/H4M-admissible LocalDiag-origin;

  3. strict C1P Edge-origin;

  4. impossible;

  5. or genuinely free-affine.

  Only cases 1–4 close HGO.2 structurally.
  —

BAOC.2. Setup: Base B1 and B3 Coordinate Source                                    Lemma B1 supplies a parent product
block
                                  𝑟             𝑠
                                       𝑎𝑖 +         𝑏𝑗 = 𝑁,           𝑟, 𝑠 ≤ 2𝐽0 .                              (B1)
                                 ∏︁            ∏︁

                                 𝑖=1          𝑗=1

  Lemma B3 supplies a finite grouping set. For a grouping

                                                     Γ = (𝐼, 𝐽)

  one has grouped variables

                                       𝑢𝐼 =                       𝑣𝐼 =
                                               ∏︁                         ∏︁
                                                     𝑎𝑖 ,                       𝑎𝑖 ,
                                               𝑖∈𝐼                        𝑖∈𝐼
                                                                           /

  and

                                       𝑢′𝐽 =                      𝑣𝐽′ =
                                               ∏︁                         ∏︁
                                                     𝑏𝑗 ,                        𝑏𝑗 ,
                                               𝑗∈𝐽                        𝑗 ∈𝐽
                                                                            /




                                                            287
   with grouped parent relation

                                           𝑢𝐼 𝑣𝐼 + 𝑢′𝐽 𝑣𝐽′ = 𝑁.                                    (B3)
    At the affine-origin bookkeeping level, a selected grouped variable is treated as a coordinate in a
finite free coordinate system attached to (ℬ, Γ). Thus the base coefficient vectors are coordinate
vectors

                                                     𝑒𝛼

    for surviving grouped variables and residual product variables.
    This is a change of coordinates, not an assertion that 𝑒𝛼 is a linear form in the original factor
variables 𝑎𝑖 , 𝑏𝑗 . Product grouping is multiplicative, and the affine catalogue starts after a grouping
has been chosen and the descendant has entered an affine/WACLE regime.
    —

BAOC.3. Setup: Allowed Coefficient Transport Rules The B1/B3/F3/F4/E5 lemmas
support the following transport grammar for homogeneous coefficient vectors.

T1. Fixing and projection When some grouped variables are fixed by dyadic slicing, congruence
slicing, or conditioning, they become constants. Homogeneous vectors are projected to the remaining
active coordinates.
     If

                                             𝑧 = (𝑧free , 𝑧fixed )

   and

                                 𝐿(𝑧) = ℓfree · 𝑧free + ℓfixed · 𝑧fixed + 𝑐,

   then the transported vector is

                                                 ℓ ↦→ ℓfree .

T2. Controlled CRT restriction F3 controlled CRT absorption replaces a lattice coset by a
subcoset

                        Λ′ = {𝑧 ∈ Λ : 𝐿0 (𝑧) ≡ 𝑎 (mod 𝑞)},            𝑞 ≤ (log 𝑁 )𝐶 .

   Choosing coordinates

                                               𝑧 = 𝑧0 + 𝑇 𝑧 ′

   on the sublattice transports

                                                 ℓ ↦→ 𝑇 𝑡 ℓ.                                     (CRT)
    E5 proves that controlled content remains controlled. The particular matrix 𝑇 is part of the
skeleton provenance data.


                                                     288
T3. Fixed divisor quotient        If a fixed divisor condition

                                               𝑑 | 𝐿(𝑧)

   is absorbed and the quotient form survives, then on the restricted lattice

                                           𝐿𝑑 (𝑧) = 𝐿(𝑧)/𝑑.

   On homogeneous vectors this gives
                                                   1 𝑡
                                              ℓ ↦→   𝑇 ℓ,                                   (FDQ)
                                                   𝑑
   where the right side is integral on the restricted coordinate lattice. E5/F4 prove content does
not increase. The triple (𝑑, 𝑇, ℓ) is recorded as part of the quotient-origin data.

T4. Variable quotient residual         For an ordinary quotient equation

                                              𝐿(𝑧) = 𝑑𝑠,                                          (VQ)
   F4 either routes to Edge, LocalDiag, CKP, or leaves a central-long affine GoodAWACK quotient
form.
   If 𝑠 survives as an active quotient form, then in the extended active coordinate system its
homogeneous vector ℓ𝑠 satisfies

                                             𝑑ℓ𝑠 = 𝑇 𝑡 ℓ𝐿 .                                     (VQT)
    This relation is part of the origin record. If it forces local dependence, the atom is LocalDiag; if
it exposes balanced multiplicative structure, it is CKP; if it gives a strict C1P budget, it is Edge.
Otherwise it may feed GoodAWACK.

T5. Bounded affine regrouping          E5 permits bounded affine changes and regrouping:

                                            𝑧 = 𝑧0 + 𝐴𝑧 ′ ,

   where relevant coefficients and minors are bounded by powers of log 𝑁 . The vector transport is

                                               ℓ ↦→ 𝐴𝑡 ℓ.                                       (AFF)
   This preserves controlled content, but by itself it does not decide true complexity.

T6. Primitive slicing      Primitive slicing chooses a long one-dimensional fibre

                                             𝑧 = 𝑧0 + 𝑢𝑣.

   On the fibre a marked form becomes

                                 𝐿(𝑧0 + 𝑢𝑣) = 𝑔𝑢 + 𝑏,         𝑔 = ℓ(𝑣).

   For BAOC, the pre-slicing vector ℓ remains the object used in the TC1/HighTC tensor test.
The fibre coefficient 𝑔 is used analytically by E7/E9.


                                                 289
T7. Auxiliary bounded forms Auxiliary bounded or smooth coefficient forms inherit their
homogeneous vectors through the same transport rules T1–T6.
   No routing step is allowed to introduce an affine form without one of these provenance operations.
   —

BAOC.4. Statement and Proof: Transport-Level Catalogue

Theorem H.14 (Theorem BAOC.1. Transport-level affine-origin catalogue). Every active affine
form 𝐿𝜌 in a terminal GoodAWACK skeleton produced by

                                          𝐵1 → 𝐵3 → 𝐹 3/𝐹 4

   has a homogeneous vector ℓ𝜌 generated from the B1/B3 grouped-coordinate source by finitely
many applications of T1–T7.
   Equivalently, for every terminal skeleton S there exists a finite provenance expression

                                            ℓ𝜌 ∈ 𝒞tr (ℬ, Γ, r)

   for each active form, where 𝒞tr is the transport closure generated by T1–T7.

Proof. Start with the B1 parent block. By Lemma B1, the only initial variables are finitely many
product variables 𝑎𝑖 , 𝑏𝑗 , with 𝑟, 𝑠 ≤ 2𝐽0 .
    Choose the B3 grouping history Γ. By Lemma B3, the grouping set is finite and each grouping
replaces products of parent variables by grouped variables 𝑢𝐼 , 𝑣𝐼 , 𝑢′𝐽 , 𝑣𝐽′ . At the affine bookkeeping
level these grouped variables supply the base coordinate vectors.
    F3 allows only controlled CRT absorption, F4 large-divisor decision, finite grouping selec-
tion/elimination, terminal LocalDiag detection, terminal Edge detection, and terminal labelling.
Controlled CRT absorption transports coefficient vectors by T2.
    F4 handles fixed divisor and variable quotient equations. F4.3–F4.4 give fixed divisor absorption
and quotient content control, which are T3. F4.5–F4.11 give the exhaustive variable quotient
decision, which is T4 together with the alternatives Edge, LocalDiag, CKP, and GoodAWACK. If
the quotient relation creates forced local dependence, balanced multiplicative structure, or strict
saving, the atom is no longer terminal GoodAWACK; it is LocalDiag, CKP, or Edge. Hence any
quotient form that reaches GoodAWACK is exactly a T4 quotient form.
    E5 records the permitted affine regrouping, primitive slicing, and content-stability operations.
These are T5 and T6. Auxiliary forms inherit the same transport data, giving T7.
    Since the F3 measure is well-founded and every routing history is finite, only finitely many
transport steps occur. Therefore every terminal GoodAWACK homogeneous vector lies in the
transport closure 𝒞tr (ℬ, Γ, r). This proves the transport catalogue.
   —


BAOC.5. Scope Boundary Relative to NoFAH BAOC is a provenance grammar, not the
final no-free-affine closure theorem.
    By itself it does not decide:

   1. whether a given affine regrouping is tagged or untagged in the E10M sense;

   2. whether a HighTC relation among the ℓ𝜌 ⊙ ℓ𝜌 is origin-degenerate;


                                                  290
   3. whether a formal free-affine pattern is impossible for actual B1 descendants.

   Consequently, BAOC alone cannot exclude the formal free-affine pattern

                  ℓ0 = (1, 0),   ℓ1 = (1, 1),    ℓ2 = (1, 2),   ℓ3 = (1, 3).              (4AP-vectors)
   These vectors have bounded coefficients and controlled content. Their tensors satisfy

                                       𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.

    The BAOC transport grammar alone does not contain a rule saying that a bounded affine family
of this shape cannot be produced by T1–T7.
    Thus BAOC proves provenance, while NoFAH is supplied by E10YMX/E10L.
    —

BAOC.6. Structural Closure Needed for NoFAH The structural closure must add a tagged-
origin classification to the transport grammar.

Lemma H.15 (Lemma BAOC.2. Tagged B1 affine-origin closure). For every parent block ℬ,
grouping Γ, and terminal routing history r, the transport closure T1–T7 can be refined to a finite-
parametric list

                                 F(ℬ, Γ, r) = {ℱ𝜈 (p) : 𝜈 ∈ 𝒩 , p ∈ 𝒫𝜈 },

   where each ℱ𝜈 (p) gives:

  1. a concrete active coordinate lattice;

  2. concrete transport matrices 𝑇, 𝐴;

  3. concrete quotient-origin relations;

  4. the full active affine vector list {ℓ𝜌 };

  5. the marked set ℳ;

  6. the terminal rejection data for Edge, CKP, LocalDiag, and LongAP/Local.

    Moreover, for every listed family, every HighTC certificate is origin-degenerate, impossible,
or routed by the E10YMX no-untagged-AFF finite-grammar closure, with its allowed origin tags
supplied by E10M/E10K.
    Then NoFAH follows.

Proof that the tagged closure implies NoFAH. Let S be an actual terminal GoodAWACK skeleton
with a HighTC certificate. The tagged closure places its coefficient vectors in one of the listed
families ℱ𝜈 (p). The final clause says that the certificate is origin-degenerate, impossible, or excluded
by the no-untagged-AFF closure. Therefore S is not FreeAffineHighTC. This proves NoFAH.
   By HGO2R, NoFAH implies full HGO.2.
   —




                                                     291
BAOC.7. Scope Boundary: Alternative Route Through B3/H4 There is a different
possible strong input:

                        Every B3 affine-dependence flag is H4-canonical.

   If this were proved, then many free-affine patterns, including the 4AP identity

                                    𝐿0 − 3𝐿1 + 3𝐿2 − 𝐿3 = 0,

   could be safely routed to LocalDiag.
   This alternative is not used here. Lemma H4 deliberately admits only canonical local projections
tagged by the parent B1 cell and routing history. A bare affine identity among oscillatory forms is
not enough.
   Thus the B3/H4 route would require a separate admission theorem. The proof instead uses the
E10YMX finite-grammar closure, which is later consumed by E10L.
   —

BAOC.8. Output for Branch B           The Branch B chain is:

             GoodAWACK = TC1 ⊔ OriginDegenerateHighTC ⊔ FreeAffineHighTC.

   The TC1 part is closed by Lemma TNG, which packages TGT, MRT, TTD, ROC, BRS, TTH,
and X9L-GT in the near-global form. The second part is rerouted by HGO2R.
   The remaining part is

                                       𝑅FreeAffineHighTC (𝑁 ).

    BAOC records this class as a structural catalogue boundary. The class is not left to BAOC
alone: E10M proves that no untagged rank-dropping affine origin survives, E10K converts this into
AFF-origin completeness, and E10L assembles the resulting GoodAWACK cancellation.
    —
                             Remark H.16 (BAOC.9. Output).
 BAOC proves the transport catalogue; the no-free-affine closure is supplied by E10YMX/E10L.
  Established output:

                                          ℓ𝜌 ∈ 𝒞tr (ℬ, Γ, r)

   for every terminal GoodAWACK active affine form.
   Structural closure:

                      The free-affine class is discharged by E10YMX/E10L.

   Structural completion block:


E10YMX packages the enumeration and classification of the relevant rank-dropping affine origins.

   This is the structural task supplied by the E10X closure chain.


                                                292
H.7      E10G catalogue schema
H.7.1       E10G. Strong BAOC Catalogue and Reduction
E10G.0. Statement and Role Lemma E10G supplies a finite catalogue schema and identifies
the FreeAffineHighTC obstruction that is discharged by the finite GoodAWACK grammar closure
Lemmas E10Y and E10X. It is not used as an independent proof of strong BAOC. Its role is to
reduce the formal catalogue class to the actual-origin closure theorem E10Y/E10X, after which
E10K gives AFF-origin completeness and E10L assembles the GoodAWACK estimate.
   E10G treats the strong form of BAOC isolated by Lemma BAOC.
   The desired structural statement is:

  every actual terminal GoodAWACK affine-vector family has no FreeAffineHighTC certificate.

    Equivalently, for every terminal GoodAWACK skeleton

                                        S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )

    produced by

                                                       𝐵1 → 𝐵3 → 𝐹 3/𝐹 4,

    every HighTC relation

                                                      𝑐𝜌 (ℓ𝜌 ⊙ ℓ𝜌 ) = 0,           𝑐𝑚 ̸= 0
                                                ∑︁

                                                  𝜌

    with 𝑚 ∈ ℳS should be forced into one of:

                                   CKP,            LocalDiag,             Edge,          Impossible.

    The outcome is a precise split.


A finite-parametric catalogue compiler is available, and the decisive actual-origin rigidity is supplied by E10YMX.

    More concretely:

   1. B1/B3/F3/F4/E5 give a finite-parametric transport catalogue schema for all terminal
      GoodAWACK coefficient vectors.
   2. For every fixed catalogue cell, the TC1/HighTC tensor test is a finite symbolic row-reduction
      problem.
   3. The broad affine-regrouping/CRT transport interface admits formal 4AP-like FreeAffine-
      HighTC vector patterns, so the proof routes those formal witnesses to the actual-origin closure
      theorem E10Y/E10X.

   Thus E10G supplies the finite catalogue schema and identifies the closure input supplied by
E10Y/E10X:


prove transport rigidity for the actual affine-regrouping matrices, or route every resulting free affine dependence to H4-canonical LocalDiag.


    —

                                                                    293
E10G.1. Setup: Source Data Already Proved We use only the already established source
lemmas.

B1 source      Lemma B1 supplies a typed finite-convolution parent block
                                                         𝑟                       𝑠
             𝑃𝐴 (𝑎) + 𝑃𝐵 (𝑏) = 𝑁,           𝑃𝐴 (𝑎) =                 𝑃𝐵 (𝑏) =                𝑟, 𝑠 ≤ 2𝐽0 .         (B1)
                                                        ∏︁                      ∏︁
                                                              𝑎𝑖 ,                    𝑏𝑗 ,
                                                        𝑖=1                     𝑗=1

   The parent block has finitely many dyadic scale and coefficient-type choices.

B3 source      Lemma B3 supplies a finite grouping set

                                            𝒢(ℬ),         |𝒢(ℬ)| ≪𝐽0 1,

   and preliminary labels:

                TypeI/Edge,        LongAP/Local,          CKP,       BranchB,         LocalDiag flag.

   For the BranchB/GoodAWACK path, all short, purely local, CKP-balanced, and forced-
dependence candidates must have failed or have been terminally routed away.

F3/F4 source        Lemma F3 defines terminal GoodAWACK by:

  1. central-long affine WACLE structure;

  2. bounded affine complexity;

  3. smooth weight of polylogarithmic complexity;

  4. no forced local diagonal relation;

  5. no unresolved ordinary large divisor condition;

  6. at least one marked affine Liouville-type form with controlled content;

  7. long active fibre directions.

   Lemma F4 proves the exhaustive ordinary-divisor decision:

                OrdinaryLargeDivisor =⇒ Edge ⊔ LocalDiag ⊔ CKP ⊔ GoodAWACK,

   or else the corrected F3 measure strictly decreases.
   The GoodAWACK quotient case is precisely the case where the divisor/quotient ambiguity has
been absorbed or resolved and the remaining object is central-long affine with controlled content.

E5 source      Lemma E5 proves controlled content stability under:


CRT,   fixed divisor absorption,   primitive slicing,   affine regrouping,   Cauchy/cube,     local diagonal extraction.

   For the present catalogue, the important point is that E5 controls content but does not enumerate
the affine transport matrices.
   —

                                                         294
E10G.2. Strong catalogue cells Fix a parent block ℬ, a B3 grouping Γ ∈ 𝒢(ℬ), and an F3/F4
routing history r that ends in terminal GoodAWACK.
   The BAOC grammar can be sharpened into the following finite-parametric catalogue schema.

Cell C0. Base grouped-coordinate cell        After choosing (ℬ, Γ), the active pre-routing coordinate
module is

                                   𝑉0 = Z𝑘0 ,       𝑘0 ≤ 𝐾0 (𝐽0 ).

   The base homogeneous vectors are coordinate vectors

                                                𝑒𝛼 ∈ 𝑉0*

   attached to surviving grouped variables and residual product variables.
   This cell is finite for fixed 𝐽0 .

Cell C1. Projection/fixing cell      Fixing dyadic, congruence, or auxiliary variables replaces

                                        𝑉 ∼
                                          = 𝑉free ⊕ 𝑉fixed

   by 𝑉free . Homogeneous vectors are transported by the projection
                                                        *
                                        𝜋free : 𝑉 * → 𝑉free .

   This cell cannot create HighTC relations not already present after restriction; it only deletes
coordinates.

Cell C2. Controlled CRT cell        A controlled congruence restriction

                              𝐿0 (𝑧) ≡ 𝑎 (mod 𝑞),          𝑞 ≤ (log 𝑁 )𝐶 ,

   chooses coordinates

                                          𝑧 = 𝑧0 + 𝑇 𝑧 ′

   on the sublattice. Homogeneous vectors are transported by

                                             ℓ ↦→ 𝑇 𝑡 ℓ.                                   (CRT-T)
   The determinant and relevant minors of 𝑇 are polylogarithmically controlled by F3/E5. The
actual-origin classification of such matrices is supplied by E10YMX.




                                                  295
Cell C3. Fixed-divisor quotient cell          For

                                                𝑑 | 𝐿(𝑧)

   absorbed on a restricted lattice, the quotient form is

                                         𝐿𝑑 (𝑧) = 𝐿(𝑧)/𝑑.

   On coefficient vectors:
                                                 1 𝑡
                                              ℓ ↦→ 𝑇 ℓ,                                     (FDQ-T)
                                                 𝑑
   where the right side is integral on the chosen restricted coordinate lattice.
   By F4.4/E5.2, content does not increase:

                                                contΛ (𝐿)
                             contΛ𝑑 (𝐿/𝑑) =                  ≤ contΛ (𝐿).
                                              (contΛ (𝐿), 𝑑)

   This cell is already controlled by F4 at the transport-catalogue level.

Cell C4. Variable quotient residual cell            For a quotient equation

                                              𝐿(𝑧) = 𝑑𝑠,                                        (VQ)
   F4 routes to Edge, LocalDiag, CKP, or GoodAWACK.
   If it reaches GoodAWACK, then neither short-volume Edge, nor forced local dependence, nor
balanced CKP applies. The surviving quotient vector satisfies an origin relation

                                              𝑑ℓ𝑠 = 𝑇 𝑡 ℓ𝐿 .                                 (VQ-T)
    The relation (VQ-T) must be retained as part of the strong catalogue cell.
    Any HighTC certificate using (VQ-T) in a way that determines one active form from another is
origin-degenerate and is already handled by HGO2R.

Cell C5. Bounded affine regrouping cell              E5 allows bounded affine changes and regrouping:

                                           𝑧 = 𝑧0 + 𝐴𝑧 ′ ,

   with coefficients and relevant minors bounded by powers of log 𝑁 . Homogeneous vectors
transform by

                                               ℓ ↦→ 𝐴𝑡 ℓ.                                   (AFF-T)
    This is the decisive cell. E5 proves controlled content under such transformations. The
classification of which matrices 𝐴 can arise from actual routing is supplied by E10YMX rather than
by E5 itself.
    Therefore C5 requires the actual-origin classification supplied by E10YMX before it can be used
inside clean terminal GoodAWACK.




                                                    296
Cell C6. Primitive slicing cell       Primitive slicing writes a marked form on a long fibre as

                                          𝐿(𝑧0 + 𝑢𝑣) = 𝑔𝑢 + 𝑏.

    For the true-complexity verification, the relevant vector is the pre-slicing vector ℓ, not merely
the one-dimensional coefficient 𝑔. This cell supplies the analytic E7/E9 interface but does not by
itself decide HighTC.
    —

E10G.3. Catalogue compiler theorem
Lemma H.17 (Lemma E10G.1. Finite-parametric strong-catalogue schema). Every terminal
GoodAWACK skeleton produced by

                                          𝐵1 → 𝐵3 → 𝐹 3/𝐹 4

   belongs to a finite-parametric catalogue cell obtained by composing C0–C6.
   More explicitly, for each terminal skeleton there are:

  1. a finite B1/B3 source cell (ℬ, Γ);
  2. a finite F3/F4 routing word r;
  3. controlled integer matrices 𝑇𝑗 , 𝐴𝑗 ;
  4. quotient-origin equations 𝑑ℓ𝑠 = 𝑇 𝑡 ℓ𝐿 ;
  5. a finite active list {ℓ𝜌 };
  6. a nonempty marked subset ℳ;
  7. terminal rejection data recording why Edge, CKP, LocalDiag, and LongAP/Local did not apply.

    For every fixed choice of this symbolic data, TC1/HighTC is decided by a finite rational row-
reduction on

                                             𝑄𝜌 = ℓ𝜌 ⊙ ℓ𝜌 .

Proof. B1 and B3 give finitely many parent/grouping source cells. The number of parent variables
and groupings is bounded in terms of 𝐽0 .
    F3 has a well-founded routing measure M♯ , and its generic routing steps are limited to con-
trolled CRT absorption, F4 large-divisor decision, square-divisor routing, finite grouping selection/
elimination, terminal LocalDiag detection, terminal Edge detection, and terminal class labelling.
Hence every routing word r is finite, with length bounded by the initial obstruction data.
    F4 handles every ordinary divisor or quotient predicate. Its fixed-divisor and variable-quotient
branches are exactly C3 and C4 when the residual reaches GoodAWACK; otherwise the atom is
terminal Edge, LocalDiag, or CKP.
    E5 supplies the content-stability rules for CRT restriction, fixed divisor absorption, affine
regrouping, primitive slicing, and Cauchy/cube operations. At the homogeneous-vector level these
are C1–C6.
    Thus every active terminal vector ℓ𝜌 is produced by a finite composition of C0–C6, with quotient-
origin relations retained. Since the number of active forms and ambient rank are bounded in terms
of 𝐽0 , the tensor list

                                                  297
                                        {ℓ𝜌 ⊙ ℓ𝜌 } ⊂ Sym2 (Q𝑘 )

   has bounded size. Therefore, after fixing a catalogue cell, TC1/HighTC is a finite rational row-
reduction problem. Lemma proved.
   —


E10G.4. Rigidity input for strong BAOC The compiler lemma implies NoFAH once it is
combined with the following rigidity statement, supplied by the master closure Lemma E10X and
the AFF-OC consequence E10K.

Lemma H.18 (Lemma E10G.2. Transport-rigidity NoFAH). For every catalogue cell C0–C6 that
is actually produced by the B1/B3/F3/F4 routing history, every HighTC tensor relation

                                       𝑐𝜌 (ℓ𝜌 ⊙ ℓ𝜌 ) = 0,    𝑐𝑚 ̸= 0
                                  ∑︁

                                   𝜌

   with 𝑚 ∈ ℳ is origin-degenerate:

  1. it uses a repeated/proportional source vector;

  2. or it uses a fixed divisor/gcd/quotient-origin relation;

  3. or it exposes a B3 CKP-balanced multiplicative grouping;

  4. or it forces a strict C1P Edge predicate;

  5. or it is incompatible with the parent cell and routing history.

   If E10G.2 holds, then HGO2R gives:

                       HighTC =⇒ CKP ⊔ LocalDiag ⊔ Edge ⊔ Impossible,

   and hence

                                       𝑅FreeAffineHighTC (𝑁 ) = 0.

    Branch B closes using Lemma TNG for the global TC1 near-global testing chain plus the origin-
degenerate rerouting.
    —

E10G.5. Scope Boundary: Free-Affine Class Inside the Catalogue Interface E10G.2 is a
reduction target for the master closure Lemma E10X, not a consequence of the catalogue compiler
alone.
   The free-affine class is not F4’s fixed-divisor or variable-quotient analysis. Those parts carry
explicit origin equations and are exactly the cases handled by HGO2R.
   The obstruction is the combination of:

  1. terminal GoodAWACK accepting any bounded-complexity central-long affine system with
     controlled content and a marked Liouville form, after negative tests fail;

  2. E5 allowing bounded affine regrouping with controlled coefficients/minors;

                                                  298
  3. the catalogue schema alone does not classify the actual matrices 𝐴, 𝑇 ;

  4. the catalogue schema alone does not prove that every affine dependence generated by such
     matrices is H4-canonical LocalDiag.

   Because of C5, the catalogue schema still admits the following vector family:

             ℓ0 = (1, 0),    ℓ1 = (1, 1),        ℓ2 = (1, 2),      ℓ3 = (1, 3).       (4AP-vectors)
   Indeed, these arise from the two-coordinate source (𝑥, 𝑟) by the bounded affine forms

                                  𝑥,    𝑥 + 𝑟,     𝑥 + 2𝑟,      𝑥 + 3𝑟.

   All coefficients are 𝑂(1), all contents are 1, and the affine complexity is bounded.
   Their quadratic tensors satisfy:

                                    𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.                               (4AP-Q)
   If any 𝐿𝑖 is marked, the marked tensor lies in the span of the others. Thus the pattern is
HighTC.
   At the level of the GoodAWACK interface alone, (4AP-vectors) need not be:

  1. Edge by short volume or content;

  2. CKP by balanced multiplicative origin;

  3. LocalDiag in the H4-canonical sense;

  4. impossible from the parent B1 equation.

    Therefore the catalogue compiler passes this formal class to the E10X finite-grammar and origin-
completeness closure.
    This is the same structural interface example as in the NoFAH B1-origin verification, but now
localized to a precise catalogue cell: bounded affine regrouping before the E10X rigidity theorem is
applied.
    —

E10G.6. Closure Route The proof uses Route R1 below, as formalized in E10YMX/E10L.
Routes R2 and R3 are recorded only as alternative sufficient inputs.

Route R1. Affine-regrouping rigidity Strengthen C5 from arbitrary bounded affine regrouping
to an explicit rigid list of matrices generated by the actual B1/B3/F3/F4 operations.
    One sufficient target would be:

                              𝐴 ∈ 𝒜rigid (ℬ, Γ, r),       |𝒜rigid | ≪𝐽0 1,

   where every listed matrix is built from coordinate projection, signed permutation, controlled
diagonal quotient, CRT basis choice with recorded congruence origin, and incidence maps coming
from quotient equations.
   Then one must row-reduce each resulting family and prove that every HighTC relation is origin-
degenerate or impossible.

                                                   299
Route R2. B3/H4 canonical-local admission Prove that every B3 affine-dependence flag
surviving into a terminal GoodAWACK-looking affine system is actually LPI/H4M-admissible
canonical LocalDiag.
   This would route patterns such as

                                                     𝐿0 − 3𝐿1 + 3𝐿2 − 𝐿3 = 0

    to LocalDiag, but only if the resulting local term is genuinely admitted by H4’s tagged canonical
local-projection interface.
    This route must not use the broad word "affine dependence" alone; it needs an H4 admission
proof.

Route R3. Higher-order analytic input An actual 4AP-like free-affine catalogue cell outside
the E10YMX actual-origin closure would require a separate analytic estimate at that cell.
    Such an estimate would have to control the surviving HighTC family, for example through a
𝑈 -level or nilsequence orthogonality input with complexity strong enough for the exact catalogue
  3

cell.
    This route reintroduces a higher-order analytic block, but now with a sharply specified target
rather than the whole GoodAWACK class.
    —
                              Remark H.19 (E10G.7. Output).
         E10G proves the finite catalogue schema used by the E10YMX/E10L closure.
    What is proved here:

Every terminal GoodAWACK atom belongs to a finite-parametric catalogue schema C0–C6, and each fixed cell has a finite tensor row-reduction test.


    Structural closure:

                             The C5/free-affine class is discharged by E10YMX/E10L.

    Completion block:

                    E10YMX provides the required affine-regrouping origin completeness.

H.8       E10H matrix-origin reduction
H.8.1       E10H. Matrix Rigidity Reduction for Strong BAOC
E10H.0. Statement and Role Lemma E10H is a reduction: it localizes the structural issue
left by E10G to CRT/AFF matrix-origin rigidity. The resulting reduction is closed by the finite
GoodAWACK grammar Lemma E10X, whose proof uses E10I, E10J, E10Y, E10M, and E10K.
    E10H treats the next block after Lemma E10G.
    The target isolated there was:


E10G-Rigidity: enumerate the actual affine-regrouping/CRT matrices allowed by B1/B3/F3/F4.

    The purpose of E10H is to isolate the precise rigidity statement needed from the source lemmas.
    The outcome is a sharper reduction:

                                                                     300
All non-matrix transport operations are origin-safe; the remaining target is CRT/AFF matrix-origin rigidity.

   More precisely, the source lemmas prove enough to control:

  1. fixing/projection;

  2. fixed divisor quotient;

  3. variable quotient residuals, provided quotient-origin equations are retained;

  4. primitive slicing as an analytic fibre operation;

  5. Cauchy/cube shifts as linear-part preserving operations.

   The part isolated for the matrix-origin step is the enumeration of:

  1. the basis matrices used in controlled CRT sublattices;

  2. the bounded affine regrouping matrices admitted by E5;

  3. the full active vector list after these matrices are composed.

   Thus E10H reduces E10G-Rigidity to a concrete matrix-origin lemma stated in Section E10H.7,
which is discharged by E10X.
   —

E10H.1. Rigidity statement           Let S be a terminal GoodAWACK skeleton from

                                          𝐵1 → 𝐵3 → 𝐹 3/𝐹 4.

   Write the active affine forms as

                                 𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 ,       𝑄𝜌 = ℓ𝜌 ⊙ ℓ𝜌 .

   The E10G-Rigidity statement isolated by this reduction is:

Lemma H.20 (Lemma E10H.Rig). For every actual terminal GoodAWACK skeleton S, the
coefficient vectors ℓ𝜌 lie in an explicitly enumerated finite-parametric matrix family

                                                ℛ(ℬ, Γ, r),

   where every matrix in the family carries one of the following origin tags:

  1. coordinate projection/fixing;

  2. CRT sublattice basis tied to a controlled congruence;

  3. fixed divisor quotient;

  4. variable quotient residual;

  5. B3 grouping incidence;

                                                    301
  6. primitive slicing/fibre selection;
  7. Cauchy/cube shift.
   Moreover, every HighTC tensor relation
                                              𝑐𝜌 𝑄𝜌 = 0,           𝑐𝑚 ̸= 0
                                       ∑︁

                                          𝜌

   for a marked 𝑚 ∈ ℳ is origin-degenerate or impossible.
   If this lemma holds, then E10G and HGO2R imply:
                                       𝑅FreeAffineHighTC (𝑁 ) = 0.
   —

E10H.2. Safe transport operations We first separate the transport operations that are already
safe.

S1. Fixing and projection         Projection deletes fixed coordinates:
                                          ℓ = (ℓfree , ℓfixed ) ↦→ ℓfree .
    This operation cannot introduce a new untagged origin. If it creates equality, proportionality, or
forced collision between surviving forms, F3 routes the atom to LocalDiag. Otherwise it merely
lowers ambient rank.
    Projection may turn a previously TC1 system into HighTC by collapsing coordinates. But then
the collapse itself is recorded as a fixing/projection origin. If the resulting relation is caused by the
collapse, it is not FreeAffine; if it is not caused by the collapse, it must already be present in the
pre-projection system.
    Thus projection is origin-safe.

S2. Fixed divisor quotient        For a fixed divisor condition
                                                    𝑑 | 𝐿(𝑧),
   F4/E5 replace the lattice by
                                  Λ𝑑 = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 0 (mod 𝑑)}
   and the quotient form by
                                                   𝐿𝑑 = 𝐿/𝑑.
   At the vector level:
                                                          1 𝑡
                                                   ℓ ↦→     𝑇 ℓ,
                                                          𝑑
   and F4.4/E5.2 prove
                                                    contΛ (𝐿)
                              contΛ𝑑 (𝐿/𝑑) =                     ≤ contΛ (𝐿).
                                                  (contΛ (𝐿), 𝑑)
    Any HighTC relation whose support uses the fixed quotient in a way that determines one active
form from another is origin-degenerate by HGO2R, cases D1/D2. If it does not use the quotient
origin, the fixed quotient is merely a controlled vector transport.
    Thus fixed divisor quotient is safe except for the CRT matrix 𝑇 , which is separated below.

                                                       302
S3. Variable quotient residual        For

                                              𝐿(𝑧) = 𝑑𝑠,

   F4 gives the exhaustive alternative:

                             Edge ⊔ LocalDiag ⊔ CKP ⊔ GoodAWACK.

   If the quotient reaches GoodAWACK, the quotient vector satisfies an explicit origin equation

                                             𝑑ℓ𝑠 = 𝑇 𝑡 ℓ𝐿 .                               (VQ-origin)
   If a HighTC certificate uses this equation to force dependence, then it is origin-degenerate:

  1. short quotient/divisor gives Edge;

  2. forced determination gives LocalDiag;

  3. balanced multiplicative quotient gives CKP;

  4. incompatibility gives Impossible.

    Therefore variable quotient residuals are safe, again modulo the same CRT/basis matrix-origin
target 𝑇 .

S4. Primitive slicing      Primitive slicing writes a marked form on a one-dimensional fibre as

                                         𝐿(𝑧0 + 𝑢𝑣) = 𝑔𝑢 + 𝑏.

    For TC1/HighTC classification, the relevant object remains the pre-slicing vector ℓ. The fibre
coefficient 𝑔 is used by E7/E9 analytically. Primitive slicing therefore does not create a new HighTC
tensor relation among the pre-slicing vectors.
    Thus primitive slicing is not the rigidity obstruction.

S5. Cauchy/cube shifts        Cauchy/cube operations introduce shifts:

                                     𝐿(𝑧 + 𝜔ℎ) = 𝐿(𝑧) + ℓ(𝜔ℎ).

   The linear part in 𝑧 remains ℓ. If cube operations create equality, proportionality, or forced local
dependence, E5/F3 route to LocalDiag. Otherwise at least one marked controlled-content form
survives.
   Hence Cauchy/cube operations are linear-part safe.
   —

E10H.3. The remaining matrix operations The only remaining operations capable of creating
new free affine tensor patterns are:




                                                 303
M1. Controlled CRT basis choice             F3 controlled CRT absorption imposes

                                𝐿(𝑧) ≡ 𝑎 (mod 𝑞),             𝑞 ≤ (log 𝑁 )𝐶 ,

   and replaces the lattice coset by

                                  Λ′ = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 𝑎 (mod 𝑞)}.

   To express Λ′ in free coordinates, one chooses a basis

                                             𝑧 = 𝑧0 + 𝑇 𝑧 ′ .

   Vectors transform by

                                                ℓ ↦→ 𝑇 𝑡 ℓ.

   The F3/E5 statements control the index and content growth. They do not by themselves classify
the possible matrices 𝑇 , nor do they constrain the resulting rows beyond polylogarithmic content/
minor bounds.

M2. Bounded affine regrouping            E5 permits an integer affine map
                                                      ′
                                              𝑇 : Z𝑟 → Z𝑟

   with coefficients and relevant minors bounded by powers of log 𝑁 , and records only:

                                   cont(𝐿 ∘ 𝑇 ) ≤ (log 𝑁 )𝐶 cont(𝐿).

   If 𝑇 is unimodular, content is preserved exactly.
   This proves content stability, but not rigidity. It does not say that 𝑇 must be a coordinate
projection, signed permutation, diagonal quotient, incidence matrix, or any other finite rigid matrix
family.
   Thus M1/M2 are exactly where E10G-Rigidity requires the E10X actual-origin classification.
   —

E10H.4. Interface Example: Formal 4AP-like Witness Before imposing the actual-
descendant constraint supplied by E10X, the broad M1/M2 interface permits, at the coefficient-
vector level, the following formal situation.
    Take four source coordinate forms 𝑌0 , 𝑌1 , 𝑌2 , 𝑌3 on Z4 , and restrict/regroup to a two-dimensional
affine sublattice

                                       𝑌𝑖 = 𝑥 + 𝑖𝑟,       0 ≤ 𝑖 ≤ 3.

   Equivalently, use the matrix

                                                  1       0
                                                  ⎛        ⎞
                                                ⎜1        1⎟
                                             𝑇 =⎜           ⎟,
                                                ⎜           ⎟
                                                ⎝1        2⎠
                                                  1       3

   so that

                                                   304
                                             𝑇 𝑡 𝑒𝑖 = (1, 𝑖).
   The transported vectors are
                                      ℓ𝑖 = (1, 𝑖),         0 ≤ 𝑖 ≤ 3.
   They have bounded coefficients and content 1. Their quadratic tensors satisfy

                                     𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.                                   (4AP-Q)
   No equality or proportionality among the 𝐿𝑖 = 𝑥 + 𝑖𝑟 is forced. The domain can be central-long:
                        𝑥 ≍ 𝑋,      𝑟 ≍ 𝑅,       𝑥 + 3𝑟 ≍ 𝑋,            𝑋, 𝑅 ≥ 𝑁 𝜃 .
   This is a FreeAffineHighTC pattern unless the routing history records an origin that makes it:
  1. CKP;
  2. H4-canonical LocalDiag;
  3. strict C1P Edge;
  4. impossible.
     The broad M1/M2 matrix hypotheses alone do not decide whether a matrix of this kind is an
actual terminal descendant. Therefore this formal pattern is not used as a terminal GoodAWACK
cell. Its role is to show why the matrix-origin reduction in E10H must be paired with the finite-
grammar theorem E10X.
     E10X supplies that actual-origin information. If such a 4AP-like rank-dropping transport is
produced by actual B1/B3/F3/F4 routing, then its rank drop is tagged and the cell leaves clean
terminal GoodAWACK by the CKP, LocalDiag, Edge, Impossible, or post-terminal analytic route.
If it is untagged, then it is not an admissible actual descendant under the F3-complete routing
interface.
     —

E10H.5. Why H4 admission is not automatic                    One might try to use the linear identity
                                     𝐿0 − 3𝐿1 + 3𝐿2 − 𝐿3 = 0
   to label the 4AP pattern LocalDiag.
   This is not justified by Lemma H4.
   H4 admits a local/main term only if it equals the tagged canonical local projection
                                             Loc𝑄 𝑅ℬ,𝜏 (𝑁 )
    up to 𝑜(𝑁 ).
    A bare affine identity among oscillatory forms does not prove such an equality. In particular, a
Liouville-weighted four-term affine pattern is a nonlocal oscillatory configuration, not automatically
a local density projection modulo 𝑄.
    Therefore the H4 route would require the additional theorem:

        every B3/F3 affine-dependence flag surviving from actual routing is H4-canonical.
   The proof does not use this automatic-H4 implication. Instead, it uses E10YMX to exclude the
untagged actual terminal occurrence before E10L assembles the clean GoodAWACK estimate.
   —

                                                     305
E10H.6. Rigidity reduction theorem
Lemma H.21 (Lemma E10H.1. Reduction to matrix-origin rigidity). For terminal GoodAWACK
skeletons in the proof tree, every HighTC certificate is origin-degenerate or impossible unless it is
supported entirely after applying M1/M2 matrix transports in a way that does not use fixed-divisor,
variable-quotient, repeated/proportional, CKP-balanced, C1 Edge, or parent-incompatibility origins.
    Equivalently:

             FreeAffineHighTC ⊆ HighTC produced by CRT/AFF matrix transport.

Proof. By E10G, every terminal GoodAWACK vector is produced by the catalogue cells C0–C6.
    Cells corresponding to fixing/projection, fixed divisor quotient, variable quotient residual,
primitive slicing, and Cauchy/cube shifts are S1–S5 above. In each case, either the operation
preserves the relevant linear parts, deletes coordinates with recorded origin, or carries an explicit
quotient/local origin.
    If a HighTC relation uses any of these origins in an essential way, then HGO2R routes it to
CKP, LPI/H4M-admissible LocalDiag, strict C1P Edge, or Impossible. If it does not use those
origins, then those operations are irrelevant to its free-affine character.
    The only operations left capable of producing a new untagged affine tensor relation are M1
controlled CRT basis choice and M2 bounded affine regrouping. Therefore every remaining FreeAffine-
HighTC certificate must come from the matrix-origin part. Lemma proved.
   —


E10H.7. Matrix-origin closure lemma The structural conclusion is stated directly at the
matrix level.
Lemma H.22 (Lemma E10H.2. CRT/AFF matrix-origin rigidity). Let 𝑇tot be the total coeffi-
cient transport matrix obtained by composing all controlled CRT basis choices and bounded affine
regroupings in an actual terminal GoodAWACK routing history.
    Then one of the following holds:

  1. 𝑇tot belongs to an explicitly enumerated rigid family whose tensor row-reduction has no
     FreeAffineHighTC relation;

  2. any HighTC relation created by 𝑇tot is tagged by a quotient/gcd/divisor/local origin and is
     origin-degenerate;

  3. the corresponding tagged atom is H4-canonical LocalDiag;

  4. the routing cell is empty or violates B1/B3/F3/F4 admissibility.

   E10X supplies this conclusion for actual terminal GoodAWACK descendants. With that input,
E10H.2 gives:

                                     𝑅FreeAffineHighTC (𝑁 ) = 0.

    The formal 4AP-like cell of E10H.4 is not an obstruction to E10H.2: it is not a constructed
actual B1 descendant. E10X proves that an actual untagged occurrence of that type is impossible in
the routing tree.
    —

                                                306
                              Remark H.23 (E10H.8. Output).
              E10H reduces FreeAffineHighTC to CRT/AFF matrix-origin rigidity.
    What is proved here:

              FreeAffineHighTC is reduced to the CRT/AFF matrix-origin rigidity problem.

    Structural closure:

                          The remaining rank-dropping AFF issue is discharged by E10X.

    Completion block:

            MOR: Matrix-Origin Rigidity for controlled CRT and bounded affine regrouping.

    This block is supplied by E10X, using E10I, E10J, E10Y, E10M, and E10K.

H.9       E10I matrix-origin rigidity
H.9.1        E10I. Matrix-Origin Rigidity Verification
E10I.0. Statement and Role Lemma E10I supplies the MOR reduction to untagged rank-
dropping AFF. That final class is discharged by the finite GoodAWACK grammar Lemma E10X.
   Therefore the class isolated by E10I is not a hidden gap; it is the input passed to the E10J–
E10Y–E10M–E10K closure packaged by E10X.
   E10I continues the MOR block isolated in Lemma E10H.
   The target was:

        MOR: prove matrix-origin rigidity for controlled CRT and bounded affine regrouping.

    The outcome is a further reduction.

controlled CRT and full-rank affine coordinate changes are tensor-safe; E10I reduces the matrix target to untagged rank-dropping affine slicing/regrouping.


    At the stage of E10I alone, the proof has reduced rather than proved

                                                         𝑅FreeAffineHighTC (𝑁 ) = 0,

   because the remaining matrix target is the following narrower statement, which is discharged by
E10X:

        FreeAffineHighTC ⊆ HighTC created by rank-dropping AFF transport without origin.

    —




                                                                          307
E10I.1. Linear algebra fact: tensor tests are invariant under rational isomorphism Let
𝑉, 𝑊 be finite-dimensional rational vector spaces and let
                                            𝑆 : 𝑉 * → 𝑊*
   be an injective linear map. It induces
                     Sym2 (𝑆) : Sym2 (𝑉 * ) → Sym2 (𝑊 * ),     ℓ ⊙ ℓ ↦→ 𝑆ℓ ⊙ 𝑆ℓ.
   If 𝑆 is injective, then Sym2 (𝑆) is injective.
   Consequently, for any finite family {ℓ𝜌 } ⊂ 𝑉 * and any marked index 𝑚,
                                 ℓ𝑚 ⊙ ℓ𝑚 ∈ spanQ {ℓ𝜌 ⊙ ℓ𝜌 : 𝜌 ̸= 𝑚}
   if and only if
                              𝑆ℓ𝑚 ⊙ 𝑆ℓ𝑚 ∈ spanQ {𝑆ℓ𝜌 ⊙ 𝑆ℓ𝜌 : 𝜌 ̸= 𝑚}.
Proof. Choose bases. An injective linear map 𝑆 has a left inverse over Q on its image. Hence the
induced map on symmetric tensors also has a left inverse on its image, so Sym2 (𝑆) is injective.
    Applying Sym2 (𝑆) to a rational linear relation among the tensors preserves the relation. Con-
versely, if a relation holds after applying Sym2 (𝑆), injectivity implies the same relation held before
applying it.
    This proves the equivalence.
   —


E10I.2. Controlled CRT is tensor-safe In F3, a controlled CRT restriction replaces a lattice
coset by
                        Λ′ = {𝑧 ∈ Λ : 𝐿(𝑧) ≡ 𝑎 (mod 𝑞)},        𝑞 ≤ (log 𝑁 )𝐶 .
    If nonempty, Λ′ is a finite-index subcoset of Λ. Its difference lattice has the same rank as the
original difference lattice.
    Choosing coordinates on Λ′ gives
                                            𝑧 = 𝑧0 + 𝑇 𝑧 ′ ,
    where 𝑇 is a full-rank square matrix over Q after choosing bases of the original and restricted
difference lattices. Homogeneous vectors transform by
                                              ℓ ↦→ 𝑇 𝑡 ℓ.
   Since 𝑇 𝑡 is injective over Q, Section E10I.1 shows that the TC1/HighTC tensor test is invariant
under this coordinate choice.
Lemma H.24 (Lemma E10I.1. CRT basis choice does not create FreeAffineHighTC). Controlled
CRT basis choice cannot turn a TC1 family into a FreeAffineHighTC family, nor can it create a
new untagged tensor relation. Any HighTC relation after CRT was already present before CRT,
transported by an injective symmetric-square map.
Proof. Immediate from E10I.1 and the full-rank finite-index nature of controlled CRT restrictions.
   Thus the matrix-origin residual is not ordinary CRT basis choice.
  —


                                                 308
E10I.3. Full-rank affine regrouping is tensor-safe                   E5 permits bounded affine regrouping:

                                            𝑧 = 𝑧0 + 𝐴𝑧 ′ .

   If this regrouping is a full-rank coordinate change between equal-rank active parameter lattices,
then 𝐴 is invertible over Q. Homogeneous vectors transform by

                                               ℓ ↦→ 𝐴𝑡 ℓ.

   Again 𝐴𝑡 is injective, so the tensor test is invariant.
Lemma H.25 (Lemma E10I.2. Full-rank AFF maps are not the obstruction). Any bounded
affine regrouping whose linear part is full-rank on the active affine span preserves the TC1/HighTC
classification.
    In particular, unimodular changes, finite-index basis changes, signed permutations, and diagonal
controlled quotient coordinate changes cannot create a new FreeAffineHighTC certificate.
Proof. Apply E10I.1 to 𝑆 = 𝐴𝑡 .
  —


E10I.4. Rank-dropping AFF maps are the only remaining matrix danger The tensor
test is not invariant under rank-dropping maps.
    If
                                                      ′
                                            𝐴 : Q𝑘 → Q𝑘

   has rank 𝑘 ′ < 𝑘, then
                                                                 ′
                                         𝐴𝑡 : (Q𝑘 )* → (Q𝑘 )*

   need not be injective. Distinct quadratic tensors in Sym2 ((Q𝑘 )* ) may collapse to dependent
tensors after restriction to the lower-dimensional slice.
   This is exactly how a 4AP-like pattern appears.
   Take source coordinate forms

                                             𝑌0 , 𝑌1 , 𝑌2 , 𝑌3

   on Q4 , and restrict to the two-dimensional slice

                                     𝑌𝑖 = 𝑥 + 𝑖𝑟,          0 ≤ 𝑖 ≤ 3.

   The resulting vectors are

                                      ℓ𝑖 = (1, 𝑖),         0 ≤ 𝑖 ≤ 3,

   and

                                     𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.

   This tensor relation is not produced by an invertible coordinate change. It is produced by a
rank-dropping affine slice.
   Thus any structural MOR proof must control rank-dropping AFF maps.
   —

                                                     309
E10I.5. What B1/B3/F3/F4/E5 contribute to rank-dropping maps The source lemmas
do not introduce a free list of rank-dropping affine maps. They provide product data, routing
operations, and stability transports whose actual rank-dropping occurrences are classified by the
E10Y/E10M grammar and then packaged by E10X.

B1    Lemma B1 gives product variables and dyadic cells, not affine matrix parametrizations.

B3    Lemma B3 gives a finite set of product groupings:

                                      𝑢𝐼 =                      𝑣𝐼 =
                                             ∏︁                        ∏︁
                                                   𝑥𝑖 ,                      𝑥𝑖 ,
                                             𝑖∈𝐼                       𝑖∈𝐼
                                                                        /

     and preliminary labels. It does not give a matrix list for later affine parameterizations.

F3 Lemma F3 permits controlled CRT absorption and terminal routing. CRT is full-rank and
tensor-safe by E10I.1. F3 does not list affine slices of the form

                                               𝑌𝑖 = 𝑥 + 𝑖𝑟.

F4 Lemma F4 handles fixed-divisor and variable-quotient origins. These are origin-tagged and
already routed by HGO2R when they cause HighTC. F4 does not enumerate untagged rank-dropping
AFF maps.

E5    Lemma E5 is the stability source that explicitly permits a broad affine map:
                                                           ′
                                               𝑇 : Z𝑟 → Z𝑟

     with coefficients and relevant minors bounded by powers of log 𝑁 .
     E5 proves content stability:

                                   cont(𝐿 ∘ 𝑇 ) ≤ (log 𝑁 )𝐶 cont(𝐿).

    It does not say that 𝑇 is full-rank on the active affine span, nor that every rank-dropping 𝑇
carries a quotient/local/CKP/Edge origin.
    Therefore the E5 content-stability statement is not the MOR closure mechanism. The proof
uses E10X to classify the actual rank-dropping occurrences created by the B1/B3/F3/F4 routing
tree and reads E5 only as a stability lemma for those authorized transports.
    —

E10I.6. MOR reduction theorem

Lemma H.26 (Lemma E10I.3. Reduction to rank-dropping AFF origin). For actual terminal
GoodAWACK skeletons supported by the proof tree,

                                          FreeAffineHighTC

    can only arise from a rank-dropping bounded affine regrouping/slicing map whose rank drop is
not already recorded as:


                                                          310
  1. fixing/projection;

  2. controlled CRT finite-index restriction;

  3. fixed-divisor quotient;

  4. variable quotient residual;

  5. forced LocalDiag;

  6. CKP-balanced grouping;

  7. strict C1P Edge;

  8. parent incompatibility.

Proof. By E10H, all non-matrix operations are origin-safe.
    By E10I.2, controlled CRT basis choices are full-rank finite-index coordinate changes and cannot
create new tensor dependence.
    By E10I.3, full-rank affine regroupings are tensor-safe.
    Thus any remaining FreeAffineHighTC certificate must be created by the only matrix operation
not covered by these safe cases: a rank-dropping AFF map. If the rank drop is tagged by one of the
origins 1–8, then HGO2R reroutes it. Therefore the surviving case is precisely an untagged rank-
dropping AFF origin. Lemma proved.
   —


E10I.7. Rank-drop closure lemma             The remaining structural input is a rank-drop origin
lemma.
Lemma H.27 (Lemma E10I.4. No untagged rank-dropping AFF in terminal GoodAWACK). Let
S be an actual terminal GoodAWACK skeleton. Every rank-dropping affine map used to produce its
active affine system is one of:

  1. a recorded fixing/projection already covered by the skeleton origin map;

  2. a quotient/divisor/gcd-origin map covered by F4;

  3. a local/collision map routed to H4-canonical LocalDiag;

  4. a CKP-balanced grouping;

  5. a strict C1P Edge configuration;

  6. an impossible/empty cell.

   Equivalently, no untagged rank-dropping AFF map may survive into terminal GoodAWACK.
   If E10I.4 is proved, then

                                     𝑅FreeAffineHighTC (𝑁 ) = 0.

    E10X supplies this lemma for actual terminal GoodAWACK skeletons through the E10Y/E10M
finite-grammar classification.
    —

                                                311
                             Remark H.28 (E10I.8. Output).
              MOR is partially proved: CRT and full-rank AFF are tensor-safe.
   Completion theorem:

                No untagged rank-dropping AFF map in terminal GoodAWACK.

   Structural closure:

                                   This task is discharged by E10X.

H.10     E10J rank-dropping AFF origin verification
H.10.1    E10J. Rank-Dropping AFF Origin Verification
E10J.0. Statement and Role Lemma E10J proves that tagged rank drops are origin-degenerate
or already routed, and reduces the remaining case to the affine-origin completeness theorem packaged
by E10X and proved through the E10Y/E10M/E10K finite-grammar chain for actual terminal
GoodAWACK skeletons.
    E10J treats the next block isolated in Lemma E10I.
    The target was:

         RDA: no untagged rank-dropping AFF map survives into terminal GoodAWACK.

   The reduction proved in this file is:

                RDA reduces to an affine-regrouping origin-completeness lemma.

   What is proved:

                  every tagged rank drop is already routed or origin-degenerate.

   Completion theorem:

     exclude or classify rank drops allowed only by the broad E5 affine-regrouping interface.

   —

E10J.1. RDA statement           Let

                               S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )

   be an actual terminal GoodAWACK skeleton.
   A rank-dropping AFF map is a bounded affine transport

                                   𝑧 = 𝑧0 + 𝐴𝑧 ′ ,     rank 𝐴 < dim 𝑧,

   used to produce or represent the active affine system.
   It is tagged if its rank drop is recorded as one of:

  1. fixing/projection of inactive coordinates;

  2. fixed divisor quotient;

                                                     312
  3. variable quotient residual;

  4. controlled local/gcd dependence;

  5. CKP-balanced grouping;

  6. strict C1P Edge;

  7. impossible/empty support;

  8. primitive slicing used only analytically, while the pre-slicing vectors remain the tensor-
     verification objects.

   It is untagged if none of these origins is recorded.
   RDA asks to prove:

          no untagged rank-dropping AFF map occurs in terminal GoodAWACK.                     (RDA)
   By E10I, RDA would imply:

                                      𝑅FreeAffineHighTC (𝑁 ) = 0.

   —

E10J.2. Tagged rank drops are safe

Lemma H.29 (Lemma E10J.1. Tagged rank-dropping AFF is origin-degenerate or irrelevant). If a
rank-dropping AFF map in a terminal GoodAWACK skeleton is tagged by one of the origins listed
in E10J.1, then it cannot support a FreeAffineHighTC certificate.

Proof. We inspect the tagged cases.
    If the rank drop is fixing/projection, then the collapse is recorded in the origin map. Any
HighTC relation caused by the collapse is not free-affine; if it is not caused by the collapse, it was
already present before projection.
    If the rank drop comes from fixed divisor quotient or variable quotient residual, then F4 supplies
the quotient/divisor origin. By HGO2R, any HighTC certificate using that origin is LocalDiag,
CKP, Edge, or Impossible.
    If the rank drop is controlled local/gcd dependence, then it is precisely a LocalDiag-origin case,
admitted only when H4-canonical.
    If it is CKP-balanced, the atom is CKP and handled by G8a.
    If it is strict C1P Edge or empty support, it contributes 𝑜(𝑁 ) or zero.
    If it is primitive slicing, then by E10H and E10I the pre-slicing vectors remain the objects used
in the TC1/HighTC tensor verification; the one-dimensional fibre is an analytic E7/E9 object, not a
new terminal HighTC coefficient family.
    Thus no tagged rank drop produces FreeAffineHighTC. Lemma proved.
   —


E10J.3. Source classification for untagged rank drops We classify what the source lemmas
prove about untagged rank-dropping AFF before the finite-grammar theorem E10X is applied.


                                                 313
B1 Lemma B1 gives typed Heath–Brown product variables and dyadic cells. It does not introduce
affine matrix maps or rank-dropping affine slices.
    Thus B1 does not create an untagged rank-dropping AFF occurrence.

B3    Lemma B3 gives a finite product grouping set:
                                     𝑢𝐼 =                       𝑣𝐼 =
                                            ∏︁                         ∏︁
                                                  𝑥𝑖 ,                       𝑥𝑖 ,
                                            𝑖∈𝐼                        𝑖∈𝐼
                                                                        /

    and preliminary labels. It also says that forced equality, proportionality, repeated factor, fixed
gcd-local dependence, or affine dependence may produce a LocalDiag flag.
    But the corrected H4 interface does not accept a bare affine dependence as LocalDiag unless it is
a tagged canonical local projection. Therefore B3 is not by itself the closure theorem for untagged
rank-dropping AFF. Its role is to provide finite grouping data and tags used by E10X.

F3 Lemma F3 performs controlled CRT absorption, F4 large-divisor decision, square-divisor
routing, finite grouping selection/elimination, LocalDiag detection, Edge detection, and terminal
class labelling.
    Controlled CRT is full-rank and tensor-safe by E10I.
    F3 does not enumerate rank-dropping affine regrouping maps. Its terminal GoodAWACK
predicate is negative:
  1. central-long affine WACLE;
  2. bounded affine complexity;
  3. no forced LocalDiag;
  4. no unresolved ordinary large divisor;
  5. marked Liouville-type affine form;
  6. long active directions.
   This negative predicate must be read together with the complete F3.6 operation list. F3
labels terminal GoodAWACK descendants; it does not license a new untagged rank-dropping affine
parametrization.

F4 Lemma F4 handles ordinary divisor and quotient origins. These are tagged rank drops and are
safe by E10J.2.
    F4 handles the tagged divisor, quotient, gcd, balanced CKP, and strict C1P origins. Any rank
drop not tied to one of these origins is passed to the E10X finite-grammar closure rather than
treated as an admissible terminal generator.

BGS      Lemma BGS records
                                                         rgrp
     as affine regrouping or affine changes of variables, and includes an origin map
                                                    origS .
    This is enough to state RDA. The proof that every actual rank drop in rgrp is one of the tagged
origins in E10J.1 is supplied by E10X through the E10Y grammar theorem and E10M.

                                                         314
E5    Lemma E5 permits an affine map
                                                     ′
                                             𝑇 : Z𝑟 → Z𝑟

     with coefficients and relevant minors bounded by powers of log 𝑁 , and proves only:

                                   cont(𝐿 ∘ 𝑇 ) ≤ (log 𝑁 )𝐶 cont(𝐿).

     It does not state:

  1. 𝑇 must be full-rank on the active affine span;

  2. every rank drop of 𝑇 must be fixing/projection;

  3. every rank drop of 𝑇 must have quotient/divisor/gcd/CKP/Edge/LocalDiag origin;

  4. every affine dependence created by 𝑇 is H4-canonical LocalDiag.

   Therefore E5 is a stability lemma, not the closure theorem for RDA. The proof reads E5 in the
E10X-clean sense: it may transport already authorized full-rank or tagged data, but it is not an
additional terminal generator of untagged rank-dropping AFF.
   —

E10J.4. Formal rank-dropping AFF witness At the broad E5/BGS interface, before applying
E10X’s actual-descendant restriction, one can write the following formal rank-dropping AFF cell.
   Start with four independent source coordinate forms

                                             𝑌0 , 𝑌1 , 𝑌2 , 𝑌3 .

     Apply the rank-dropping affine parametrization

                                     𝑌𝑖 = 𝑥 + 𝑖𝑟,          0 ≤ 𝑖 ≤ 3.

     Equivalently, the transport matrix is

                                                 1        0
                                                   ⎛        ⎞
                                               ⎜1         1⎟
                                             𝐴=⎜            ⎟.
                                               ⎜            ⎟
                                               ⎝1         2⎠
                                                 1        3

     The resulting homogeneous vectors are

                                               ℓ𝑖 = (1, 𝑖).

     They satisfy the HighTC tensor relation

                                     𝑄0 − 3𝑄1 + 3𝑄2 − 𝑄3 = 0.

   All contents are 1, all coefficients are 𝑂(1), and the affine complexity is bounded. A central-long
domain can be chosen:

                          𝑥 ≍ 𝑋,    𝑟 ≍ 𝑅,        𝑥 + 3𝑟 ≍ 𝑋,           𝑋, 𝑅 ≥ 𝑁 𝜃 .

                                                   315
    At the level of the broad terminal GoodAWACK wording alone, this formal cell is not automati-
cally:

  1. Edge;

  2. CKP;

  3. H4-canonical LocalDiag;

  4. impossible.

    Thus broad interface language alone does not classify this formal rank-dropping AFF cell.
    This is not claimed to be an actual B1 descendant. It is an interface example showing why RDA
is proved through E10YMX rather than through the broad E5/BGS wording alone. If this pattern
is generated by actual B1/B3/F3/F4 routing, then E10X classifies its rank drop and routes it away
from clean terminal GoodAWACK; if it remains untagged, it is not an admissible actual terminal
skeleton.
    —

E10J.5. RDA reduction theorem

Lemma H.30 (Lemma E10J.2. RDA reduces to affine-regrouping origin completeness). Assume
the following origin-completeness statement:


    Every rank-dropping affine regrouping recorded in rgrp is tagged by one of E10J.1(1)–(8).
                                                                                       (AFF-OC)
   Then RDA holds.

Proof. Let S be a terminal GoodAWACK skeleton. By E10I, any FreeAffineHighTC certificate
must arise from a rank-dropping AFF map.
   By AFF-OC, every such rank drop is tagged by one of the origins in E10J.1.
   By Lemma E10J.1, tagged rank drops are origin-degenerate or irrelevant for FreeAffineHighTC.
   Therefore no FreeAffineHighTC certificate remains. This proves RDA.
  —


E10J.6. AFF-origin completeness closure lemma            The closure block is an origin-completeness
upgrade for affine regrouping.

Lemma H.31 (Lemma E10J.3. AFF-origin completeness). In every actual B1/B3/F3/F4 terminal
GoodAWACK routing history, every affine regrouping or affine change of variables recorded in

                                                  rgrp

   has linear part 𝐴 satisfying exactly one of:

  1. 𝐴 is full-rank on the active affine span;

  2. the rank drop is a recorded fixing/projection;


                                                  316
   3. the rank drop is induced by fixed divisor or quotient-origin data;
   4. the rank drop is induced by a forced local/gcd/collision relation and is H4-canonical LocalDiag;
   5. the rank drop exposes CKP-balanced grouping;
   6. the rank drop gives strict C1P Edge;
   7. the cell is impossible.
    If E10J.3 is proved, then:
                                                 𝑅FreeAffineHighTC (𝑁 ) = 0.
    E10YMX supplies E10J.3 for the actual terminal GoodAWACK cells.
    —
                             Remark H.32 (E10J.7. Output).
          RDA is reduced to AFF-origin completeness, which is supplied by E10YMX.
    What is proved:
           RDA follows from AFF-origin completeness, and all tagged rank drops are safe.
    Structural closure:
            AFF-origin completeness is discharged by E10X and assembled in E10K/E10L.

    Completion block:
                                 AFF-OC: Affine-regrouping origin completeness.

    This block is supplied by E10X and assembled in E10K/E10L.

H.11       E10Y GoodAWACK routing grammar completeness
H.11.1       E10Y. Completeness of the GoodAWACK Routing Grammar
E10Y.0. Statement and Role Lemma E10Y is a structural completeness theorem for the
GoodAWACK routing grammar. It proves that every operation which can generate or modify an
actual terminal GoodAWACK affine skeleton is already represented in the finite B1/B3/F3/F4
routing grammar, with E5 used only for controlled content transport.
   The lemma concerns only actual B1/B3/F3/F4/E5-generated descendants. It does
not classify arbitrary bounded affine systems and it does not assert that every formal affine
parametrization is reachable. Its assertion is:


every skeleton-generating pre-terminal operation in an actual B1-origin GoodAWACK descendant is one of the operations certified below.
                                                                                            (E10Y)
    Consequently, an unlisted rank-dropping affine regrouping cannot enter a terminal GoodAWACK
skeleton as a hidden operation. Post-terminal analytic operations may estimate a fixed terminal
object, but they do not generate a new terminal GoodAWACK skeleton.
    Logical dependencies are B1, B3, F3, F3A, F3T, F4, and the content-stability calculations of E5.
E10Y is used by E10M, E10X, E10K, E10YMX, and E10L. Non-logical reproducibility records may
be maintained separately for the same finite operation list, but they are not inputs to E10Y.
    —

                                                                317
E10Y.1. Setup

Actual routing record        An actual routing record is a tuple
                                       r = (𝑉, 𝒞, ℒ, 𝒬, 𝜏, orig, 𝑊 )
   where:
   1. 𝑉 is the finite list of active variables inherited from B1;
   2. 𝒞 records dyadic, congruence, content, gcd and divisor restrictions;
   3. ℒ is the finite list of affine forms visible on the current cell;
   4. 𝒬 records fixed-divisor, quotient and local tags;
   5. 𝜏 is the current routing tag;
   6. orig records the origin of every rank-changing operation;
   7. 𝑊 is the bounded or polylogarithmic weight data transported with the cell.
   An actual terminal GoodAWACK skeleton is the terminal value of such a routing record along a
descendant of
                                        𝐵1 −→ 𝐵3 −→ 𝐹 3/𝐹 4.

Pre-terminal operation A pre-terminal operation is a transformation of an actual routing
record before terminal class labelling. It is actual-generated if it is invoked by the B1, B3, F3
or F4 routing construction, or by the E5 content-stability calculation applied to a record already
produced by those routing layers. This definition is external to the E10Y grammar: it refers to
the construction of descendants in B1/B3/F3/F4/E5, not to the list of E10Y transition classes.
Thus "actual-generated" means "lying in the image of the independently defined B1/B3/F3/F4/E5
construction"; it does not mean "allowed because E10Y allows it."
Lemma H.33 (Lemma E10Y.0. Source-to-record extraction). Every Branch B descendant that is
actually produced by B1/B3/F3/F4/E5 and then fed to the GoodAWACK terminal class carries a
finite routing record
                                       r = (𝑉, 𝒞, ℒ, 𝒬, 𝜏, orig, 𝑊 )
of the kind defined above. Each rank-relevant operation occurrence in that descendant is represented
either by a transformation of this record before terminal labelling or by a post-terminal analytic
operation after terminal labelling.
Proof. B1 fixes a finite-depth Heath–Brown product block, its variables, its dyadic cell and its
coefficient data. B3 replaces this by one of finitely many grouped product records. F3 and F4 act
only through their recorded routing decisions: congruence restrictions, divisor or quotient choices,
local/gcd relations, Edge or CKP routing, continuation tags, and terminal class labels. E5 is applied
only to a record already carrying those variables, constraints and origin tags. Hence every actual
Branch B descendant has a finite record of the displayed form. Any later TC1/HighTC, coarea,
Fourier, BRS/X16, Davenport/AP, Cauchy–Schwarz or local-projection step is performed only
after the terminal routing record has already been fixed, and is therefore recorded as post-terminal
analytic use rather than as a new pre-terminal operation.
    This proves the extraction claim.


                                                   318
Skeleton-generating operation A pre-terminal operation is skeleton-generating if it changes at
least one of

                                 𝑉,       𝒞,      ℒ,        𝒬,      orig,

   in a way that can affect the terminal GoodAWACK affine skeleton.

Post-terminal analytic non-generator A post-terminal analytic non-generator is an operation
performed after a terminal routing object has been fixed. Such an operation may form test functions,
apply Cauchy–Schwarz, take Fourier transforms, slice a fixed testing family, or estimate an auxiliary
sum. It does not create a new B1/B3/F3/F4 descendant and it does not add a new terminal
GoodAWACK skeleton.

Terminal tensor-test vectors          For a terminal GoodAWACK skeleton S, the terminal affine
forms are

                                      ℒS = {𝐿𝜌 (𝑧) = ℓ𝜌 · 𝑧 + 𝑐𝜌 }.

The TC1/HighTC test is applied to the corresponding terminal vectors ℓ𝜌 and tensors 𝑄𝜌 = ℓ𝜌 ⊗ ℓ𝜌 .
Post-terminal operations may restrict domains, average over fibres, or introduce auxiliary testing
variables, but they may not replace the terminal list {ℓ𝜌 } by a new list and then treat the new list
as a fresh GoodAWACK routing descendant. Any operation that would change the terminal tensor-
test vectors for routing purposes must already occur as a pre-terminal operation in the routing
record.

Rank drop and tag An affine transformation is full-rank on the active affine span if its linear
part is injective on the difference space generated by the current active forms, up to the finite-index
restrictions already recorded in 𝒞. For terminal GoodAWACK records, E5-clean full-rank transport
also has trivial kernel on the span of the terminal tensor-test vectors. It is rank-dropping if this
injectivity fails on the active span or on the terminal tensor-test span.
    Throughout this lemma, a bounded affine map means an affine map whose coefficients, denomina-
tors and induced lattice index are controlled by the fixed routing complexity and the polylogarithmic
parameter hierarchy. Thus "bounded" is not a new qualitative assumption; it is the quantitative
bounded-complexity condition already present in the B1/B3/F3/F4/E5 routing record.
    A rank drop is tagged if orig records one of:


Fix/Proj,   CRT,    FixedDiv,   VarQuot,       LocalDiag,    CKP,     Edge,   PostTerminalNonGenerator.

   It is untagged if it is present only as a free affine regrouping or formal affine parametrization.
   —

E10Y.2. Phase Separation

Lemma H.34 (Lemma E10Y.1. Pre-terminal and post-terminal phases are disjoint). Every actual
B1-origin descendant has a finite pre-terminal routing phase followed by a terminal analytic phase.
Operations from the terminal analytic phase do not generate new terminal GoodAWACK skeletons.




                                                   319
Proof. B1 supplies typed Heath–Brown product variables, dyadic cells and exact convolution weights.
B3 supplies finitely many product-grouping candidates and preliminary structural labels. F3 and
F4 then perform the finite routing decisions that determine whether the descendant is Edge, CKP,
GoodAWACK, LongAP/Local or LocalDiag.
    Once a GoodAWACK terminal object has been labelled, the later TC1/HighTC, Cauchy–Schwarz,
Fourier, coarea, Shiu/BRS, Davenport/AP and local projection arguments operate on that fixed
terminal object or on a testing family derived from it. None of those arguments returns to B3, F3
or F4 to create an additional descendant, and none introduces a new terminal routing class.
   Thus the routing construction has two separated phases:

            B1/B3/F3/F4 pre-terminal routing       then   terminal analytic estimation.

   The second phase estimates, reroutes or discards already fixed terminal data; it is not a skeleton-
generation mechanism.
   —


E10Y.3. Initial B1/B3 Sources

Lemma H.35 (Lemma E10Y.2. B1 and B3 introduce no free rank-dropping affine regrouping). B1
and B3 supply finitely many initial sources for the routing grammar, but neither layer introduces an
arbitrary rank-dropping affine regrouping into a terminal GoodAWACK skeleton.

Proof. In B1 the variables are the product variables of the fixed-depth Heath–Brown decomposition,
together with dyadic restrictions and exact convolution weights. This layer introduces product
coordinates and cells; it does not apply a bounded affine map to the active affine span.
    In B3 the construction enumerates finitely many product-grouping candidates. The operation is
a finite selection among grouped product-coordinate descriptions. If the grouping exposes a short-
volume, quotient, local, CKP-balanced or Edge structure, that information is recorded as a routing
feature and passed to F3/F4. If it does not, the descendant remains a candidate for terminal routing.
    Therefore B1/B3 create the finite set of start states for the later grammar, but they do not add
a hidden rank-dropping AFF operation.
    —


E10Y.4. Exhaustive Pre-Terminal Routing

Lemma H.36 (Lemma E10Y.3. F3/F4 exhaust the skeleton-generating pre-terminal operations).
Let r be an actual routing record after B3. Every skeleton-generating pre-terminal operation applied
before terminal class labelling is one of the F3/F4 operations recorded in the following list:

  1. controlled CRT absorption;

  2. F4 large-divisor or quotient decision;

  3. square-divisor routing;

  4. finite grouping selection or elimination;

  5. terminal LocalDiag detection;


                                                 320
   6. terminal Edge detection by a C1P predicate;

   7. terminal class labelling into Edge, CKP, GoodAWACK, LongAP/Local or LocalDiag.

Proof. Lemma F3A proves that Section F3.6 is complete for generic F3 routing-level operations.
Lemma F3T expands this list into the complete finite routing table by B1 type, B3 grouping, dyadic
regime, divisor/conductor state, coefficient type, terminal class and exclusion reason. Lemma F4
supplies the ordinary divisor and quotient decision used inside the second item.
    The F3T table has no sixth terminal class and no row whose operation is "arbitrary affine
regrouping." Each non-terminal row either continues the finite routing procedure with a recorded tag
or is eliminated as incompatible, empty, Edge, local, CKP or already terminal. Hence any actual-
generated operation that can change the terminal skeleton before labelling is represented by the
seven operations above.
   —


E10Y.5. Full-Rank Transport and Tagged Rank Drop

Lemma H.37 (Lemma E10Y.4. E5 does not add an independent skeleton generator). E5 content
stability may transport content, coefficients and auxiliary variables along an already generated routing
record. It does not create an additional terminal GoodAWACK skeleton from an external affine
system.
    More precisely, every E5-compatible affine transport is one of:

   1. full-rank on the active affine span and on the terminal tensor-test span, with controlled bounded-
      minor/content loss;

   2. rank-dropping with an origin tag already supplied by B1/B3/F3/F4;

   3. post-terminal analytic slicing after the terminal object has already been fixed.

Proof. The content-stability calculation of E5 is applied only after the current routing record has
already supplied the variables, congruence restrictions, divisor data and origin information being
transported. A full-rank transport preserves the active affine rank, is injective on the terminal
tensor-test span, and changes content only by the controlled bounded-minor factors recorded in E5.
    If the transport is not full-rank on the active affine span, or if it has a kernel on the terminal
tensor-test span, then it is not E5-clean full-rank transport. The lost rank must then come from a
restriction already present in the routing record: fixing/projection, CRT compatibility, fixed divisor
quotient, variable quotient residual, local/diagonal dependence, CKP-balanced structure, Edge
predicate or post-terminal post-terminal analytic slicing. These are exactly the tags recorded in the
origin component of the routing record.
    Thus E5 is a stability principle for transports whose source is already known. It is not a separate
mechanism for adjoining an untagged rank-dropping affine map to a GoodAWACK terminal skeleton.
   —




                                                  321
E10Y.6. No Feedback from Analytic Tests

Lemma H.38 (Lemma E10Y.5. Terminal analytic operations do not feed back into the routing
grammar). Let S be a terminal GoodAWACK skeleton. The TC1/HighTC split, global testing
construction, regular/singular testing dichotomy, BRS/X16 short-image analysis, Davenport/AP
estimate, Cauchy–Schwarz, cube expansion, Fourier expansion and local projection arguments do
not create an additional terminal GoodAWACK skeleton.

Proof. Each listed operation is invoked after the terminal skeleton has been selected. Its input is a
fixed terminal skeleton, a fixed terminal testing family, or a sum derived from those fixed data. The
output is one of the following:

   1. an 𝑜(𝑁 ) analytic estimate;

   2. a routing-away conclusion to Edge, CKP, LongAP/Local or LocalDiag already present in the
      F3/F4 terminal alternatives;

   3. a proof that the GoodAWACK contribution belongs to the HighTC finite grammar closure;

   4. a local-main projection used only in the final assembly.

    None of these outputs is a new B1/B3/F3/F4 descendant. Therefore post-terminal analytic
tests cannot supply a missing pre-terminal operation and cannot produce an untagged terminal
GoodAWACK rank drop.


Lemma H.39 (Lemma E10Y.5b. Terminal tensor-vector immutability). Once a terminal
GoodAWACK skeleton S is fixed, the affine vectors ℓ𝜌 and tensors 𝑄𝜌 = ℓ𝜌 ⊗ ℓ𝜌 used in the
TC1/HighTC test are immutable under post-terminal analytic operations. Post-terminal slicing,
averaging, Cauchy–Schwarz, cube expansion, TC1 testing, BRS/X16 estimates and Davenport/AP
estimates may restrict or test the fixed terminal object, but they may not replace the terminal affine
skeleton by a new one.

Proof. By definition, the TC1/HighTC split is applied to the terminal list ℒS produced by B1/B3/
F3/F4/E5 before post-terminal estimation begins. A later analytic operation has one of two effects.
It either restricts the summation domain or introduces auxiliary variables used to test, average, or
estimate the already fixed terminal data. Neither effect changes the routing record r, the origin
map, or the terminal class label.
    If a proposed post-terminal step replaced the list {ℓ𝜌 } by a new rank-relevant list and then
used that new list as a terminal GoodAWACK skeleton, the step would be a skeleton-generating
operation rather than a post-terminal analytic operation. By Lemma E10Y.0 and Lemma E10Y.3,
such a step would have to appear in the pre-terminal routing record and be classified by F3/F4/E5.
Therefore it cannot occur as a hidden post-terminal feedback operation.
   —


E10Y.7. Apparent Operations Table The following table records the status of all operation
types that can appear syntactically in the GoodAWACK branch.




                                                 322
       Apparent operation                              Mathematical status
       Dyadic refinement                               B1/B3 cell restriction; no affine rank drop
       Product grouping                                B3 finite candidate source; not a free affine
                                                       map
       Controlled CRT restriction                      F3 operation; full-rank on the active differ-
                                                       ence lattice or incompatible
       Fixed divisor quotient                          F4/E5-compatible tagged quotient
       Variable quotient residual                      F4 tagged quotient or rerouting case
       Square-divisor routing                          F3/F4 Edge or zero/short-fibre routing,
                                                       recorded by tag
       Gcd/local/proportional relation                 F4/HGO2R local or LocalDiag origin
       CKP-balanced relation                           Terminal CKP tag; not GoodAWACK
                                                       HighTC residue
       Strict saving or boundary relation              C1 Edge tag
       Full-rank affine coordinate change              E5 content-stable transport; injective on the
                                                       active affine span and terminal tensor-test
                                                       span
       Primitive or coarea slicing                     Post-terminal analytic non-generator
       Cauchy–Schwarz or cube expansion                Post-terminal analytic non-generator
       Fourier expansion or TC1 testing                Post-terminal analytic non-generator
       Local projection                                H4/D1 local-main assembly; post-terminal
                                                       non-generator
       Arbitrary rank-dropping affine                  Not actual-generated unless one of the
       reparametrization                               recorded tags is present




   This table is not an extra assumption. It is the union of Lemmas E10Y.0–E10Y.5b.
   —

E10Y.7A. Formal Transition Table This section records the routing grammar as transforma-
tions of the actual routing record

                                        r = (𝑉, 𝒞, ℒ, 𝒬, 𝜏, orig, 𝑊 ).

    The table is part of the proof of E10Y. It is not an additional assumption: each row is the formal
state-level version of a B1/B3/F3/F4/E5 operation already isolated above.




                                                     323
      Operation                    Input state                  Output state                 Rank effect                  Required tag             If not satisfied
      B1 start-state creation      no previous Branch B state 𝑉, 𝒞, 𝑊 from a fixed-depth no affine rank drop              start-state origin       not a Branch B descendant
                                                                Heath–Brown product
                                                                block
      B3 finite grouping           B1 state with product        one grouped candidate        finite selection only; no    B3 grouping origin       candidate removed or
                                   variables                    Γ, with updated ℒ and        free affine map                                       routed by F3
                                                                preliminary 𝜏
      controlled CRT absorption state with congruence           restricted lattice/coset and full-rank finite-index re-   CRT                      incompatible fibre, hence
                                   𝐿0 (𝑧) ≡ 𝑎 (mod 𝑞)           updated 𝒞                    striction on the active                               zero/empty
                                                                                             difference lattice, or empty
      F4 quotient/divisor deci-    state containing 𝑑 | 𝐿(𝑧) or updated 𝒬, 𝜏 , and origin    possible rank drop only      FixedDiv, VarQuot,       routed away or M♯ de-
      sion                         𝐿(𝑧) = 𝑑𝑠                    record                       through recorded quotient/ LocalDiag, CKP, or Edge    creases
                                                                                             local/CKP/Edge data
      square-divisor routing       state with square-divisor    Edge state, controlled       no untagged affine rank      Edge or CRT              zero/short-volume or strict
                                   predicate                    divisibility state, or empty drop                                                  C1P saving
                                                                state
      grouping selection or elimi- finite B3/F3 candidate list selected candidate, re-       finite selection; no new     B3/F3 grouping origin    candidate eliminated
      nation                                                    moved candidate, or de-      affine transformation
                                                                creased routing measure
      LocalDiag detection          state with equality, propor- terminal LocalDiag state     rank collapse leaves         LocalDiag                continue F3/F4 routing
                                   tionality, repeated form, or                              GoodAWACK
                                   forced local relation
324




      Edge detection               state satisfying a C1/C1A terminal Edge state             any collapse is absorbed     Edge                     continue F3/F4 routing
                                   strict-saving predicate                                   into a strict-saving route
      CKP detection                state exposing balanced      terminal CKP state           rank relation is a CKP       CKP                      continue F3/F4 routing
                                   bilinear Kloosterman-                                     origin, not a GoodAWACK
                                   fraction structure                                        residual
      GoodAWACK terminal           state with no Edge, CKP, terminal GoodAWACK               labelling only; no coordi-   terminal label           not a terminal
      labelling                    LocalDiag, LongAP/Local, skeleton                         nate operation                                        GoodAWACK skeleton
                                   or unresolved ordinary
                                   divisor predicate
      E5 clean transport           already generated B1/B3/ transported content/             full-rank on active and      inherited origin tag, or not E5-clean; must be
                                   F3/F4 routing record         auxiliary data on the same tensor-test spans, or inher- no tag needed in full-rank routed/tagged before ter-
                                                                record                       ited tagged rank drop        case                     minality
      post-terminal analytic non- fixed terminal skeleton       test, slice, Fourier/coarea may restrict analytic sums PostTerminalNonGenerator if it changes terminal
      generator                                                 family, or estimate of the   but cannot replace termi-                             vectors, it is not post-
                                                                fixed object                 nal tensor-test vectors                               terminal and must appear
                                                                                                                                                   above
    Consequently a rank-changing operation in an actual GoodAWACK descendant has only two
possibilities. Either it is one of the pre-terminal rows and carries the displayed origin information,
or it is a post-terminal analytic non-generator and cannot create a new terminal GoodAWACK
skeleton.
    —

E10Y.8. Bidirectional Source–Grammar Tables The following two tables make explicit that
E10Y is not taking completeness as an unnamed premise. The first table maps each independently
defined B1/B3/F3/F4/E5 operation to the E10Y transition class that covers it.


   Source operation                E10Y transition class             Rank effect
   B1 typed Heath–Brown block      B1/B3 start-state generation      no affine rank drop
   and dyadic cell
   B3 finite product grouping can- B1/B3 start-state generation      no free affine map; grouping is
   didate                                                            recorded
   F3 controlled CRT absorption    F3/F4 pre-terminal routing        finite-index restriction or incom-
                                                                     patible fibre
   F4 divisor or quotient decision   F3/F4 pre-terminal routing      tagged quotient, local, CKP,
                                                                     Edge, GoodAWACK, or de-
                                                                     creasing continuation
   F3 square-divisor routing         F3/F4 pre-terminal routing      Edge or controlled divisibility/
                                                                     CRT tag
   F3 grouping selection/            F3/F4 pre-terminal routing      finite candidate selection; no
   elimination                                                       new affine operation
   F3 LocalDiag detection            F3/F4 pre-terminal routing      terminal LocalDiag; leaves
                                                                     GoodAWACK
   F3 Edge detection                 F3/F4 pre-terminal routing      terminal Edge; leaves
                                                                     GoodAWACK
   F3 terminal labelling      F3/F4 pre-terminal routing             label only
                              E5 full-rank content-stability
   E5 full-rank content transport                                    rank preserved on active and
                              transport                              tensor-test spans
   E5 rank-dropping transport E5 tagged content-stability            allowed only with an already
                              transport                              recorded origin tag
   TC1/HighTC, BRS/X16, Dav- post-terminal analytic non-             no new routing descendant and
   enport/AP, Fourier, cube,  generator                              no replacement of terminal
   coarea, local projection                                          tensor-test vectors




   Conversely, every E10Y transition class has only the following possible sources in the proof tree.


   E10Y transition class             Possible sources                Excluded sources
   B1/B3 start-state generation      B1 and B3                       E5, F4, analytic estimates
   F3/F4 pre-terminal routing        F3.6–F3.14 and F4               arbitrary affine reparametriza-
                                                                     tion; Cauchy/cube/Fourier
                                                                     steps
   E5 full-rank content-stability    E5 applied to an already gener- external affine systems not pro-
   transport                         ated routing record             duced by B1/B3/F3/F4


                                                  325
    E5 tagged rank-dropping trans- E5 with a Fix/Proj, CRT,          untagged rank-dropping AFF
    port                           FixedDiv, VarQuot, LocalDiag,
                                   CKP or Edge origin already
                                   present
    post-terminal analytic non-    terminal analytic estimates ap- any operation that changes the
    generator                      plied after the terminal skeleton terminal tensor-test vectors for
                                   is fixed                          routing purposes




   Thus a syntactically visible operation has two tests. It must appear in the left table as a source
operation, and its E10Y class must have an allowed source in the right table. If either test fails, the
operation is not an actual-generated GoodAWACK skeleton generator.
   —

E10Y.9. Grammar Completeness Theorem

Theorem H.40 (Theorem E10Y.6. Completeness of the GoodAWACK routing grammar). Every
actual-generated operation that can generate or modify an actual terminal GoodAWACK affine
skeleton is one of the operations in Lemmas E10Y.2–E10Y.5b. Equivalently, the GoodAWACK
routing grammar has no hidden skeleton-generating operation outside:

                                     B1/B3 start-state generation,
                                     F3/F4 pre-terminal routing,
                           E5 full-rank or tagged content-stability transport,
                                 post-terminal analytic non-generators.

    Consequently, any rank-dropping affine operation visible in an actual terminal GoodAWACK
skeleton must have one of the recorded origin tags


Fix/Proj,    CRT,    FixedDiv,    VarQuot,   LocalDiag,    CKP,     Edge,   PostTerminalNonGenerator,

    or else the skeleton is not an actual B1-origin terminal GoodAWACK descendant.

Proof. Let S be an actual terminal GoodAWACK skeleton. By Lemma E10Y.0, S has a finite
routing history
                                       𝑟0 → 𝑟1 → · · · → 𝑟𝑇 = S,
where 𝑟0 is a B1/B3 start record and each transition is an actual-generated pre-terminal operation
or a terminal class labelling step. We prove by induction on 𝑡 the invariant


ℐ(𝑟𝑡 ) :   every rank-changing operation up to 𝑟𝑡 is E10Y-classified and carries an allowed origin tag.
    At 𝑡 = 0, Lemma E10Y.2 shows that B1 and B3 supply only product-coordinate and grouping
start states. No untagged rank-changing affine operation has occurred.
    Assume ℐ(𝑟𝑡 ) and consider 𝑟𝑡 → 𝑟𝑡+1 . If the transition is an F3/F4 routing operation, Lemma
E10Y.3 classifies it as controlled CRT, F4 divisor/quotient decision, square-divisor routing, finite
grouping selection or elimination, LocalDiag detection, Edge detection, or terminal class labelling.

                                                  326
Each rank-changing case either records one of the allowed tags or routes the cell away from terminal
GoodAWACK. If the transition is an E5 transport, Lemma E10Y.4 says that it is either full-rank
on the active and terminal tensor-test spans, or else rank-dropping with an already recorded origin
tag. Thus ℐ(𝑟𝑡+1 ) holds.
    After terminal labelling, Lemmas E10Y.5 and E10Y.5b apply. Post-terminal analytic operations
may restrict or test the fixed terminal object, but they do not produce a new routing descendant
and do not replace the terminal tensor-test vectors. Hence they cannot violate the invariant by
adding a hidden skeleton-generating operation.
    These cases exhaust the route from B1 to the terminal GoodAWACK object. Therefore no
additional skeleton-generating operation can occur. In particular, an untagged rank-dropping affine
regrouping is not a permissible operation in the actual GoodAWACK routing grammar.
    —


Parameter check H.41 (E10Y.10. Parameter Check and Output Form). The theorem introduces
no new analytic parameter and no new error term. Its only finiteness input is the fixed-depth B1
decomposition, the finite B3 grouping list, the finite F3T routing table and the finite F4 divisor/
quotient decision tree. All content losses are those already controlled in E5.
   The output supplied to E10X is:

    the finite grammar used in E10X is complete for actual terminal GoodAWACK skeletons.

   The output supplied to E10M/E10K/E10YMX/E10L is:


a rank-dropping affine regrouping in terminal GoodAWACK is admissible only with an allowed origin tag.

H.12     E10M no untagged rank-dropping AFF
H.12.1    E10M. No Untagged Rank-Dropping AFF in Terminal GoodAWACK
E10M.0. Statement and Role Lemma E10M is the no-untagged-rank-drop theorem behind the
E10K interface cleanup. Lemma E10Y proves that the GoodAWACK routing grammar is complete
for actual B1-origin terminal skeletons. Lemma E10X supplies the finite grammar invariant, while
the reader-facing master closure is Lemma E10YMX, which packages E10Y, E10M, E10X, and
E10K into the finite GoodAWACK grammar interface.
    The finality of the generator list used below is proved in E10Y. Non-logical reproducibility
records may be kept separately for future maintenance, but they are not logical prerequisites for the
theorem below.
    The issue isolated by E10I–E10K is the following residual:

 could a terminal GoodAWACK skeleton contain an untagged rank-dropping affine regrouping?

    The answer is no, provided the terminal object is required to be an actual descendant of the
active routing tree

                                        𝐵1 → 𝐵3 → 𝐹 3/𝐹 4




                                                 327
    and not merely a formal affine pattern allowed by broad wording in E5, BGS, BAOC or E10G.
    Thus the result below is not a new analytic estimate. It is the structural version of the F3-
complete routing interpretation: every terminal GoodAWACK skeleton must be generated by the
finite operation list already proved in F3.6 and by the F4 large-divisor decision procedure.
    Logical dependencies are B1, B3, F3, F4, E5, BGS, E10Y, E10I, E10J, and HGO2R. E10M is
used by E10X, E10K, E10YMX, and E10L.

Role inside the E10X master closure E10M is the central structural input for the rank-
dropping AFF obstruction isolated in E10H–E10J. Packaged by E10X, it discharges the active
descendants of:

  1. E10H.2, by proving that a formal 4AP-like matrix witness cannot be an untagged actual
     terminal GoodAWACK cell;

  2. E10I.4, by proving that the only rank-dropping AFF residual left after the CRT/full-rank
     safety reductions has no untagged actual occurrence;

  3. E10J.3, by proving AFF-origin completeness for actual B1/B3/F3/F4 terminal GoodAWACK
     routing histories.

   The formal 4AP-like matrix family

                                     𝑌𝑖 = 𝑥 + 𝑖𝑟,      0 ≤ 𝑖 ≤ 3,

    is therefore not deleted from the proof. It is treated in E10X as an interface witness at the
broad E5/BGS/BAOC level. E10M proves the decisive structural claim: if such a rank-dropping
configuration is produced by actual routing, then its rank drop is tagged by one of the permitted
origins; if it is untagged, it violates the E10Y-certified F3-complete routing interface and is not an
admissible terminal GoodAWACK skeleton.
    —

E10M.1. Setup: Definitions

Actual terminal GoodAWACK skeleton                  An actual terminal GoodAWACK skeleton is a
skeleton

                             S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )

   which occurs as the terminal record of a descendant produced by the E10Y-certified routing
grammar, namely:

  1. the typed Heath–Brown product variables and dyadic cells of Lemma B1;

  2. the finite product-grouping candidates of Lemma B3;

  3. the routing operations of Lemma F3, Section F3.6;

  4. the large-divisor decision procedure of Lemma F4;

  5. the content-stability transports of Lemma E5, read in the clean sense of E10Y/E10L/E10M.


                                                 328
Rank-dropping AFF occurrence A rank-dropping AFF occurrence is a bounded affine map
used in the skeleton record whose linear part drops rank on the active affine span.
   Such an occurrence is tagged if its rank drop is explicitly caused by one of:

  1. fixing or projection;

  2. congruence compatibility or an inconsistent fibre;

  3. fixed divisor quotient;

  4. variable quotient residual;

  5. local/diagonal or gcd origin;

  6. CKP-balanced origin;

  7. strict C1P Edge origin;

  8. post-terminal analytic slicing which is not used to generate the terminal tensor-test vectors.

    It is untagged if it is recorded only as a free affine regrouping or affine parametrization, with no
origin in the routing record.
    —

E10M.2. Statement and Proof

Theorem H.42 (Theorem E10M.1. No untagged rank-dropping AFF). Let S be an actual terminal
GoodAWACK skeleton. Then every rank-dropping AFF occurrence in its terminal record is tagged.
   Equivalently:


    no untagged rank-dropping affine regrouping survives into terminal GoodAWACK.               (E10M)

Proof. We trace all places where a rank drop could enter an actual terminal GoodAWACK skeleton.
By Lemma E10Y, this trace exhausts all actual-generated skeleton-generating operations.
    B1. By Lemma B1, the starting objects are product variables, dyadic cells and exact Heath–
Brown convolution factors. B1 introduces no affine matrix map and no rank-dropping affine
slice.
    B3. By Lemma B3, the next operation is finite product-grouping selection. B3 may record
grouping alternatives and preliminary labels, but it does not introduce a free affine parametrization.
If a grouping exposes short factors, CKP-balanced structure, canonical local structure, forced
dependence or an Edge predicate, the descendant is routed to the corresponding terminal class
rather than being left as untagged GoodAWACK data.
    F3. By Lemma F3, Section F3.6, the generic routing-level operations are exactly:

  1. controlled CRT absorption;

  2. F4 large-divisor decision;

  3. square-divisor routing;

  4. finite grouping selection/elimination;


                                                  329
   5. terminal LocalDiag detection;

   6. terminal Edge detection by C1P predicates;

   7. terminal class labelling into CKP, GoodAWACK, LongAP/Local, Edge, LocalDiag.

    Cauchy/cube operations and Fourier expansion are explicitly post-terminal proof subroutines,
not generic routing operations. Hence F3 contains no operation whose effect is "add an arbitrary
rank-dropping affine regrouping to the terminal skeleton."
    Controlled CRT absorption is finite-index and full-rank on the difference lattice. It may change
content by bounded or polylogarithmic factors controlled by E5, but it does not create a new rank-
dropping affine relation. If the CRT conditions are incompatible, the fibre is impossible.
    Square-divisor routing is either terminal C1 Edge when the square divisor is large, or a controlled
CRT/divisibility restriction when the square divisor is small. In the second case it is subsumed by
controlled absorption; in the first case it leaves GoodAWACK. Thus it does not create a free affine
rank drop.
    Finite grouping selection/elimination chooses among B3’s already finite product-grouping candi-
dates. It either keeps a candidate with its recorded origin data or removes it with a strict routing-
measure decrease. It is not a new affine slice.
    Terminal LocalDiag and terminal Edge detections are tagged by their defining local or C1P
predicates. They leave the GoodAWACK class once detected.
    Terminal GoodAWACK labelling is only a label. It records that after the F3/F4 decisions no
Edge, CKP, LongAP/Local, LocalDiag or unresolved large-divisor predicate remains. It is not itself
a coordinate operation.
    F4. By Lemma F4, every ordinary large-divisor or quotient predicate is decided exhaustively. A
rank drop coming from a fixed divisor quotient, variable quotient residual, gcd/local dependence,
proportional or repeated forms, or quotient-determined forms is therefore tagged by the F4 origin
record. The resulting descendant is routed to Edge, CKP, LocalDiag, LongAP/Local, GoodAWACK
with the ambiguity resolved, or to a measure-decreasing continuation.
    Thus F4 may create tagged rank-drop data, but it does not admit a free rank-dropping AFF
occurrence with no origin.
    E5. Lemma E5 supplies content stability for the allowed transports. The phrase "affine
regrouping" in E5 cannot be read as an additional terminal-routing operation, because F3.6 is the
exhaustive list of such operations. Therefore E5 may be used only for:

   1. full-rank coordinate changes;

   2. tagged fixing/projection or quotient/local transports;

   3. tagged CKP, Edge or impossible-origin reductions;

   4. post-terminal analytic slicing after the terminal tensor-test vectors have already been fixed.

    This is precisely the E5-clean interpretation recorded in E10L.
    Finally, post-terminal primitive slicing, Cauchy/cube and Fourier operations are analytic sub-
routines inside estimates. They are not allowed to generate a new terminal GoodAWACK skeleton
after the TC1/HighTC tensor test has been declared.
    All actual rank-dropping AFF occurrences are therefore tagged by one of the origins listed in
E10M.1. An untagged rank-dropping AFF occurrence would have to come from an operation not



                                                 330
present in B1, B3, F3.6 or F4, or from reading E5 as an extra terminal generator. Both alternatives
contradict the E10Y-certified F3-complete routing interface. The theorem follows.
   —


E10M.3. Finite Classification Table: AFF Occurrence Origins The proof above is
summarized by the following finite classification table. The table records every mathematical source
in the B1/B3/F3/F4/E5-clean routing grammar where a reader might suspect that a rank-dropping
affine regrouping is introduced.
    The table is exhaustive by E10Y and the structural source analysis in the proof above. Separate
reproducibility records, if consulted, are outside the proof and are not used to prove the table.




                                                331
      Source/interface   Phrase or operation          Can drop rank?              If yes, tag source            Terminal generator?          Why no untagged AFF
                                                                                                                                             survives
      B1                 Heath–Brown product          No                          None needed                   Yes, as initial product data B1 creates product coor-
                         variables; dyadic cells                                                                                             dinates and weights, not
                                                                                                                                             affine regrouping maps.
      B3                 finite product grouping;     No as a new affine map      Existing B1 grouping          Yes, only as candidate       B3 selects among finite
                         preliminary labels                                       record                        selection                    product groupings; ex-
                                                                                                                                             posed dependence routes
                                                                                                                                             to CKP/LocalDiag/Edge/
                                                                                                                                             GoodAWACK labels with
                                                                                                                                             origin data.
      F3, F3.6           controlled CRT absorption    No on the difference lattice CRT compatibility / im-      Yes                          CRT restriction is finite-
                                                                                   possible fibre                                            index/full-rank on the
                                                                                                                                             active span; inconsistency
                                                                                                                                             is tagged impossible.
      F3, F3.6/F3.9      square-divisor routing       No untagged affine rank     C1 square-divisor Edge or     Yes                          Large square divisors are
                                                      drop                        controlled divisibility tag                                terminal Edge; small
                                                                                                                                             square divisors become
                                                                                                                                             controlled absorption
                                                                                                                                             and inherit the CRT/
                                                                                                                                             divisibility tag.
      F3, F3.6           finite grouping selection/   No as a new slice           B3 grouping origin            Yes                          Selection records or re-
332




                         elimination                                                                                                         moves a candidate; it is
                                                                                                                                             not an additional affine
                                                                                                                                             operation.
      F3, F3.6           LocalDiag detection          Yes only by forced equal-   LocalDiag tag                 Yes, but leaves              Once detected, the atom
                                                      ity/local dependence                                      GoodAWACK                    is terminal LocalDiag, not
                                                                                                                                             terminal GoodAWACK.
      F3, F3.6           Edge detection               Yes only through strict     C1 Edge tag                   Yes, but leaves              Once detected, the atom
                                                      saving predicate                                          GoodAWACK                    is terminal Edge and is
                                                                                                                                             handled by C1.
      F3, F3.6           CKP detection                Yes only through gcd/       CKP tag                       Yes, but leaves              Once CKP-balanced struc-
                                                      balanced grouping                                         GoodAWACK                    ture appears, the atom is
                                                                                                                                             terminal CKP and is han-
                                                                                                                                             dled by G8a.
      F3, F3.6           GoodAWACK labelling          No                          None needed                   Yes                          The label records the
                                                                                                                                             absence of other terminal
                                                                                                                                             predicates; it performs no
                                                                                                                                             coordinate operation.
      F4                 fixed divisor quotient       Yes                         F4 fixed-divisor origin       Yes                          The quotient origin is
                                                                                                                                             recorded; untagged use
                                                                                                                                             is forbidden by the F4
                                                                                                                                             decision procedure.
      F4                      variable quotient residual   Yes                           F4 quotient-residual origin   Yes   The residual is routed to
                                                                                                                             Edge/CKP/LocalDiag/
                                                                                                                             LongAP/GoodAWACK
                                                                                                                             with ambiguity resolved, or
                                                                                                                             to a decreasing continua-
                                                                                                                             tion.
      F4                      repeated/proportional        Yes                           local/diagonal or C1/CKP Yes        Forced dependence is ter-
                              forms                                                      origin                              minally routed away or
                                                                                                                             recorded as tagged origin
                                                                                                                             data.
      E5                      affine regrouping/content    Only if read too broadly      E10M-clean full-rank or       No    E5 is a stability lemma
                              stability                                                  tagged transport                    for transports already
                                                                                                                             created by B1/B3/F3/F4;
                                                                                                                             it is not an extra terminal
                                                                                                                             generator.
      BGS                     skeleton record / 𝑟grp       Records possible rank         inherited origin tag          No    BGS records terminal
                                                           behavior                                                          data produced upstream;
                                                                                                                             it does not create a new
                                                                                                                             operation.
      BAOC                    weak transport catalogue,    Interface only                inherited B1/B3/F3/F4         No    BAOC is catalogue/
                              C5/T5 interface examples                                   origin                              grammar support in the
                                                                                                                             proof tree; broad catalogue
333




                                                                                                                             classes are discharged by
                                                                                                                             E10YMX.
      E10G                    bounded AFF cell /           Reduction only                E10H–E10K chain               No    E10G isolates the free-
                              FreeAffineHighTC inter-                                                                        affine class; it does not
                              face example                                                                                   authorize a new terminal
                                                                                                                             AFF map.
      E10H                    matrix-origin rigidity re-   Reduces to CRT/AFF            matrix-origin reduction tag No      E10H localizes the issue
                              duction                      issue                                                             to E10I–E10K; it does
                                                                                                                             not generate a terminal
                                                                                                                             skeleton.
      E10I                    CRT and full-rank AFF        Full-rank only, except        MOR/RDA reduction tag         No    E10I proves safe cases and
                              safety                       reduced residual                                                  passes only rank-dropping
                                                                                                                             AFF to E10J/E10M.
      E10J                    tagged rank drops            Yes                           origin-degenerate or routed No      E10J proves tagged rank
                                                                                         tag                                 drops are safe and reduces
                                                                                                                             only the untagged possibil-
                                                                                                                             ity to E10M.
      post-terminal Cauchy/   analytic slicing after termi- May restrict analytic sums   PostTerminalNonGenerator No         These steps estimate a
      cube/Fourier steps      nal record                                                 tag                                 fixed terminal atom and
                                                                                                                             cannot create a new termi-
                                                                                                                             nal GoodAWACK skeleton.
    Therefore the only conceivable source of an untagged rank-dropping AFF would be to read one
of the record/stability documents as adding a new terminal operation outside F3.6 and F4. The
E10Y-certified F3-complete interface forbids that reading: terminal GoodAWACK skeletons are
actual descendants of B1/B3/F3/F4/E5-clean, not arbitrary formal affine systems.
    —

E10M.4. Output Consequences

Corollary H.43 (Corollary E10M.2. AFF-OC is discharged). The AFF-origin completeness
hypothesis used in Lemma E10K is a theorem for actual terminal GoodAWACK skeletons:

                                             AFF-OC.

    Thus E10K is no longer merely a conditional cleanup statement. Its F3-COMPLETE assumption
is discharged by E10Y and E10M.

Corollary H.44 (Corollary E10M.3. FreeAffineHighTC is empty in the proof tree). Combining
E10M with the reductions in E10I and E10J gives:

                                     𝑅FreeAffineHighTC (𝑁 ) = 0.

   Together with HGO2R, this leaves only the origin-degenerate HighTC cases, which route to CKP,
LocalDiag, Edge or Impossible.
   —

                                Remark H.45 (E10M.5. Output).
    E10M rules out untagged rank-dropping AFF in actual terminal GoodAWACK skeletons.
   It is cited by E10X, E10K, E10YMX and E10L. Its role is to make explicit, after E10Y, that
broad "affine regrouping" language in E5/BGS/BAOC/E10G is not an additional source of terminal
GoodAWACK affine systems.

E10M.6. Logical Dependencies Internal dependencies: B1, B3, F3, F4, E5, BGS, E10Y, E10I,
E10J, HGO2R.
   Children served: E10X, E10K, E10YMX and E10L.

H.13     E10X finite GoodAWACK grammar theorem
H.13.1    E10X. Finite GoodAWACK Grammar Closure
E10X.0. Statement and Role Lemma E10X is the finite combinatorial closure theorem for
the GoodAWACK HighTC branch. It packages the reduction chain

               BAOC → E10G → E10H → E10I → E10J → E10Y/E10M → E10K

     into a single theorem-level interface.
     The theorem is not a search assertion. Lemma E10Y proves that the finite grammar below is
complete for actual terminal GoodAWACK skeletons. Lemma E10X uses that grammar and proves
its invariant: every rank-dropping affine operation created along a derivation has an allowed origin
tag. Formal affine counterexamples at the broad BAOC/E10G/E5 interface are therefore irrelevant
unless they are derivable from

                                                334
                                             𝐵1 → 𝐵3 → 𝐹 3/𝐹 4

   with E5 used only as clean content stability.
   The theorem proved below is:


every actual terminal GoodAWACK skeleton has no untagged rank-dropping AFF source, hence no FreeAffineHighTC class.
                                                                                  (E10X)
   This is a structural theorem, not a new analytic estimate.
   Logical dependencies are B1, B3, F3, F4, E5, BGS, BAOC, HGO2R, E10G, E10H, E10I, E10J,
E10Y, E10M, and E10K. E10X is used by E10YMX, E10L and the GoodAWACK HighTC closure.
   —

E10X.1. Setup: Terminal GoodAWACK Skeletons and Grammar States An actual
terminal GoodAWACK skeleton is a record

                                 S = (ℬ, Γ, r, ΛS , ΩS , ℒS , ℳS , origS , 𝒲S )

   generated by the E10Y-certified finite grammar described below.

   1. Lemma B1 supplies typed Heath–Brown product variables, dyadic cells and exact convolution
      weights.
   2. Lemma B3 supplies a finite list of product-grouping candidates and preliminary tags.
   3. Lemma F3, Section F3.6, supplies the complete F3 routing operations: controlled CRT
      absorption, the F4 large-divisor decision, square-divisor routing, finite grouping selection
      or elimination, terminal LocalDiag detection, terminal Edge detection, and terminal class
      labelling.
   4. Lemma F4 supplies the exhaustive ordinary divisor and quotient decision, with recorded
      quotient, divisor, gcd, local, CKP, Edge, impossible, or continuation origins.
   5. Lemma E5 supplies content stability for transports already generated by the previous routing
      layers. It is not an additional terminal generator of affine systems.

   Thus terminal GoodAWACK skeletons are not arbitrary bounded affine systems. They are
actual descendants of the finite routing grammar above, and Lemma E10Y proves that this grammar
contains every actual-generated skeleton-generating operation.
   —

E10X.2. Finite GoodAWACK grammar theorem Define the finite GoodAWACK grammar
𝒢GA as follows.
   A state is a tuple

                                            s = (𝑉, ℒ, 𝒞, 𝒬, 𝒯 , 𝒪),

    where 𝑉 is the finite list of active variables inherited from B1, ℒ is the finite list of affine forms
visible on the current cell, 𝒞 is the list of controlled CRT/content restrictions, 𝒬 is the list of divisor
or quotient tags, 𝒯 is the routing tag, and 𝒪 is the origin record for every rank-changing operation
already applied. The start states are exactly the B1/B3 grouped cells.
    The transition set is finite and consists only of the following operations.

                                                       335
   Transition type                   Allowed effect on affine rank       Required origin tag or outcome
   fixing/projection                 may lower dimension by fixing       Fix/Proj
                                     variables already in 𝑉
   controlled CRT restriction        full-rank on the active span, or    CRT or empty
                                     incompatible
   fixed-divisor quotient            quotient by a recorded fixed        FixedDiv
                                     divisor
   variable quotient residual        quotient/divisor residual se-       VarQuot or rerouting tag
                                     lected by F4
   local/diagonal/gcd dependence     forced equality, proportionality,   LocalDiag
                                     repeated form, or gcd-local
                                     relation
   CKP-balanced relation             balanced bilinear Kloosterman-      CKP
                                     fraction structure
   strict saving or boundary rela-   C1 Edge, square-divisor, short-     Edge
   tion                              volume, high-frequency, small-
                                     conductor, or boundary case
   bounded affine regrouping         full-rank change on the active      inherited tag
                                     affine span, or rank-drop with
                                     recorded upstream origin
   primitive/post-terminal slicing   occurs only after terminal          PostTerminalNonGenerator
                                     tensor-test vectors are fixed
   E5 auxiliary inheritance          transports content or auxiliary     inherited tag; no terminal gen-
                                     variables already generated         erator
                                     upstream
   terminal labelling                labels a terminal cell              Edge, CKP, GoodAWACK,
                                                                         LocalDiag, or LongAP/Local



   Every transition is one of the operations authorized in F3.6/F3T or one of the F4 quotient
outcomes. E5 transitions are allowed only when their input state already has an origin record; they
cannot create a terminal GoodAWACK state from an arbitrary external affine system. By E10Y, no
additional actual-generated skeleton-generating transition exists.
Theorem H.46 (Theorem E10X.2A. Finite grammar invariant). For every state s reachable in
𝒢GA , every rank-dropping affine operation visible in s carries one of the following origin tags:


Fix/Proj,   CRT,       FixedDiv,   VarQuot,   LocalDiag,    CKP,     Edge,      PostTerminalNonGenerator.

    Consequently, a reachable terminal GoodAWACK state has no untagged rank-dropping AFF
operation.
Proof. We argue by induction on the length of the grammar derivation.
   At length zero, the state is a B1/B3 grouped cell. Its affine forms are the original product-
coordinate forms and their grouped descendants, and no rank-dropping affine operation has yet
been applied. The assertion is therefore vacuous.
   Assume the assertion for a reachable state s, and apply one transition.

  • A fixing or projection transition records the tag Fix/Proj.

                                                   336
  • A controlled CRT restriction is either incompatible, hence not terminal, or records the tag
    CRT with controlled content.

  • A fixed divisor quotient records FixedDiv.

  • A variable quotient residual records the F4 quotient tag VarQuot.

  • A local, diagonal, gcd, repeated-form, or proportionality transition records LocalDiag.

  • A balanced bilinear multiplicative transition records CKP.

  • A strict saving, boundary, square-divisor, short-volume, high-frequency, or small-conductor
    transition records Edge.

  • A post-terminal analytic slicing transition is allowed only after the terminal tensor-test vectors
    are fixed; it records PostTerminalNonGenerator and cannot create a new terminal affine
    skeleton.

  • An E5 transition only transports controlled content, CRT data, or auxiliary variables already
    present in the input state. By definition of the E5-clean interface in E10X.1, it inherits the
    existing origin record and does not introduce a new untagged rank-dropping map.

   Thus the invariant is preserved by every transition. Since the transition set is finite and every
terminal GoodAWACK skeleton is, by Lemma E10Y, a reachable terminal state of 𝒢GA , no terminal
GoodAWACK skeleton contains an untagged rank-dropping AFF operation. The theorem is proved.


Corollary H.47 (Corollary E10X.2B. No free affine HighTC generator). Any formal affine con-
figuration that cannot be derived from 𝒢GA is not an actual terminal GoodAWACK skeleton. In
particular, a FreeAffineHighTC pattern can remain only as a formal interface witness unless it is
produced by a grammar derivation; if it is produced by such a derivation, Theorem E10X.2A supplies
an allowed origin tag and HGO2R/E10K route it to an already handled class.
    —

E10X.3. Scope of the Mathematical Proof               The mathematical proof of E10X is exactly:

  1. the autonomous routing-record completeness theorem E10Y;

  2. the finite grammar 𝒢GA specified in E10X.1–E10X.2;

  3. the induction invariant in Theorem E10X.2A;

  4. the no-untagged-rank-drop theorem E10M;

  5. the AFF-origin-completeness consequence E10K.

   Thus the proof of E10X is internal to the mathematical lemmas listed above and does not require
any additional premise.
   —




                                                337
Parameter check H.48 (E10X.4. Parameter Check: Stability of the Formal Grammar). The
E10X finite-grammar theorem is valid for the grammar specified in E10X.1–E10X.2. If the formal
transition set 𝒢GA is changed, then Theorem E10X.2A must be rechecked for the changed transition
set.
     The present proof uses only the transition table displayed in E10X.2. In particular:

  1. E5 is used only as content stability for transports already generated by B1/B3/F3/F4;

  2. post-terminal analytic slicing is not allowed to replace the terminal tensor-test vectors;

  3. no extra rank-changing, quotient, projection, affine, CRT, diagonal, local, CKP, or Edge
     transition is available unless it appears in E10X.2 with an allowed origin tag.

   Thus the invariant proof is a finite mathematical induction over the displayed grammar, not a
maintenance assertion about a list of files.
   —

E10X.5. Output: No Untagged Rank-Dropping AFF By Lemma E10X.2A, the grammar
invariant already proves that reachable terminal GoodAWACK states have no untagged rank-
dropping AFF. By Lemma E10Y, this grammar is complete for actual terminal GoodAWACK
skeletons. By Lemma E10M, every rank-dropping AFF occurrence found in an actual terminal
GoodAWACK skeleton is one of the tagged grammar cases.
    The allowed tags are:

  1. fixing or projection;

  2. congruence compatibility or impossible fibre;

  3. fixed divisor quotient;

  4. variable quotient residual;

  5. local, diagonal, or gcd origin;

  6. CKP-balanced origin;

  7. strict C1P Edge origin;

  8. post-terminal analytic slicing that does not generate the terminal tensor-test vectors.

  Therefore an untagged rank-dropping affine regrouping cannot occur in an actual terminal
GoodAWACK skeleton:

                                       No-Untagged-AFF.

   —




                                                338
E10X.6. Interface Example: Formal 4𝐴𝑃 -Like Family The files E10G, E10H, E10I and
E10J use the formal family
                                     𝑌𝑖 = 𝑥 + 𝑖𝑟,        0 ≤ 𝑖 ≤ 3,
   whose coefficient vectors ℓ𝑖 = (1, 𝑖) satisfy

                             ℓ0 ⊙ ℓ0 − 3ℓ1 ⊙ ℓ1 + 3ℓ2 ⊙ ℓ2 − ℓ3 ⊙ ℓ3 = 0.                     (4AP)
    This example is admissible as a formal interface test: it shows that a broad phrase such as
"bounded affine regrouping" is too large if it is read without the actual B1/B3/F3/F4 routing origin.
    It is not a terminal GoodAWACK obstruction. Indeed, if such a family arises from a full-rank
affine change on the active affine span, E10I shows that the TC1/HighTC tensor test is invariant
and no new FreeAffineHighTC certificate is created.
    If it arises from a rank-dropping map with fixing, projection, quotient, local, CKP, Edge,
impossible, or post-terminal analytic origin, then HGO2R reroutes the resulting HighTC certificate
to an already handled class.
    If it arises only from an untagged rank-dropping affine parametrization, then it violates the
E10Y-certified routing grammar and is not an actual terminal GoodAWACK skeleton by E10Y/
E10M.
    Thus the 4𝐴𝑃 -like example remains in the proof as a sharp interface test, while E10X proves
that it has no untagged actual terminal occurrence.
    —

E10X.7. Proof: AFF-Origin Completeness and FreeAffineHighTC Lemma E10K derives
AFF-origin completeness from E10M:


every rank-dropping affine map in an actual terminal GoodAWACK skeleton has an allowed origin tag.
                                                                                    (AFF-OC)
    By E10J, AFF-OC implies RDA, the rank-dropping AFF origin statement. By E10I, RDA
eliminates the remaining matrix-origin class after CRT and full-rank AFF safety. By E10H and
E10G, this eliminates the broad catalogue FreeAffineHighTC class:
                                      𝑅FreeAffineHighTC (𝑁 ) = 0.

    Together with HGO2R, every HighTC GoodAWACK certificate is therefore either origin-
degenerate and routed to CKP, LocalDiag, Edge, or Impossible, or it is an empty FreeAffineHighTC
class.
    —

E10X.8. Output for E10L          The structural input inserted into Theorem E10L.4 is:
                           E10X =⇒ E10K =⇒ 𝑅FreeAffineHighTC (𝑁 ) = 0.
   The TC1 contribution is handled independently by TNGTTHM, namely the chain
         𝐵1-origin coarea → TTH-SC → MRT/TTD → ROC/BRS → TTH → X9L-GT.
   Thus E10L closes GoodAWACK without X8:
                                      𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).
   —

                                                   339
E10X.9. Logical dependencies Internal dependencies: B1, B3, F3, F3A, F3T, F4, E5, BGS,
HGO2R, BAOC, E10G, E10H, E10I, E10J, E10Y, E10M, and E10K. Non-logical verification records
are not logical prerequisites.
    Children served: E10YMX, E10L, E10G, E10H, E10I, E10J, the GoodAWACK manuscript
section, and the E10 finite-grammar appendix.

H.14         E10K affine-origin completeness
H.14.1         E10K. AFF-Origin Completeness
E10K.0. Statement and Role Lemma E10K proves AFF-origin completeness using E10Y and
E10M.
   Lemma E10Y proves that the GoodAWACK routing grammar is complete for actual B1-origin
terminal skeletons. Lemma E10M proves that actual terminal GoodAWACK skeletons contain no
untagged rank-dropping AFF occurrence. Therefore E10K is the AFF-OC consequence of E10Y plus
E10M. Lemma E10X packages this implication as the finite GoodAWACK grammar theorem used by
E10L. Any non-logical verification supplement is retained outside this proof only as reproducibility
support.
   The target was:

            AFF-OC: every rank-dropping affine regrouping in rgrp has an allowed origin tag.

     The outcome is closure by E10Y grammar completeness and the E10M no-untagged-AFF lemma:

                                              AFF-OC follows from E10Y plus E10M.

     Lemma F3 contains the key fact:

                    generic F3 routing operations do not include arbitrary affine regrouping.

   Therefore an untagged rank-dropping affine map cannot be a terminal-routing operation. E10Y
records completeness of the actual B1/B3/F3/F4/E5 operation list, and E10M proves the no-
untagged-rank-drop theorem on that list.
   However, E5, BGS, BAOC, and E10G use broader language around "affine regrouping." That
language is read through the following normalized interface:


affine regrouping may be used only as full-rank coordinate change, tagged projection/fixing, tagged quotient/local operation, or post-terminal analytic slicing.


    With this normalization, made explicit in E10Y and E10M, AFF-OC holds and hence the
structural FreeAffineHighTC obstruction disappears.
    Logical dependencies are B1, B3, F3, F4, E5, BGS, E10G, E10H, E10I, E10J, E10Y, E10M, and
HGO2R. E10K is used by E10X and E10L.
    —




                                                                             340
E10K.1. Setup: Complete Terminal-Routing Operations Lemma F3, Section F3.6, states
that F3 has only the following generic routing-level operations:

  1. controlled CRT absorption;

  2. F4 large-divisor decision;

  3. square-divisor routing;

  4. finite grouping selection/elimination;

  5. terminal LocalDiag detection;

  6. terminal Edge detection by C1P predicates;

  7. terminal class labelling into CKP, GoodAWACK, LongAP/Local, Edge, LocalDiag.

   It also explicitly says that Cauchy/cube and Fourier expansion are not generic F3 routing
operations, but post-terminal proof subroutines.
   We use the corresponding reading:

         arbitrary rank-dropping affine regrouping is not a generic F3 routing operation.
                                                                                (F3-COMPLETE)
   This is not an extra mathematical estimate. It is a bookkeeping consequence of the finite
operation list in F3.6.
   —

E10K.2. Setup: Allowed Meanings of Affine Regrouping Under F3-COMPLETE, every
occurrence of "affine regrouping" in the Branch B infrastructure must be interpreted as one of the
following.

A1. Full-rank coordinate change          The linear part is full-rank on the active affine span. By
E10I, this is tensor-safe:

                                           TC1/HighTC

   is invariant under the induced rationally injective map on symmetric tensors.

A2. Tagged fixing/projection Some coordinates are fixed by dyadic slicing, conditioning,
congruence compatibility, or an already recorded routing restriction.
    This is rank-dropping, but the rank drop is tagged. If it creates a HighTC relation, the relation
is caused by recorded projection data and is not FreeAffine.

A3. Tagged F4 quotient/divisor/local origin           The rank drop is produced by:

  1. fixed divisor quotient;

  2. variable quotient residual;

  3. fixed gcd/local dependence;


                                                341
  4. repeated/proportional forms;

  5. quotient-determined active forms.

   These are exactly F4/BGS origin-degenerate cases and are routed by HGO2R.

A4. Tagged CKP or Edge origin            The rank drop exposes:

  1. B3 CKP-balanced finite-convolution structure;

  2. strict C1P Edge saving;

  3. empty/impossible support.

   These are terminally handled outside GoodAWACK.

A5. Post-terminal analytic slicing Primitive slicing or Cauchy/cube operations may reduce
dimension inside E10’s proof.
    They are not terminal-routing operations generating the GoodAWACK skeleton. For the TC1/
HighTC test, the pre-slicing affine vectors remain the objects being tested; this is the E10H/E10I
interface.
    —

E10K.3. Statement and Proof: AFF-OC after E10Y and E10M

Theorem H.49 (Theorem E10K.1. AFF-origin completeness). By Lemma E10Y, the terminal
GoodAWACK skeleton is generated by the complete GoodAWACK routing grammar. By Lemma
E10M, actual terminal GoodAWACK skeletons contain no untagged rank-dropping AFF occurrence.
   Then every rank-dropping affine map recorded in

                                                rgrp

   for an actual terminal GoodAWACK skeleton has one of the allowed origin tags A2–A5.
   Equivalently:

            there is no untagged rank-dropping AFF map in terminal GoodAWACK.

Proof. Let S be an actual terminal GoodAWACK skeleton produced by

                                      𝐵1 → 𝐵3 → 𝐹 3/𝐹 4.

    This is now a direct consequence of E10Y and E10M. For completeness, we recall the mechanism.
    By Lemma B1, the starting data are product variables and dyadic cells. No affine rank-dropping
map is introduced at B1.
    By Lemma B3, the grouping choices are finite product groupings. They are recorded as grouping
alternatives. If they reveal short factors, CKP-balanced structure, local AP structure, or forced
dependence, the atom is routed to Edge, CKP, LongAP/Local, or LocalDiag. If not, the residual
may feed BranchB/GoodAWACK, but B3 has not introduced an arbitrary rank-dropping affine
map; it has only selected product groupings and tags.



                                                342
   By F3.6, terminal-routing operations are exactly controlled CRT absorption, F4 large-divisor
decision, square-divisor routing, finite grouping selection/elimination, terminal LocalDiag detection,
terminal Edge detection, and terminal class labelling.
   Controlled CRT absorption is finite-index/full-rank on the difference lattice and is tensor-safe
by E10I.
   F4 large-divisor decision produces either:

  1. Edge;

  2. LocalDiag;

  3. CKP;

  4. GoodAWACK after fixed quotient/divisor ambiguity is resolved.

    Any rank drop caused by fixed divisor, variable quotient, gcd-local dependence, or quotient-
determined forms is therefore tagged by F4 origin data.
    Finite grouping selection/elimination does not create an untagged affine slice. It only chooses
among the finitely many B3 product groupings and either terminally routes the atom or eliminates
the grouping.
    Terminal LocalDiag and Edge detections are tagged by their definitions.
    Terminal GoodAWACK labelling is not a new operation. It only declares that after the above
decisions no unresolved Edge, CKP, LongAP/Local, LocalDiag, ordinary divisor, or grouping
alternative remains.
    Therefore any rank-dropping affine map that remains in the terminal GoodAWACK skeleton
must have been one of:

  1. a recorded fixing/projection;

  2. an F4 quotient/divisor/local origin;

  3. a CKP/Edge/impossible origin;

  4. post-terminal analytic slicing not used as the terminal tensor object.

   These are exactly A2–A5.
   Thus no untagged rank-dropping AFF map survives terminal GoodAWACK. E10Y proves that
the preceding list is complete for actual B1-origin terminal skeletons, and E10M proves that each
rank-dropping occurrence in that complete list is tagged. AFF-OC follows. The theorem is proved.
   —


E10K.4. Output for RDA and FreeAffineHighTC                  By E10J, RDA follows from AFF-OC.
  Therefore, using E10Y and E10M:

                                                RDA

   holds.
   By E10I, RDA eliminates the remaining MOR obstruction.
   By E10H, this eliminates the remaining matrix-origin obstruction.
   By E10G and HGO2R, this eliminates the residual FreeAffineHighTC branch:

                                                 343
                                     𝑅FreeAffineHighTC (𝑁 ) = 0.

   Combining with the TC1 global-testing route and origin-degenerate HighTC rerouting gives:

                                     𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 )

   without using X8.
   —

Parameter check H.50 (E10K.5. Parameter Check: Interface Normalization Supplied by E10Y/
E10M/E10K and Consumed by E10L). The proof above depends on reading F3.6 as complete
for actual F3 routing and on excluding post-terminal analytic operations from the list of skeleton
generators. E10Y makes this grammar-completeness statement explicit, E10M proves the no-
untagged-rank-drop theorem on that grammar, and E10K packages the resulting AFF-origin-
completeness interface. E10L consumes the already normalized E10Y/E10M/E10K interface when
estimating the terminal GoodAWACK contribution.
    The broader language in the auxiliary Branch B documents is read as follows to avoid reintro-
ducing an untagged AFF operation:

E5 cleanup    Lemma E5 uses the phrase:

                                         affine regrouping

    among "allowed F3 operations."
    E5 is a conditional content-stability lemma for transports whose origin tags have already been
recorded by B1/B3/F3/F4. It does not obtain its meaning from E10L, and it does not introduce a
new terminal GoodAWACK generator.
    Equivalently, in the structural grammar language:

  1. full-rank affine coordinate changes preserve content;

  2. rank-dropping affine maps are allowed only when tagged by fixing/projection, quotient/divisor/
     local origin, CKP, Edge, impossible, or post-terminal analytic slicing;

  3. Cauchy/cube and primitive slicing are post-terminal E10 proof operations, not terminal-
     routing operations creating new GoodAWACK skeletons.

BGS cleanup      Lemma BGS records

                                                rgrp

   as affine regrouping or affine changes of variables.
   In the clean skeleton record rgrp records only:

  1. B3 product grouping choices;

  2. full-rank coordinate changes;

  3. tagged rank drops of the types A2–A5.



                                                344
BAOC/E10G cleanup The weak BAOC grammar and E10G catalogue do not serve as indepen-
dent terminal generators of arbitrary rank-dropping bounded affine maps. Their broad C5/T5 cell
is normalized by the E10Y/E10M/E10K interface before E10L uses it:

    1. full-rank AFF, tensor-safe;

    2. tagged rank-dropping AFF, origin-degenerate or post-terminal analytic;

    3. forbidden untagged rank-dropping AFF.

      —
                                Remark H.51 (E10K.6. Output).
        AFF-OC is proved for actual terminal GoodAWACK skeletons by E10Y and E10M.
      Mathematical consequence:

                                       𝑅FreeAffineHighTC (𝑁 ) = 0

      inside the active B1/B3/F3/F4/E5 routing tree.
      This is the structural input used by E10L to close the HighTC class without X8.

E10K.7. Logical Dependencies Internal dependencies: B1, B3, F3, F4, E5, BGS, E10G, E10H,
E10I, E10J, E10Y, E10M, and HGO2R.
   Children served: E10L.


I      Final Assembly and Handoff Details
I.1       I1 final weighted assembly
I.1.1      I1. Final Weighted Assembly
I1.0. Statement and Role Lemma I1 is the final weighted assembly theorem. It combines
the exact B1 decomposition, the B3/F3/F4 terminal routing, the Edge/Local/CKP/GoodAWACK
terminal estimates, and the H4M local bridge to prove

                                       𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )

   for all sufficiently large even 𝑁 . The Branch B input is Lemma E10L, which proves
𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ) without using X8.
   Logical dependencies: PAR, GEB, B1, B3, F3, F4, C1, D1, G8a, E10L, H4M, and the H4
component imported through H4M. Outputs served: G1 and G0H.
   —

I1.1. Setup: Inputs        Let 𝑁 be a sufficiently large even integer. The proof-level inputs are:

                                        𝐵1,    𝐵3,    𝐹 3,   𝐹 4,

                         𝐶1,   𝐷1,    𝐺8𝑎,    𝐸10𝐿,     𝐻4𝑀,        𝑃 𝐴𝑅,   𝐺𝐸𝐵.

      The external/standard inputs still visible through these inputs are:


                                                  345
   1. X1, the Heath–Brown identity used in B1;

   2. X9L-GT, the averaged linear/Fourier Liouville input used by E10L through the TC1 coarea
      route after TTH supplies the near-global Davenport/AP range;

   3. X10, the DFI Kloosterman-fraction input used inside CKPX10M, with the smooth-weight
      derivative interface supplied by CKPD before G8a imports the nonzero-frequency conclusion;

   4. X16, only through the BRS/X16 carrier-slice interface supplied by X16BRS and X16C.

   The I1 proof does not use X8.
   —

I1.2. Statement: Theorem I1

Theorem I.1 (Theorem I1). For all sufficiently large even 𝑁 ,

                         𝑅Λ (𝑁 ) =               Λ(𝑛1 )Λ(𝑛2 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).
                                       ∑︁

                                     𝑛1 +𝑛2 =𝑁


   Here
                                      ∏︁ 𝑝 − 1                    ∏︁ (︂           1
                                                                                      )︂
                       S(𝑁 ) = 2𝐶2                ,       𝐶2 =            1−            .
                                      𝑝|𝑁
                                            𝑝−2                   𝑝>2
                                                                               (𝑝 − 1)2
                                      𝑝>2

   —

I1.3. Setup: Exact B1 Decomposition                   By Lemma B1, for fixed sufficiently large 𝐽0 ≥ 𝐽* ,

                                      𝑅Λ (𝑁 ) =               𝑐ℬ 𝑅ℬ (𝑁 ),
                                                       ∑︁

                                                      ℬ∈B𝐽0

   where

                                         #B𝐽0 ≪𝐽0 (log 𝑁 )4𝐽0 .

   This decomposition is exact. No error term is introduced at this stage.
   —

I1.4. Setup: Terminal Routing By Lemma B3, each typed B1 block enters one of the
preliminary routing families:

                     TypeI/Edge,        LongAP/Local,               BranchB,         CKP.

     By Lemma F3, together with the exhaustive large-divisor decision in Lemma F4, every descendant
is finally routed into one of the terminal tagged classes:

             Edge,      LongAP/Local,             CKP,          GoodAWACK,             LocalDiag.

   These terminal classes are disjoint at the tagged routing-history level and exhaust all descendants.
The exact identity used here is Lemma F3.15: for each parent B1 block ℬ,

                                                       346
                                    𝑅ℬ (𝑁 ) =              𝑅ℬ,𝜏 (𝑁 ),
                                                  ∑︁

                                                𝜏 ∈𝒯 (ℬ)

before any terminal estimate is applied. Therefore the total weighted sum decomposes as
       𝑅Λ (𝑁 ) = 𝑅Edge (𝑁 ) + 𝑅LongAP (𝑁 ) + 𝑅CKP (𝑁 ) + 𝑅GoodAWACK (𝑁 ) + 𝑅LocalDiag (𝑁 ).
   —

I1.5. Proof: Edge Contribution By Lemma C1A, every terminal Edge atom carries one of the
strict C1P saving mechanisms. Lemma C1 estimates all atoms satisfying these mechanisms. Hence,
after summing over the polylogarithmic family of B1/B3/F3 descendants,

                                        𝑅Edge (𝑁 ) = 𝑜(𝑁 ).

   Ordinary divisor labels are not counted as Edge unless a strict C1P saving predicate is verified;
otherwise F4 routes them to CKP, LocalDiag, or GoodAWACK.
   —

I1.6. Proof: LongAP/Local Contribution By Lemma D1, including the coefficient-exclusion
Lemma D1.2A, every tagged LongAP/Local atom contains only controlled local AP/congruence
data and equals the explicit LPI local projection of the same tagged B1/F3 cell plus an error 𝑜(𝑁 ).
In particular, the only local replacement is Λ(𝑛) ↦→ Λ𝑄 (𝑛 mod 𝑄). Thus

                               𝑅LongAP (𝑁 ) = 𝑀LongAP (𝑁 ) + 𝑜(𝑁 ).

    The local main term 𝑀LongAP (𝑁 ) is passed to H4M with its parent B1 tag and routing-history
tag; H4M imports the detailed H4 local algebra.
    —

I1.7. Proof: CKP Contribution By Lemma G8a, every tagged CKP atom equals its LPI-
admissible canonical local projection plus an error 𝑜(𝑁 ). Therefore

                                  𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ).

    The nonzero-frequency CKP contribution is handled by CKPX10M, whose external analytic
input is the DFI theorem X10. Lemma G8a imports this nonzero-frequency conclusion and combines
it with the CKP zero-frequency local term, which is admitted by LPI and assembled by H4M.
Equivalently, the CKP local component is imported into I1 through H4M.
    —

I1.8. Proof: Branch B / GoodAWACK Contribution                          By Lemma E10L,

                                    𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).

   Its proof route is:


TC1 split + TC1 Fourier closure + HighTC rerouting + AFF-OC/E10K =⇒ 𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ).
   The GoodAWACK contribution does not use X8. The Branch B external input is the citation-
grade X9L-GT averaged Liouville/Fourier estimate in the near-global Davenport/AP range.
   —

                                                 347
I1.9. Proof: LocalDiag Contribution Terminal LocalDiag atoms are not error terms. They
are canonical local/main terms admitted by the H4M local bridge. Let

                                            𝑀LocalDiag (𝑁 )

   be their total tagged local contribution. These terms are included in the local main sum together
with LongAP/Local and CKP zero-frequency terms.
   —

I1.10. Proof: Local/Main Compatibility            Collect all canonical local terms:

                      𝑀local (𝑁 ) = 𝑀LongAP (𝑁 ) + 𝑀CKP (𝑁 ) + 𝑀LocalDiag (𝑁 ).

    There is no fourth local summand. By Lemma H4M, every auxiliary local-looking term produced
by controlled CRT absorption, fixed-divisor quotienting, or primitive local slicing is a tagged subterm
of one of the three displayed classes. Endpoint and smooth-boundary localizations are C1 Edge
errors and are not part of 𝑀local . Moreover every active local/main term satisfies the explicit tagged
admission condition
                                local
                               𝑀ℬ,𝜏   (𝑁 ) = Loc𝑄 𝑅ℬ,𝜏 (𝑁 ) + 𝑜ℬ,𝜏 (𝑁 ),
where Loc𝑄 is the single Λ𝑄 -replacement inside the same parent B1 block and routing cell. H4M
packages the H4 reconstruction of the local Goldbach model by tagged linearity over the exact B1/
F3 partition, the no-double-counting lemma, and the finite CRT local factor calculation. Thus there
is no branch-specific local surrogate and

                                    𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   —

I1.11. Proof: Final Summation          Using the terminal decomposition and the branch estimates:

       𝑅Λ (𝑁 ) = 𝑅Edge (𝑁 ) + 𝑅LongAP (𝑁 ) + 𝑅CKP (𝑁 ) + 𝑅GoodAWACK (𝑁 ) + 𝑅LocalDiag (𝑁 ),

                          𝑅Edge (𝑁 ) = 𝑜(𝑁 ),     𝑅GoodAWACK (𝑁 ) = 𝑜(𝑁 ),

             𝑅LongAP (𝑁 ) = 𝑀LongAP (𝑁 ) + 𝑜(𝑁 ),       𝑅CKP (𝑁 ) = 𝑀CKP (𝑁 ) + 𝑜(𝑁 ),

   and LocalDiag contributes only canonical local terms. GEB records that the branch 𝑜(𝑁 ) terms
above, including all polylogarithmic terminal summations, CKP derivative losses, TC1 Davenport/
AP losses, X16/BRS slice-floor losses, and H4M local-bridge boundary terms, combine to a single
𝑜(𝑁 ) remainder. Hence

                                    𝑅Λ (𝑁 ) = 𝑀local (𝑁 ) + 𝑜(𝑁 ).

   By H4M,

                                    𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Therefore

                                     𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   This proves Theorem I1.
   —

                                                 348
I1.12. Output: Positivity Handoff to G1/G2                       For even 𝑁 ,

                                               S(𝑁 ) ≥ 2𝐶2 > 0.

      Therefore I1 implies

                                                     𝑅Λ (𝑁 ) > 0

   for all sufficiently large even 𝑁 , once the 𝑜(𝑁 ) error is smaller than 𝐶2 𝑁 . This is only the
weighted positivity statement; the genuine prime representation uses G2 to remove nontrivial prime
powers and G1/G0H to convert positive genuine prime-pair weight into an actual prime pair.
   The final passage from the weighted asymptotic to a representation by two primes uses:

     1. Lemma G2, prime powers negligible;
     2. Lemma G1, passage from the genuine prime-pair asymptotic to strong Goldbach.

      —
                                      Remark I.2 (I1.13. Output).
               I1 proves the final weighted assembly using E10L as the Branch B input.
      Thus

                                          𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

      Together with G2 and G1, this proves the root Goldbach statement for all sufficiently large even
𝑁.

I1.14. Logical Dependencies External dependencies: X1 through B1, X9L-GT through E10L/
TTH, and X10 through CKPX10M/CKPD.
   Internal dependencies: PAR, GEB, B1, B3, F3, F4, C1, D1, G8a, E10L, H4M, and H4 through
H4M.
   Children served: G1 and G0H.

I.2       G2 prime powers negligible
I.2.1      G2. Prime Powers Negligible Lemma
G2.0. Statement and Role            Lemma G2 is needed to pass from the weighted asymptotic

                             𝑅Λ (𝑁 ) =               Λ(𝑛1 )Λ(𝑛2 ) = S(𝑁 )𝑁 + 𝑜(𝑁 )
                                           ∑︁

                                         𝑛1 +𝑛2 =𝑁

   to an actual representation of 𝑁 as a sum of two primes. The von Mangoldt function is supported
not only on primes, but also on prime powers:

                                                     log 𝑝,   𝑛 = 𝑝𝑘 , 𝑘 ≥ 1,
                                                {︃
                                     Λ(𝑛) =
                                                     0,       otherwise.

    Therefore we have to show that the contribution of representations in which at least one summand
is a nontrivial prime power 𝑝𝑘 , 𝑘 ≥ 2, is small compared with the main term ≍ 𝑁 .
    Logical dependencies: elementary prime-power counting. If combined with I1, the output served
is G1/G0H.
    —

                                                          349
G2.1. Setup: Prime-Prime and Prime-Power Decomposition                                Define

                                𝑅𝑝𝑝 (𝑁 ) =                  log 𝑝1 log 𝑝2 .
                                                 ∑︁

                                              𝑝1 +𝑝2 =𝑁
                                             𝑝1 ,𝑝2 prime

  The sum is over ordered positive prime pairs, matching the ordered convention for 𝑅Λ (𝑁 ).
  This is the genuine prime-prime contribution.
  Denote the contribution of nontrivial prime powers by

                                  𝑅pow (𝑁 ) = 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ).

  Then 𝑅pow (𝑁 ) consists of pairs

                                          𝑛1 + 𝑛 2 = 𝑁

  such that at least one of 𝑛1 , 𝑛2 has the form

                                          𝑝𝑘 ,        𝑘 ≥ 2.

  The lemma aims to prove:

                             𝑅pow (𝑁 ) = 𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).

  —

G2.2. Proof: Counting Nontrivial Prime Powers                     Let

                              𝒫2 (𝑁 ) = {𝑝𝑘 ≤ 𝑁 : 𝑝 prime, 𝑘 ≥ 2}.

  Then

                                        #𝒫2 (𝑁 ) ≪ 𝑁 1/2 .

  Indeed, if 𝑝𝑘 ≤ 𝑁 and 𝑘 ≥ 2, then

                                             𝑝 ≤ 𝑁 1/2 .

  For each prime 𝑝 ≤ 𝑁 1/2 , the number of possible exponents 𝑘 ≥ 2 is at most

                                             𝑂(log 𝑁 ).

  A crude bound therefore gives

                                     #𝒫2 (𝑁 ) ≪ 𝑁 1/2 log 𝑁.

  A sharper elementary bound removes the extra logarithm:

             #𝒫2 (𝑁 ) ≤ #{𝑝2 ≤ 𝑁 } +         #{𝑝𝑘 ≤ 𝑁 } ≪ 𝑁 1/2 +
                                        ∑︁                                    ∑︁
                                                                                    𝑁 1/𝑘 ≪ 𝑁 1/2 .
                                       𝑘≥3                                    𝑘≥3

  Indeed, the dominant contribution is from squares.
  —

                                                 350
G2.3. Proof: Weighted Bound for Bad Pairs             For every 𝑛 ≤ 𝑁 , we have

                                          Λ(𝑛) ≤ log 𝑁.

   If a representation counted in 𝑅pow (𝑁 ) has 𝑛1 ∈ 𝒫2 (𝑁 ), then 𝑛2 = 𝑁 − 𝑛1 is determined. Its
contribution is at most

                                     Λ(𝑛1 )Λ(𝑛2 ) ≤ (log 𝑁 )2 .

   Thus the contribution of pairs with first coordinate a nontrivial prime power is

                              ≪ #𝒫2 (𝑁 )(log 𝑁 )2 ≪ 𝑁 1/2 (log 𝑁 )2 .

   The same estimate holds for pairs with second coordinate a nontrivial prime power. Hence, by
the union bound,

                                   𝑅pow (𝑁 ) ≪ 𝑁 1/2 (log 𝑁 )2 .

   Therefore

                                        𝑅pow (𝑁 ) = 𝑜(𝑁 ).

   —

G2.4. Output: Consequence for the Genuine Prime-Prime Sum                  Since

                                 𝑅Λ (𝑁 ) = 𝑅𝑝𝑝 (𝑁 ) + 𝑅pow (𝑁 ),

   and

                                        𝑅pow (𝑁 ) = 𝑜(𝑁 ),

   we get

                                   𝑅𝑝𝑝 (𝑁 ) = 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).

   Using I1,

                                   𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ),

   we obtain

                                   𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

   Thus the weighted contribution from genuine prime pairs has the same main term as the full
von Mangoldt sum. The singular series is the H4M/I1 Goldbach singular series; G2 only removes
nontrivial prime-power support and does not alter the local factor.
   —




                                                351
G2.5. Statement and Proof: Lemma G2
Lemma I.3 (Lemma G2). Let

                                  𝑅𝑝𝑝 (𝑁 ) =                  log 𝑝1 log 𝑝2 .
                                                   ∑︁

                                                𝑝1 +𝑝2 =𝑁
                                               𝑝1 ,𝑝2 prime

      Then

                           𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).

      Consequently, if

                                     𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ),

      then

                                     𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

Proof. The difference 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) is the nonnegative contribution of representations where
at least one summand is a nontrivial prime power 𝑝𝑘 , 𝑘 ≥ 2. There are 𝑂(𝑁 1/2 ) such possible
summands up to 𝑁 , and for each such summand the other summand is uniquely determined. Since
Λ(𝑛) ≤ log 𝑁 , each weighted contribution is at most (log 𝑁 )2 . Therefore the total contribution is

                                     𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).

      The consequence follows by subtracting this negligible term from I1. Lemma proved.
      —


          Remark I.4 (G2.6. Output). 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑂(𝑁 1/2 (log 𝑁 )2 ) = 𝑜(𝑁 ).
   Hence the genuine prime-prime weighted sum has the same main term as 𝑅Λ (𝑁 ). This is the
input from G2 used by G1 and G0H.

G2.7. Logical Dependencies External dependencies: elementary prime-power counting and the
bound Λ(𝑛) ≤ log 𝑁 .
   Internal dependencies: I1 only for the stated consequence 𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).
   Children served: G1 and G0H.

I.3     G1 weighted asymptotic to primes
I.3.1     G1. Passage from Weighted Asymptotic to Strong Goldbach
G1.0. Statement and Role Theorem G1 is the final passage from the weighted asymptotic to
the strong Goldbach statement for all sufficiently large even integers.
   From I1 we have:

                                     𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

  The singular series here is the Goldbach Euler product obtained in H4 from the finite local
model, not a branch-specific local surrogate.
  From G2 we have:

                                                   352
                                        𝑅𝑝𝑝 (𝑁 ) = 𝑅Λ (𝑁 ) + 𝑜(𝑁 ),

     where

                                   𝑅𝑝𝑝 (𝑁 ) =                    log 𝑝1 log 𝑝2 .
                                                      ∑︁

                                                   𝑝1 +𝑝2 =𝑁
                                                  𝑝1 ,𝑝2 prime

  The sum is over ordered positive prime pairs, matching the convention in 𝑅Λ (𝑁 ) =
 𝑛1 +𝑛2 =𝑁 Λ(𝑛1 )Λ(𝑛2 ).
∑︀

  We have to prove that, for all sufficiently large even 𝑁 ,

                                               𝑅𝑝𝑝 (𝑁 ) > 0.

     This immediately implies the existence of primes 𝑝1 , 𝑝2 such that

                                              𝑁 = 𝑝1 + 𝑝2 .

     Logical dependencies: I1 and G2. Output served: G0 and G0H.
     —

G1.1. Setup: Positivity of the Singular Series                    For even 𝑁 , the Goldbach singular series is
                                                            ∏︁ 𝑝 − 1
                                         S(𝑁 ) = 2𝐶2                       ,
                                                           𝑝|𝑁
                                                                  𝑝−2
                                                           𝑝>2

     where
                                            ∏︁ (︂           1
                                                                      )︂
                                    𝐶2 =            1−                     > 0.
                                            𝑝>2
                                                         (𝑝 − 1)2

     Each factor
                                                     𝑝−1
                                                     𝑝−2

     is positive for 𝑝 > 2. Therefore

                                                  S(𝑁 ) > 0

     for every even 𝑁 .
     Moreover, since each factor (𝑝 − 1)/(𝑝 − 2) > 1, we have the uniform lower bound

                                            S(𝑁 ) ≥ 2𝐶2 > 0

     for even 𝑁 .
     —




                                                      353
G1.2. Proof: Positivity of the Genuine Prime-Pair Weighted Sum                            By I1 and G2,

                                    𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

  Since S(𝑁 ) ≥ 2𝐶2 > 0, we have

                                          S(𝑁 )𝑁 ≥ 2𝐶2 𝑁.

  The error term 𝑜(𝑁 ) satisfies, for sufficiently large 𝑁 ,

                                           |𝑜(𝑁 )| ≤ 𝐶2 𝑁.

  Hence for sufficiently large even 𝑁 ,

                               𝑅𝑝𝑝 (𝑁 ) ≥ 2𝐶2 𝑁 − 𝐶2 𝑁 = 𝐶2 𝑁 > 0.

  Thus

                                            𝑅𝑝𝑝 (𝑁 ) > 0.

  —

G1.3. Proof: From Positivity to a Prime Representation                         By definition,

                                 𝑅𝑝𝑝 (𝑁 ) =                  log 𝑝1 log 𝑝2 .
                                                  ∑︁

                                               𝑝1 +𝑝2 =𝑁
                                              𝑝1 ,𝑝2 prime

  Each summand is nonnegative, and it is strictly positive exactly when there is a representation

                                              𝑁 = 𝑝1 + 𝑝2

  with 𝑝1 , 𝑝2 prime.
  If no such representation existed, then the sum would be empty and

                                            𝑅𝑝𝑝 (𝑁 ) = 0.

  But for sufficiently large even 𝑁 we have shown

                                            𝑅𝑝𝑝 (𝑁 ) > 0.

  Therefore at least one prime pair exists.
  —




                                                  354
G1.4. Statement and Proof: Theorem G1

Theorem I.5 (Theorem G1). Assume I1 and G2:

                                       𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ),

      and

                                       𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑜(𝑁 ).

      Then every sufficiently large even integer 𝑁 can be represented as a sum of two primes:

                                              𝑁 = 𝑝1 + 𝑝2 .

Proof. By G2,

                                       𝑅𝑝𝑝 (𝑁 ) = 𝑅Λ (𝑁 ) + 𝑜(𝑁 ).

      By I1,

                                      𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

    For even 𝑁 , the singular series satisfies S(𝑁 ) ≥ 2𝐶2 > 0. Therefore 𝑅𝑝𝑝 (𝑁 ) > 0 for all sufficiently
large even 𝑁 . Since 𝑅𝑝𝑝 (𝑁 ) is a sum of positive weights log 𝑝1 log 𝑝2 over prime representations
𝑝1 + 𝑝2 = 𝑁 , positivity implies that at least one such representation exists. The theorem is proved.
    —


Remark I.6 (G1.5. Output). I1 and G2 imply strong Goldbach for all sufficiently large even 𝑁.
  The only ingredients used at this final stage are:

   1. the asymptotic 𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ) from I1;

   2. the prime-power removal 𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑜(𝑁 ) from G2;

   3. positivity of the Goldbach singular series for even 𝑁 .

G1.6. Logical Dependencies External dependencies: standard positivity of the singular series
Euler product.
   Internal dependencies: I1 and G2.
   Children served: G0 and G0H.

I.4     G0 final handoff verification
I.4.1       G0H. Final Handoff from I1/G2 to Strong Goldbach
G0H.0. Statement and Role            Lemma G0H records the final proof-tree handoff

                                       𝐼1 + 𝐺2 =⇒ 𝐺1 =⇒ 𝐺0.

   It proves that the weighted von Mangoldt asymptotic in I1 implies the existence of a genuine
prime representation after the prime-power contribution is removed by G2.
   Logical dependencies: I1, G2, and G1. Output served: G0.

                                                   355
G0H.1. Setup: Ordered-Pair Conventions All Goldbach sums below are over ordered positive
pairs.

                                  𝑅Λ (𝑁 ) :=                   Λ(𝑛1 )Λ(𝑛2 ),
                                                     ∑︁

                                                  𝑛1 +𝑛2 =𝑁
                                                   𝑛1 ,𝑛2 ≥1

   and

                                𝑅𝑝𝑝 (𝑁 ) :=                      log 𝑝1 log 𝑝2 .
                                                      ∑︁

                                                   𝑝1 +𝑝2 =𝑁
                                                  𝑝1 ,𝑝2 prime

   This convention matches B1, I1, G2, and G1. Using unordered pairs would only change the
normalization by a bounded factor, but the proof tree uses the ordered convention throughout.

G0H.2. Setup: Input from I1         I1 proves that for all sufficiently large even 𝑁 ,

                                    𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).                                            (1)
    Here S(𝑁 ) is the singular series reconstructed in H4 from the finite local model and then used
in I1:
                                  ∏︁ 𝑝 − 1                       ∏︁ (︂         1
                                                                                       )︂
                    S(𝑁 ) = 2𝐶2               ,        𝐶2 =              1−                 > 0.
                                  𝑝|𝑁
                                        𝑝−2                      𝑝>2
                                                                            (𝑝 − 1)2
                                  𝑝>2


   The positivity of 𝐶2 follows from convergence of 𝑝>2 (𝑝 − 1)−2 .
                                                               ∑︀

   For even 𝑁 , every factor (𝑝 − 1)/(𝑝 − 2) with 𝑝 > 2 is > 1, hence

                                          S(𝑁 ) ≥ 2𝐶2 > 0.                                               (2)

G0H.3. Setup: Input from G2             G2 proves

                          𝑅Λ (𝑁 ) − 𝑅𝑝𝑝 (𝑁 ) = 𝑂 𝑁 1/2 (log 𝑁 )2 = 𝑜(𝑁 ).                                (3)
                                                       (︀                   )︀


   The support of the difference consists exactly of ordered pairs (𝑛1 , 𝑛2 ) with 𝑛1 + 𝑛2 = 𝑁 ,
Λ(𝑛1 )Λ(𝑛2 ) ̸= 0, and at least one coordinate a nontrivial prime power 𝑝𝑘 , 𝑘 ≥ 2.
   Indeed, if both coordinates are primes, the pair is counted in 𝑅𝑝𝑝 ; otherwise any nonzero
Λ-contribution has at least one nontrivial prime-power coordinate.
   The elementary count is:

              #{𝑝𝑘 ≤ 𝑁 : 𝑝 prime, 𝑘 ≥ 2} ≤ 𝜋(𝑁 1/2 ) +                             𝜋(𝑁 1/𝑘 ) ≪ 𝑁 1/2 .
                                                                          ∑︁

                                                                     3≤𝑘≤log2 𝑁

   For each selected nontrivial prime power, the other coordinate is determined, and the weight is
at most (log 𝑁 )2 . A union bound over the two coordinates gives (3). Double-counting pairs where
both coordinates are nontrivial prime powers is harmless because this is only an upper bound.




                                                      356
G0H.4. Proof: Positivity of the Genuine Prime-Pair Sum                 Combining (1) and (3),

                            𝑅𝑝𝑝 (𝑁 ) = 𝑅Λ (𝑁 ) + 𝑜(𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).                           (4)
    By (2),

                                          S(𝑁 )𝑁 ≥ 2𝐶2 𝑁.

    The total 𝑜(𝑁 ) error in (4) is eventually bounded in absolute value by 𝐶2 𝑁 . Therefore, for all
sufficiently large even 𝑁 ,

                                𝑅𝑝𝑝 (𝑁 ) ≥ 2𝐶2 𝑁 − 𝐶2 𝑁 = 𝐶2 𝑁 > 0.                                (5)

G0H.5. Proof: Positivity Implies a Prime Representation Every summand in 𝑅𝑝𝑝 (𝑁 ) is
nonnegative, and it is strictly positive for every actual prime pair because log 𝑝 > 0 for every prime
𝑝. If no prime pair 𝑝1 + 𝑝2 = 𝑁 existed, then 𝑅𝑝𝑝 (𝑁 ) would be an empty sum and hence would
equal 0.
    But (5) gives 𝑅𝑝𝑝 (𝑁 ) > 0. Hence there exists at least one ordered pair of primes (𝑝1 , 𝑝2 ) such
that

                                            𝑁 = 𝑝1 + 𝑝2 .

    This is exactly strong Goldbach for sufficiently large even 𝑁 .
Parameter check I.7 (G0H.6. Parameter and Interface Checks).             1. The ordered-pair conven-
    tion is consistent across B1, I1, G2 and G1.

    2. The use of G2 is essential: positivity of 𝑅Λ (𝑁 ) alone would not exclude the possibility that
       the mass came from prime powers, while (3) excludes this at 𝑜(𝑁 ) scale.

    3. No cancellation is hidden in G2, since Λ(𝑛) ≥ 0.

    4. The lower bound S(𝑁 ) ≥ 2𝐶2 is uniform for even 𝑁 , so the final positivity is not vulnerable
       to the size of the prime divisors of 𝑁 .

    5. The theorem obtained is only the ledger target 𝐺0: sufficiently large even 𝑁 . No finite
       verification for small even 𝑁 is included here.
       Remark I.8 (G0H.7. Output). The final handoff from I1 and G2 to G0 is proved.
   I1 plus G2 gives 𝑅𝑝𝑝 (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ). Since S(𝑁 ) ≥ 2𝐶2 > 0 for even 𝑁 , the genuine
prime-pair sum is positive for all sufficiently large even 𝑁 . Therefore G1 derives the existence of a
prime representation, and the root target G0 follows.

G0H.8. Logical Dependencies External dependencies: elementary positivity of the singular
series Euler product and Λ(𝑛) ≥ 0.
    Internal dependencies: I1, G2, and G1.
    Children served: G0.


J     Dependency Ledger and Synchronization Notes
This appendix records the active dependency ledger used by the proof.

                                                 357
J.1   Proof tree and ledger
Proof Strategy. The proof route is:

  1. B1 expands 𝑅Λ (𝑁 ) by a fixed-depth Heath–Brown typed dyadic decomposition.

  2. B3/F3P/F3/F3T/F4 route every descendant into intrinsic terminal classes: Edge, LongAP/
     Local, CKP, GoodAWACK, LocalDiag.

  3. C1P defines the strict Edge predicates; C1A verifies Edge admission; C1 proves admitted
     strict Edge terms are 𝑜(𝑁 ).

  4. LPI first defines the common tagged local projection interface. D1 then excludes surviving
     nonlocal arithmetic coefficients from LongAP/Local atoms. B1LD identifies the B1 local-
     density normalization, G8a supplies the CKP zero-frequency local input, and H4 evaluates
     the finite tagged local algebra. H4M packages these inputs into the single local bridge
     𝑀local (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ), with no fourth residual local class.

  5. G8a handles CKP: zero frequency is local through G8a/LPI, while nonzero frequencies are
     reduced to DFI and summed by the CKPX10M master theorem.

  6. E10L handles GoodAWACK through TC1/HighTC splitting, the TNGTTHM near-global-or-
     routed TC1 theorem, and the E10YMX finite-grammar closure.

  7. H4M imports the H4 finite local-factor computation and assembles all admitted local/main
     pieces into S(𝑁 )𝑁 + 𝑜(𝑁 ).

  8. I1 gives 𝑅Λ (𝑁 ) = S(𝑁 )𝑁 + 𝑜(𝑁 ).

  9. G2 removes prime powers and G1/G0H hand off to a genuine prime pair.

   The X16-Core input for step 6 is proved by the Shiu/AP divisor-correlation argument X16C.
The CKP/X10 derivative check required for step 5 is supplied by the CKPD appendix and packaged
with DFI matching and excluded-range routing by CKPX10M.
   —


Compact Proof Tree The following diagram is a reader map for the active proof dependencies.
An arrow A-->Bmeans that A is used as an input for proving, routing, or assembling B. The ledger
table below remains the authoritative parent/child record.




                                              358
                                                                                                                    X12
                                                                    C1
       PAR     GEB                                    C1A
                                                                   Edge                                             G2
                                                                                                               prime powers
                                                                                                                                            G0
                                                                                                                              G1   G0H
                             F3                                                                       I1                                 Goldbach
      X1, X2   B1    B3              F4                D1
                          F3A, F3T                                  H4                        weighted asymptotic
                                                LPI             local/main
                                          local interface
                                                       B1LD

                                                                   CKPD
                                                      G1a          X10
                                                                                    G4a
                                                      G2a
                                                                                    G8a
                                                      G3a
                                                                   X10ER
                                                              excluded ranges
                                                                                                    E10Y
                                                                   BGS
                                                                                                    E10M
                                                       E5         HGO2R          E10G–E10J
                                                                                                    E10X
                                                                   BAOC
                                                                                                    E10K
                                                                                                                  E10L
                                                                                                               GoodAWACK
                                                                                    TNG
359




                                                                                  TGT+MRT
                                                                                TTD+ROC+BRS
                                                                   X16
                                                                                                    TTH
                                                                   X16C
                                                                                                   X9L-GT
                                                                  X16BRS
   Text fallback:

G0
+-- G0H
    +-- G1
    +-- G2
    +-- I1
+-- G1
    +-- G2
    +-- I1
        +-- PAR -> GEB
        +-- Decomposition/routing:
        |   +-- B1/X1/PAR -> B3 -> F3P -> F3/F3A/F3T -> F4
        |   +-- F3F4M master routing theorem packages the F3/F4 partition
        |   +-- E5 stability and transport compatibility
        +-- Edge:
        |   +-- C1P -> C1A -> C1
        +-- Local/Main:
        |   +-- F3F4M + LPI -> D1 + B1LD + G8a + H4 -> H4M
        +-- CKP:
        |   +-- G1a -> G2a -> G3a -> CKPD + G4a/X10 -> CKPX10M
        |   +-- CKPX10M routes excluded nonzero CKP ranges through X10ER -> C1P/C1A/C1
        |   +-- h=0 goes through G8a/LPI
        |   +-- G8a + B1LD -> H4M
        +-- GoodAWACK:
            +-- E10L
                 +-- TGD
                 +-- TC1: TNGTTHM = TGT-MF + TGT + TTH-SC + MRT + TTD + ROC + BRS
                 |        + X16BRS/X16C + TTH + X9L-GT
                 +-- HighTC/grammar:
                          BGS + HGO2R + BAOC + E10G/E10H/E10I/E10J
                          + E10YMX = E10Y + E10M + E10K + E10X
                          + E5-clean interface imported from the E5 master proof

   —

Proof Ledger Table. Direction convention. In each row, ID Parent lists proof nodes whose proofs
use the row’s ID. It is an output/served-by column, not a dependency column. ID Child lists proof
nodes used to prove the row’s ID. Thus a later branch lemma appearing in the ID Parent column
of a routing lemma means only that the branch consumes the routing output; it does not mean that
the routing lemma depends on the branch estimate.




                                              360
      ID    ID Parent            ID Child               Element                 Role                    Proof status           Risk
      G0    –                    G1, G2, G0H            Strong Goldbach tar-    Final target            Derived from G1/       Publication-source in-
                                                        get                                             G0H once the active    tegration and external
                                                                                                        internal lemmas and    citation verification
                                                                                                        external dependencies
                                                                                                        are accepted
      G0H   G0, G1               I1, G2, G1             Final handoff           Ordered-pair and posi- Proved                  Low
                                                                                tivity normalization
      G1    G0, G0H              I1, G2                 Weighted asymptotic     Converts 𝑅𝑝𝑝 (𝑁 ) > 0 Proved from I1 and G2 Low
                                                        to primes               to a prime representa-
                                                                                tion
      G2    G0, G1, G0H          X12                    Prime powers negligi-   Removes non-prime       Proved                 Low
                                                        ble                     Λ-support
      I1    G1, G0H              B1, B3, F3P, F3, F4,   Final weighted assem-   Proves 𝑅Λ (𝑁 ) =        Proved from termi-     Medium / depends
                                 C1P, C1A, C1, D1,      bly                     S(𝑁 )𝑁 + 𝑜(𝑁 )          nal branch inputs;     on branch inputs and
                                 G8a, CKPX10M,                                                          CKP/X10 is imported manuscript synchro-
                                 E10L, H4M, PAR,                                                        through CKPX10M/       nization
                                 GEB                                                                    G8a, GoodAWACK
                                                                                                        through E10L, lo-
                                                                                                        cal/main through
                                                                                                        H4M, and summability
                                                                                                        through GEB
361




      PAR   B1, C1, BRS, TTH,    –                      Parameter register      Records order of        Contains the X16 and Medium / maintenance
            G3a, G8a, CKPD, X10,                                                choices, constants,     CKP/X10 derivative
            GEB, I1                                                             notation conventions, constants and the
                                                                                and a concrete consis- nonempty hierarchy
                                                                                tency witness           witness 𝐽0 = 20, 𝜂 =
                                                                                                        1/40, 𝜃 = 1/4000
      GEB   I1                   PAR, B1, B3, F3P,  Global error budget         Records terminal        Provides the readable Low-Medium / mainte-
                                 F3, F3T, F4, C1P,                              branch 𝑜(𝑁 ) sources,   global loss table at-  nance
                                 C1A, C1, D1, G8a,                              logarithmic losses,     tached to the witness
                                 CKPX10M, CKPD,                                 power savings, polylog- 𝐽0 = 20, 𝜂 = 1/40, 𝜃 =
                                 E10L, E10YMX,                                  arithmic summations, 1/4000, including the
                                 TNGTTHM, TNG,                                  and order of constants; H4M no-residual-local
                                 X16BRS, X16C, TTH,                             prime-power removal is bridge
                                 H4, H4M                                        handled separately by
                                                                                G2
      B1    B3, F3P, I1, H4, H4M, X1, X2, PAR           Typed Heath–Brown       Exact decomposition     Proved                 Low
            BGS, E10Y, E10M,                            decomposition
            E10X, E10YMX
      B3    F3P, F3, I1, H4M,     B1, X2                Block classification    Finite candidate gener- Proved at routing level Low-Medium
            BGS, E10Y, E10M,                                                    ation
            E10X, E10YMX
      F3P   F3, F3T, F3F4M, F4,    B1, B3, C1P, PAR      Intrinsic terminal    Defines Edge, CKP,       Edge is the strict C1P Low-Medium
            D1, LPI, H4, H4M, I1                         predicate catalogue   GoodAWACK, Local- saving predicate; Lon-
                                                                               Diag, and LongAP/        gAP/Local is a pos-
                                                                               Local before any termi- itive local-coefficient
                                                                               nal estimate is invoked predicate in the finite
                                                                                                        local algebra Cloc (𝑄);
                                                                                                        D1/H4M consume this
                                                                                                        predicate rather than
                                                                                                        defining it by down-
                                                                                                        stream exclusion
      F3    I1, E10L, E10Y,      B1, B3, F3P, C1P, F4, Routing exhaustion / Every descendant            Uses the intrinsic      Low-Medium
            E10X, E10YMX, F4,    E5, LPI, PAR          partition identity      reaches a tagged termi- terminal predicates
            E10M, H4, H4M, F3T,                                                nal class                fixed by F3P and the
            F3F4M                                                                                       strict Edge predicates
                                                                                                        fixed by C1P; terminal
                                                                                                        estimates are proved
                                                                                                        later by the branch
                                                                                                        lemmas
      F3A   F3, F3F4M, E10Y,     B1, B3, F3P, F3, F3T, F3.6 routing verifica-  Proves completeness      Active verification;    Low-Medium
            E10M, E10X, E10K,    F4, E5                tion                    of the F3.6 operation    E10 consumes this
            E10YMX, E10L                                                       list from the residual   interface but is not a
                                                                               obstruction set and      prerequisite
                                                                               finite grouping set
362




      F3T   I1, F3A, F3F4M, H4M, B1, B3, F3P, C1P,     Complete finite routing Expands B1/B3/F3/        Finite structural rout- Low-Medium / table
            E10Y, E10M, E10X,    F3.1–F3.15, F4, E5,   table                   F4 routing into a        ing table associated    maintenance
            E10K, E10YMX, E10L LPI, PAR                                        publication-grade table with F3; it is not a
                                                                               by B1 type, B3 group- new hypothesis for F3
                                                                               ing, dyadic regime,      and no downstream
                                                                               divisor/conductor        branch estimate is
                                                                               state, coefficient type, used
                                                                               terminal class, and
                                                                               exclusion reason
      F4    F3, F3F4M, C1A,      F3P, C1P, F3.1–F3.6, Large divisor routing    Prevents false Edge      Uses only the F3 atom Low-Medium
            C1, H4M, E10L,       E5, LPI, X6                                   labels                   interface, F3P termi-
            E10Y, E10M, E10X,                                                                           nal predicates, C1P
            E10YMX, BRS                                                                                 strict Edge predicates,
                                                                                                        and routing-measure
                                                                                                        definitions; outputs
                                                                                                        structural terminal
                                                                                                        tags that C1/D1/G8a/
                                                                                                        E10L/H4M later esti-
                                                                                                        mate or assemble
      F3F4M   H4M, I1, BGS,           B1, B3, F3P, F3, F3A, Master routing exhaus- Packages the F3/F4       Active reader-facing    Low-Medium
              HGO2R, E10Y,            F3T, F4, E5, LPI,     tion theorem           routing layer into a     compressed proof
              E10M, E10X, E10K,       C1P, PAR                                     standalone finite par- chapter; no terminal
              E10YMX, E10L,                                                        tition theorem with      estimates and no down-
              TNGTTHM                                                              explicit state space,    stream branch theorem
                                                                                   allowed transitions,     are used
                                                                                   routing measure, ter-
                                                                                   minal predicates, exact
                                                                                   partition identity, ter-
                                                                                   minal exhaustiveness,
                                                                                   and no-sixth-class con-
                                                                                   clusion
      LPI     D1, G8a, B1LD, H4,      B1, B3, F3P, finite   Preliminary local pro- Defines Λ𝑄 , Loc𝑄 ,      Uses only the already Low
              H4M, I1                 CRT local algebra     jection/admission      LPI-admissibility con- tagged B1/B3/F3P
                                                            interface              sumed by H4/H4M,         local-source vocabulary
                                                                                   and the exact local      and finite CRT local
                                                                                   source classes           algebra. It does not
                                                                                                            use F4, D1, G8a, H4,
                                                                                                            or H4M as theorem
                                                                                                            inputs; those nodes
                                                                                                            consume LPI.
      C1P     F3P, F3, F3T, F4,       B1, B3, PAR           Strict Edge predicate  Defines the seven Edge Predicate layer only: it Low-Medium
              C1A, C1, G8a,                                 catalogue              certificates E1–E7       supplies the meaning
363




              CKPX10M, X10ER,                                                      independently of G8a, of Edge; C1A verifies
              BRS, TTH, TNGT-                                                      X10, BRS, X16BRS,        source-specific admis-
              THM, E10L, H4M,                                                      X16C, H4, H4M, and       sions and C1 supplies
              GEB, I1                                                              E10L                     estimates
      C1A     C1, I1, F3T, F4, G8a,   C1P, B1, B3, F3P, F3, Edge admission ledger Verifies that every       Active admission table Low-Medium / table
              CKPX10M, X10ER,         F3T, F4, G2a, PAR                            active nonzero Edge      for C1 inputs. Down- maintenance
              BRS, TTH, TNGT-                                                      input carries a strict   stream CKP/BRS/
              THM, E10L, H4M                                                       C1P predicate E1–E7      X16/TTH/H4M nodes
                                                                                                            submit source rows
                                                                                                            checked against C1P/
                                                                                                            C1A, but they do not
                                                                                                            define Edge and are
                                                                                                            not theorem inputs to
                                                                                                            C1A.
      C1      I1, F3, F4, G8a,        C1P, X3, X15, X16,    Unified Edge estimate Terminal Edge atoms Estimates atoms sat-          Medium
              CKPX10M, E10L,          PAR                                          give 𝑜(𝑁 )               isfying C1P; paired
              BRS, TNGTTHM,                                                                                 with C1A admission
              X10ER, H4M                                                                                    for proof-tree inputs
      D1     I1, H4, H4M           B1, B3, F3P, F3, F4,   LongAP/Local         Expands F3P-LongAP/ Lemma D1.2A is now Low-Medium
                                   C1P, C1A, C1, E5,                           Local atoms into the     a direct consequence of
                                   LPI, X4                                     LPI local projection     the positive F3P local-
                                                                               and proves the associ- coefficient predicate
                                                                               ated 𝑜(𝑁 ) error         plus F3/F4 exhaustion;
                                                                                                        downstream predicates
                                                                                                        are not used to define
                                                                                                        LongAP/Local
      H4     H4M, GEB              LPI, B1, D1, G8a,     Local/main component Reconstructs the B1/ Component local alge- Low-Medium
                                   B1LD, F3P, F3, F3T, assembly                F3 tagged local Gold- bra consumed by H4M
                                   F4, C1P, C1A, C1, X4,                       bach model and builds for the final handoff;
                                   X13                                         the finite local factor  contains the explicit
                                                                               from LPI-admitted        reconstruction theo-
                                                                               terms                    rem, active branch-
                                                                                                        admission verification,
                                                                                                        dyadic recombination,
                                                                                                        no-double-counting,
                                                                                                        Euler-product calcula-
                                                                                                        tion, and the LPI no-
                                                                                                        residual-local-source
                                                                                                        rule
      H4M    I1, GEB               F3F4M, LPI, B1, B3, Master local bridge     Packages the local/      Reader-facing local       Low-Medium
                                   F3P, F3, F3T, F4, D1, theorem               main handoff into        bridge: proves the
364




                                   G8a, B1LD, H4, C1P,                         𝑀local (𝑁 ) = S(𝑁 )𝑁 + admitted local source
                                   C1A, C1                                     𝑜(𝑁 )                    set is exactly LongAP/
                                                                                                        Local, CKP ℎ = 0, and
                                                                                                        LocalDiag; 𝑀other local
                                                                                                        is only bookkeeping
                                                                                                        for explicitly LPI-
                                                                                                        admitted subterms and
                                                                                                        not a fourth branch
      B1LD   G8a, H4, H4M          LPI, B1, X4, C1       B1 local-density com- Shows B1 finite-         Supplies the CKP zero- Low-Medium
                                                         patibility            convolution local mod- frequency local-density
                                                                               els match tagwise        interface used by G8a,
                                                                               Loc𝑄 for CKP ℎ = 0       H4, and H4M
      G8a    I1, F3, E10L, H4, H4M G1a, G2a, CKPX10M, CKP theorem              CKP equals LPI-          Local-density interface Medium
                                   B1LD, C1A, C1, LPI                          admissible local projec- is supplied by B1LD;
                                                                               tion plus 𝑜(𝑁 )          nonzero-frequency
                                                                                                        cancellation, smooth-
                                                                                                        weight derivative verifi-
                                                                                                        cation, DFI matching,
                                                                                                        𝑔, ℎ-summation, and
                                                                                                        excluded nonzero-range
                                                                                                        routing are imported
                                                                                                        through CKPX10M
      G1a       G2a, G3a, CKPX10M, X14                    CKP gcd splitting       Exact 𝑢 = 𝑔𝑎, 𝑢′ = 𝑔𝑞    Proved; 𝑔 ∤ 𝑁 empty    Low
                G8a                                                               split                    layers now explicitly
                                                                                                           routed in G8a
      G2a       G3a, CKPX10M, G8a, G1a, X11, X15          Smooth AP Fourier       Separates CKP fre-       Supplies the weighted Low-Medium
                CKPD                                      expansion               quencies                 AP/Fourier identity;
                                                                                                           G8a is downstream,
                                                                                                           not an input
      G3a       G4a, CKPX10M, X10   G1a, G2a, PAR         CKP to DFI              Converts nonzero         Algebraic conversion   Medium
                                                                                  frequencies to           only; CKPD and X10
                                                                                  Kloosterman-fraction     are downstream in-
                                                                                  sums                     puts packaged by
                                                                                                           CKPX10M
      CKPD      G4a, CKPX10M, X10   G1a, G2a, G3a, C1P,   CKP/X10 smooth-         Proves DFI deriva-       Uses the CKP inter-    Medium / external
                                    C1A, C1, PAR          weight derivative ap-   tive admissibility for   face notation later    DFI source check
                                                          pendix                  𝑊𝑔,ℎ (𝑎, 𝑞)              consumed by G4a and
                                                                                                           CKPX10M; it does
                                                                                                           not use either theorem
                                                                                                           as an input. Includes
                                                                                                           the explicit DFI-X10
                                                                                                           theorem statement and
                                                                                                           supplies the smooth-
                                                                                                           weight obligation.
365




      G4a       CKPX10M             X10, CKPD, X10ER,     DFI matching            Applies external DFI     Smooth-weight in-      Medium
                                    C1P, C1A, C1                                  estimate                 terface supplied by
                                                                                                           CKPD; citation-grade
                                                                                                           theorem matching
                                                                                                           is recorded in X10
                                                                                                           and CKPD; excluded
                                                                                                           ranges are routed by
                                                                                                           X10ER
      CKPX10M   G8a, GEB, I1        B1, B3, F3P, F3,      Master CKP/X10          Packages central CKP Reader-facing CKP/         Medium
                                    F4, G1a, G2a, G3a,    nonzero-frequency       DFI matching, actual X10 theorem; ℎ = 0
                                    CKPD, G4a, X10ER,     theorem                 two-variable smooth-     is not sent to X10 and
                                    C1P, C1A, C1, PAR,                            weight derivative ver- remains the G8a/LPI
                                    X10                                           ification, 𝑔, ℎ-loss ac- local mode assembled
                                                                                  counting, and excluded by H4M
                                                                                  nonzero-frequency
                                                                                  range routing
      X10ER    G4a, CKPX10M,       G1a, G2a, C1P, C1A,   CKP excluded-range    Records that noncen-     Internal routing record Low-Medium
               CKPD, X10           C1                    verification record   tral CKP ranges are      inside the CKP pack-
                                                                               not sent to DFI/X10      age; noncentral Edge
                                                                                                        exclusions are checked
                                                                                                        against C1P and ad-
                                                                                                        mitted by C1A. The
                                                                                                        ℎ = 0 local mode is
                                                                                                        handled separately by
                                                                                                        G8a/LPI and is not an
                                                                                                        X10ER input; no sepa-
                                                                                                        rate external theorem
                                                                                                        and no separate proof
                                                                                                        file
      E10L     I1                  TGD, TNGTTHM,      Branch B /               GoodAWACK contri-        Ordinary branch          Medium
                                   E10YMX, HGO2R, C1, GoodAWACK                bution 𝑜(𝑁 )             theorem: TGD
                                   G8a, H4M                                                             splits TC1/HighTC;
                                                                                                        TNGTTHM closes
                                                                                                        TC1 through the no-
                                                                                                        rogue near-global-
                                                                                                        or-routed theorem;
                                                                                                        E10YMX closes
                                                                                                        HighTC as a finite
                                                                                                        grammar theorem with
366




                                                                                                        𝑅FreeAffineHighTC (𝑁 ) =
                                                                                                        0; HGO2R and the
                                                                                                        terminal branch
                                                                                                        estimates consume
                                                                                                        routed HighTC
                                                                                                        outputs. X16-Core is
                                                                                                        imported only through
                                                                                                        TNGTTHM/X16C.
      TGD      E10L, TGT, TNG,     F3P, F3, F4           TC1/HighTC split      True-complexity di-      Active                   Low
               TNGTTHM                                                         chotomy
      TGT-MF   TGT, TTH-SC, TTD,   C1P, C1A, C1, E5, F3, Measured Fourier      Converts the global      New standalone mea-      Low-Medium
               TNG, TNGTTHM,       F4, PAR               transfer              𝑈 2 -obstruction into    sure/Fourier bridge;
               TTH                                                             a finite probability     all normalizations and
                                                                               family of Liouville      constants are internal
                                                                               tests with fixed lower   to the fixed macro-
                                                                               bound                    template
      TTH-SC   TTD, TNG, TNGT-   TGT-MF, TGD, F3,      Structural coarea   Proves that every short Standalone closure       Medium
               THM, TTH          F4, C1P, C1A, C1, E5, closure             subtest of a released   barrier. It does not
                                 PAR                                       TC1 coarea test is ei- use TTH, TTD, ROC,
                                                                           ther non-structural     BRS, X16BRS, X16C,
                                                                           and reaggregated or ex- TNG, TNGTTHM,
                                                                           ported as a structural or E10L as theorem
                                                                           short-image certificate inputs; those down-
                                                                                                   stream nodes consume
                                                                                                   the structural certifi-
                                                                                                   cates it exports.
      TGT      TNG, TNGTTHM,     TGD, TGT-MF, MRT, TC1 global testing      Builds the global       TGT.2 plus TGT-MF Medium
               TTD, TTH          TTH, E5, X9, PAR                          TC1 testing family      supply the testing
                                                                           and closes the MRT-     construction; MRT
                                                                           admissible near-global supplies admissibility/
                                                                           branch                  PACK; TTH supplies
                                                                                                   the near-global length
                                                                                                   barrier; X9L-GT closes
                                                                                                   this regular branch.
                                                                                                   Singular or structural
                                                                                                   short-image alter-
                                                                                                   natives are handled
                                                                                                   downstream by TTD/
                                                                                                   ROC/BRS/TNG/
367




                                                                                                   TNGTTHM, not used
                                                                                                   as inputs to TGT.
      TNG      TNGTTHM           TGT-MF, TGT, TTH- TC1 near-global-or-     Packages every active Includes Theorem           Medium
                                 SC, MRT, TTD, ROC, routed component       TC1 coarea test into    TNG-A; TNGTTHM
                                 BRS, X16BRS, X16C, theorem                either the MRT/PACK is the reader-facing
                                 TTH, C1P, C1A, C1,                        + TTH near-global       master theorem con-
                                 D1, H4M, G8a, E5,                         X9L-GT branch or        sumed by E10L
                                 TGD, X9, PAR                              the TTH-SC/TTD/
                                                                           ROC/BRS/X16 routed
                                                                           branch
      TTD      TNG, TNGTTHM      TGT-MF, TGT, MRT, Testing dichotomy       Regular/singular split Consumes the TGT/         Medium
                                 TTH-SC, ROC, BRS,                                                 TGT-MF construc-
                                 TTH, C1P, C1A,                                                    tion and the upstream
                                 C1, D1, H4M, G8a,                                                 MRT, TTH-SC, ROC,
                                 X16BRS, X16C, X9                                                  BRS, and TTH out-
                                                                                                   puts. It is not an input
                                                                                                   to MRT, ROC, BRS,
                                                                                                   or TTH.
      MRT      TGT, TNG, TNGT-   TGT-MF, E5, TGD,       MRT admissibility/       Checks testing-family   Defines the regular     Medium
               THM, TTD          PAR                    PACK                     start distribution      full-rank branch for
                                                                                                         the TGT-MF testing
                                                                                                         measure. Failure of
                                                                                                         PACK is exported
                                                                                                         as a singular-origin
                                                                                                         certificate consumed by
                                                                                                         TTD/ROC/BRS; MRT
                                                                                                         does not invoke those
                                                                                                         closures.
      ROC      TTD, TNG, TNGT-   B1, BGS, BRS,       Singular-origin verifica- Routes singular starts    Depends on the          Medium-High
               THM               X16BRS, X16C, E10Y, tion                      away                      terminal-affine gram-
                                 E10M, E10K                                                              mar interface E10Y/
                                                                                                         E10M/E10K and on
                                                                                                         BRS for complemen-
                                                                                                         tary short-image cases.
                                                                                                         It does not use TTD,
                                                                                                         TNGTTHM, or E10L;
                                                                                                         TTD consumes the
                                                                                                         ROC+BRS closure.
      BRS      ROC, TTD, TNG,    B1, C1P, C1A, C1,   B1 range/slice closure      Short-image B1-origin   Quotient-tag N1 is      Medium
               TNGTTHM, TTH,     F3P, F3, F4, ROC.1,                             residual is Edge or     closed by F4.9/F4.11
               E10L              X16BRS, X16C, PAR,                              already routed          and X16-Core by
368




                                 E10Y, E10M, E10K                                                        X16C; short-image
                                                                                                         Edge satisfies C1P
                                                                                                         E6 and is admitted
                                                                                                         by C1A. BRS con-
                                                                                                         sumes structural short-
                                                                                                         image certificates ex-
                                                                                                         ported by TTH-SC
                                                                                                         through TTD/ROC,
                                                                                                         but BRS is not an
                                                                                                         input to TTH-SC. Ter-
                                                                                                         minal labels are read
                                                                                                         through F3P/F3 and
                                                                                                         untagged rank-drop
                                                                                                         exclusion comes from
                                                                                                         the terminal-affine
                                                                                                         grammar interface, not
                                                                                                         from E10L or TTD.
      X16BRS   BRS, TNG, TNGT-   X16C, X16, PAR         Carrier-slice estimate   Reduces four BRS        Shiu/AP proof sup-      Medium
               THM, TTH                                                          carriers to X16-Core    plied in X16C
      X16C      X16BRS, BRS, TNG,   X16, B1, PAR          X16-Core Shiu/AP       Controls the 𝑁 − 𝑝𝑢  Proves active X16-Core Medium / referee
                TNGTTHM, TTH                              proof                                       with 𝐶16 = 100𝐽02 +100, check of Shiu local
                                                                                 divisor correlation for
                                                                                 B1 carrier slices    𝜌16 = 1/(106 𝐽04 ), slice factors
                                                                                                      floor 𝐵16 , explicit 𝜏𝐾2

                                                                                                      Shiu admissibility, and
                                                                                                      the Cauchy–Schwarz/
                                                                                                      AP correlation step
      TTH       TGT, TTD, TNG,      TGT-MF, TTH-SC,      Near-global TC1         Supplies 𝐻 ≥         Structural consequence Medium
                TNGTTHM             BRS, E5, X16BRS,     length                  𝑋(log 𝑋)−𝐵𝜅          of TTH-SC plus BRS
                                    X16C, PAR                                                         using X16C constants;
                                                                                                      X9 is invoked only
                                                                                                      downstream after this
                                                                                                      length bound is proved.
                                                                                                      TTH does not use
                                                                                                      TGT, TTD, ROC,
                                                                                                      or MRT as theorem
                                                                                                      inputs.
      TNGTTHM   E10L, GEB, I1       TGD, F3F4M, TGT-     Master TC1 no-rogue- Every actual B1-origin Reader-facing com-         Medium
                                    MF, TGT, TTH-SC,     short-interval theorem TC1 coarea test is    pressed proof chapter
                                    TNG, MRT, TTD,                               near-global and X9L- with released-test
                                    ROC, BRS, X16BRS,                            admissible or routed records, rogue-test def-
                                    X16C, TTH, C1P,                              away before X9L-GT   inition, finite decision
                                    C1A, C1, E5, PAR, X9                                              table, TTH-SC refine-
369




                                                                                                      ment barrier, BRS/
                                                                                                      X16 short-image rout-
                                                                                                      ing, and no-third-class
                                                                                                      conclusion. It has no
                                                                                                      E10L child-dependency
                                                                                                      and imports X9 only
                                                                                                      in the near-global Dav-
                                                                                                      enport/AP form.
      BGS       E10L, E10M, E10X,   B1, B3, F3P, F3, F4, GoodAWACK skeleton Records terminal skele- Intrinsic B1/B3/F3P/ Low-Medium
                E10K, E10YMX        E5                   normal form             tons                 F3/F4/E5 normal
                                                                                                      form; the no-untagged-
                                                                                                      AFF interpretation is
                                                                                                      supplied downstream
                                                                                                      by E10Y/E10M/E10K/
                                                                                                      E10X and packaged by
                                                                                                      E10YMX
      HGO2R     E10L, E10X, E10K,   BGS, C1, G8a, H4M, Origin-degenerate         Routes origin-       Active; free-affine class Low-Medium
                E10YMX              F3P, F3, F4          HighTC rerouting        degenerate HighTC    closed by E10YMX
      BAOC      E10G, E10X, E10L,   B1, B3, F3P, F3, F4, Affine origin catalogue Weak transport cata- Active catalogue, not     Low-Medium
                E10YMX              E5                                           logue                standalone closure
      E10G   E10H, E10X, E10L,     BAOC, BGS, F3P, F3, Catalogue schema    Isolates formal          Active schema              Low-Medium
             E10YMX                F4, E5                                  FreeAffineHighTC
                                                                           interface examples and
                                                                           passes actual-origin
                                                                           closure to E10YMX
      E10H   E10I, E10X, E10L,  E10G, BGS, F3P, F3, Matrix-origin reduc-   Reduces broad 4AP-       Active reduction; not      Low-Medium
             E10YMX             F4, E5, H4M           tion                 like formal witnesses to a live obstruction after
                                                                           actual-origin rigidity   E10YMX
      E10I   E10J, E10X, E10K,  E10H, BGS, F3P, F3, MOR partial proof      CRT/full-rank AFF        Active reduction           Low-Medium
             E10L, E10YMX       E5                                         safe; remaining rank-
                                                                           dropping AFF class
                                                                           delegated to E10X/
                                                                           E10YMX
      E10J   E10X, E10K, E10L,  E10I, BGS, F3P, F3,   Rank-dropping AFF    Reduces RDA to AFF- Active reduction;               Low-Medium
             E10YMX             F4, H4M, E5           verification         origin completeness      E10YMX supplies
                                                                                                    E10J.3 for actual ter-
                                                                                                    minal GoodAWACK
                                                                                                    cells
      E10Y   E10M, E10X, E10K,  B1, B3, F3P, F3, F3A, GoodAWACK routing Proves that every           Active structural the-     Low-Medium
             E10YMX             F3T, F4, E5           grammar completeness actual-generated         orem; no new analytic
                                                                           skeleton-generating      estimate
                                                                           pre-terminal operation
370




                                                                           is in the B1/B3/F3P/
                                                                           F3/F4/E5 grammar
                                                                           and that post-terminal
                                                                           analytic tests cannot
                                                                           replace the fixed termi-
                                                                           nal tensor-test vectors
      E10M   E10X, E10K, E10YMX E10Y, F3A, B1, B3,    No untagged rank-    Excludes untagged        Active structural theo-    Low-Medium
                                F3P, F3, F4, BGS, E5 dropping AFF theorem AFF in actual ter-        rem
                                                                           minal skeletons and
                                                                           discharges the actual-
                                                                           descendant forms of
                                                                           E10H.2, E10I.4, and
                                                                           E10J.3 once packaged
                                                                           by E10YMX
      E10X   E10YMX             B1, B3, F3P, F3, F3A, Finite GoodAWACK     Uses E10Y gram-          Active finite grammar      Low-Medium
                                F3T, F4, E5, BGS,     grammar closure      mar completeness         invariant
                                HGO2R, BAOC,                               and proves the finite
                                E10G, E10H, E10I,                          grammar invariant,
                                E10J, E10Y, E10M,                          invalidation rule, 4AP
                                E10K                                       interface example, no-
                                                                           untagged theorem,
                                                                           AFF-OC implication,
                                                                           and FreeAffineHighTC
                                                                           elimination
      E10K     E10X, E10YMX        E10J, E10I, HGO2R,   AFF-origin complete-   Gives allowed tags for   Active; packaged by   Low-Medium
                                   E10Y, E10M           ness                   rank drops               E10X/E10YMX for
                                                                                                        E10L
      E10YMX   E10L, GEB, I1       B1, B3, F3P, F3, F3A, Master GoodAWACK      Proves the finite        Reader-facing HighTC Medium
                                   F3T, F4, F3F4M, E5, finite-grammar theo-    grammar theo-            theorem; source-file
                                   BGS, HGO2R, BAOC, rem                       rem in ordinary          hashes, occurrence
                                   E10G, E10H, E10I,                           statement/proof          manifests, and search
                                   E10J, E10Y, E10M,                           form: state space,       scripts are not proof
                                   E10X, E10K                                  allowed transitions,     premises
                                                                               invariant, induction,
                                                                               no untagged rank-
                                                                               dropping AFF, and
                                                                               𝑅FreeAffineHighTC (𝑁 ) =
                                                                               0
      E5       F4, BGS, E10Y, E10M, X6                  Content stability mas- Preserves controlled     Active              Low-Medium
               E10X, E10YMX                             ter proof and clean    content under al-
                                                        interface              lowed transports;
                                                                               GoodAWACK im-
                                                                               ports only the E5-
                                                                               clean interface, while
                                                                               the E5 proof body is
                                                                               maintained once in
                                                                               the routing/transport
371




                                                                               appendix
  —


External / Standard Dependencies

 ID             Type                Name                  Used by      Status
 X1             standard/formal     Heath–Brown iden-     B1           Passed in X1 verifi-
                                    tity                               cation
 X2             standard            Smooth partition of  B1, B3        Standard exact par-
                                    unity                              tition
 X3             standard/external   Type I / short-      C1            Active only inside
                                    variable estimates                 strict Edge budgets
 X4             standard            CRT and local den-   D1, H4, H4M   Standard finite local
                                    sity algebra                       algebra
 X5             standard            Cauchy–Schwarz/      TGD/TC1       Used as standard
                                    GVN machinery                      CS/GVN, not
                                                                       inverse-Gowers X8
 X6             standard            Lattice/content      E5, F4        Standard bounded-
                                    algebra                            minor/content alge-
                                                                       bra
 X9             external            Davenport/AP near- TGT, TNG, TNGT- Active only after
                                    global Liouville     THM, TTH      MRT/PACK, TTH-
                                    input                              SC, and TTH give
                                                                       the active B1-origin
                                                                       near-global coarea
                                                                       hypotheses; E10L
                                                                       imports X9 through
                                                                       TNGTTHM
 X10            external            DFI bilinear Kloost- G4a, CKPX10M, Weighted smooth
                                    erman fractions      CKPD          derivative verifica-
                                                                       tion supplied by
                                                                       CKPD; DFI-X10
                                                                       theorem statement
                                                                       explicit in X10.2,
                                                                       X10.14, and CKP-
                                                                       DER.0a; excluded-
                                                                       range routing is
                                                                       recorded by X10ER
                                                                       and packaged with
                                                                       the central nonzero-
                                                                       frequency estimate
                                                                       by CKPX10M
 X11            standard            Smooth Fourier/AP G2a, G8a         Standard
                                    expansion
 X12            standard            Prime-power bound G2               Elementary
 X13            standard            Euler product alge- H4, H4M, G1    Standard singular
                                    bra                                series algebra
 X14            standard            GCD algebra          G1a           Exact gcd splitting
 X15            standard            Smooth Fourier       G2a, C1       Standard
                                    decay




                                           372
  X16                 standard/cited      Shiu AP divisor av-   C1, BRS, X16C    BRS-specific X16C
                                          erages and fixed-                      is proved in X16C;
                                          depth divisor esti-                    Tenenbaum Ch. II.5,
                                          mates                                  Theorem 5 supplies
                                                                                 the fixed-divisor
                                                                                 second moment
                                                                                 used in X16C; Shiu/
                                                                                 local-factor use is
                                                                                 recorded in X16C
                                                                                 and the active bibli-
                                                                                 ography




    The active bibliographic list is reproduced in Appendix K and maintained in the source-layer
bibliography checklist.
    —

Package Status

            The audit-grade proof tree is synchronized with the full manuscript build.

    The active internal route now reaches the final handoff:

                                       𝐼1 + 𝐺2 =⇒ 𝐺1 =⇒ 𝐺0.

   All active proof nodes in this ledger are represented in the full audit-grade manuscript. External
checking and ordinary publication editing are not additional proof dependencies. A new proof
dependency is introduced only if a concrete failure is found in one of the active proof nodes or
external theorem invocations.


K       Bibliography
The following source register separates active proof inputs from historical and orientation references.
Detailed theorem invocations and parameter matching are recorded in the external-input proof
units. Historical references are included only for context and are not logical dependencies in the
proof ledger. The final formal reference list is generated separately from bibliography.bib.
   Active External Inputs
   X1. Heath–Brown identity
   D. R. Heath-Brown, "Prime numbers in short intervals and a generalized Vaughan identity",
Canadian Journal of Mathematics 34 (1982), no. 6, 1365–1377. DOI: 10.4153/CJM-1982-095-9.
   Use in the package: the fixed-depth Heath–Brown identity used by B1 and checked in X1.
   X9. Davenport/AP Liouville input
   H. Davenport, "On some infinite series involving arithmetical functions (II)", The Quarterly
Journal of Mathematics, os-8 (1937), no. 1, 313–320. DOI: 10.1093/qmath/os-8.1.313.
   Use in the package: the near-global Davenport/AP Liouville exponential-sum input used by
X9L-GT after TTH supplies 𝐻 ≥ 𝑋(log 𝑋)−𝐵 .
   X10. Bilinear Kloosterman-fraction input
   W. Duke, J. B. Friedlander, and H. Iwaniec, "Bilinear forms with Kloosterman fractions",
Inventiones Mathematicae 128 (1997), no. 1, 23–43. DOI: 10.1007/s002220050135.


                                                 373
    Use in the package: Theorem 2 and the smooth-weight formulation used by the CKP/X10
smooth-weight derivative appendix. This is the unique active CKP Kloosterman-fraction external
input.
    X16. Shiu/AP divisor-average input
    P. Shiu, "A Brun–Titchmarsh theorem for multiplicative functions", Journal fuer die reine und
angewandte Mathematik 313 (1980), 161–170. DOI: 10.1515/crll.1980.313.161.
    Use in the package: Shiu arithmetic-progression divisor averages used in X16C for the BRS
carrier-slice estimate.
    Active Standard Support
    Fixed divisor-function second moments
    G. Tenenbaum, "Introduction to Analytic and Probabilistic Number Theory", Graduate Studies
in Mathematics 163, American Mathematical Society, 3rd ed., 2015, Ch. II.5, Theorem 5.
    Use in the package: the standard fixed-𝐾 second moment
                                                               2
                                        𝜏𝐾 (𝑢)2 ≪𝐾 𝑈 (log 2𝑈 )𝐾 −1
                                   ∑︁

                                  𝑢≍𝑈

used in X16C.
    Historical / Orientation References
    The references in this section are not proof inputs. They are included only for historical context
and reader orientation.
    G. H. Hardy and J. E. Littlewood, "Some problems of Partitio Numerorum; III: On the expression
of a number as a sum of primes", Acta Mathematica 44 (1923), 1–70. DOI: 10.1007/BF02403921.
    Use in the package: historical background for the Hardy–Littlewood singular series prediction
for Goldbach-type problems. This paper is not invoked as a proof input.
    I. M. Vinogradov, "Representation of an odd number as a sum of three primes", Doklady
Akademii Nauk SSSR 15 (1937), 291–294.
    Use in the package: historical background for the ternary Goldbach theorem for sufficiently
large odd integers. This result is not invoked as a proof input.
    J. R. Chen, "On the representation of a large even integer as the sum of a prime and the product
of at most two primes", Scientia Sinica 16 (1973), 157–176.
    Use in the package: historical background for Chen’s almost-Goldbach theorem. This result is
not invoked as a proof input.
    H. A. Helfgott, "The ternary Goldbach problem", arXiv:1501.05438.
    Use in the package: historical background for the completed weak Goldbach theorem. This work
is not invoked as a proof input.
    R. C. Vaughan, "The Hardy–Littlewood Method", 2nd ed., Cambridge Tracts in Mathematics
125, Cambridge University Press, 1997.
    Use in the package: general background on the circle method. No theorem from this book is
invoked as an active external input.
    H. Iwaniec and E. Kowalski, "Analytic Number Theory", American Mathematical Society
Colloquium Publications 53, American Mathematical Society, 2004.
    Use in the package: general orientation for analytic-number-theory conventions. No theorem
from this book is invoked as an active external input.
    Background for Active Technology
    J.-M. Deshouillers and H. Iwaniec, "Kloosterman sums and Fourier coefficients of cusp forms",
Inventiones Mathematicae 70 (1982), no. 2, 219–288.
    Use in the package: background for the classical Kloosterman-sum technology. The active CKP/
X10 theorem is the Duke–Friedlander–Iwaniec bilinear Kloosterman-fraction input listed above.

                                                 374
    Standard Non-Bibliographic Inputs
    The proof ledger also lists elementary or standard internal IDs X11–X15: smooth Fourier/AP
expansion, prime-power counting, Euler-product algebra, gcd splitting, and smooth Fourier decay.
These are tracked in the proof tree for dependency accounting, but they do not add separate active
bibliography entries unless the final manuscript elects to cite a standard textbook for exposition.




                                               375
References
 [1] Tenenbaum, Gerald. Introduction to Analytic and Probabilistic Number Theory. American
     Mathematical Society. vol. 163. 2015. Ch. II.5, Theorem 5 is used for fixed divisor-function
     second moments.

 [2] Davenport, Harold. On some infinite series involving arithmetical functions (II). The Quarterly
     Journal of Mathematics. vol. os-8. no. 1. pp. 313–320. 1937. doi: 10.1093/qmath/os-8.1.313.
     Classical source for the near-global Liouville exponential-sum input; the active formulation is
     recorded in X9L-GT.

 [3] Heath-Brown, D. R. Prime numbers in short intervals and a generalized Vaughan identity. Cana-
     dian Journal of Mathematics. vol. 34. no. 6. pp. 1365–1377. 1982. doi: 10.4153/CJM-1982-095-9.
     Source for the fixed-depth Heath–Brown identity used in X1/B1.

 [4] Shiu, P. A Brun–Titchmarsh theorem for multiplicative functions. Journal für die reine und
     angewandte Mathematik. vol. 313. pp. 161–170. 1980. doi: 10.1515/crll.1980.313.161.
     Source for the Shiu AP divisor-average input used in X16C.

 [5] Duke, William and Friedlander, John B. and Iwaniec, Henryk. Bilinear forms with
     Kloosterman fractions. Inventiones Mathematicae. vol. 128. no. 1. pp. 23–43. 1997. doi:
     10.1007/s002220050135. Active CKP/X10 input: Theorem 2 with the smooth-weight formu-
     lation.

 [6] Deshouillers, Jean-Marc and Iwaniec, Henryk. Kloosterman sums and Fourier coefficients of
     cusp forms. Inventiones Mathematicae. vol. 70. no. 2. pp. 219–288. 1982. Background source
     for classical Kloosterman-sum technology; the active CKP/X10 theorem is Duke–Friedlander–
     Iwaniec.

 [7] Hardy, G. H. and Littlewood, J. E. Some problems of “Partitio Numerorum”; III: On the
     expression of a number as a sum of primes. Acta Mathematica. vol. 44. pp. 1–70. 1923. doi:
     10.1007/BF02403921. Historical and orientation reference only; not an active proof input.

 [8] Vinogradov, I. M. Representation of an odd number as a sum of three primes. Doklady Akademii
     Nauk SSSR. vol. 15. pp. 291–294. 1937. Historical and orientation reference for the ternary
     Goldbach theorem; not an active proof input.

 [9] Chen, J. R. On the representation of a large even integer as the sum of a prime and the product
     of at most two primes. Scientia Sinica. vol. 16. pp. 157–176. 1973. Historical and orientation
     reference for Chen’s almost-Goldbach theorem; not an active proof input.

[10] Helfgott, H. A. The ternary Goldbach problem. 2015. arXiv:1501.05438. Historical and orientation
     reference for the completed weak Goldbach theorem; not an active proof input.

[11] Vaughan, R. C. The Hardy–Littlewood Method. Cambridge University Press. vol. 125. 1997.
     General historical and methodological orientation; not an active proof input.

[12] Iwaniec, Henryk and Kowalski, Emmanuel. Analytic Number Theory. American Mathematical
     Society. vol. 53. 2004. General orientation for analytic-number-theory conventions; not an active
     proof input.




                                                 376
