﻿        CKP/X10/X16 Analytic Independent Theorem Package

                               Denis Saltykov (ds1678@gmail.com)

                                              May 2026


Contents
1 CKP/X10/X16 Analytic Independent Theorem Package                                                     3
  1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   3

2 Introduction and Statement of Results                                                                3
  2.1 Purpose of the Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       3
  2.2 Package Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      3
       2.2.1 Theorem CKP-X10-X16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           3
  2.3 What This Package Imports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        4

3 Parameters, Notation, and Error Bookkeeping                                                          4

4 External Analytic Inputs                                                                             4
  4.1 X10: DukeвЂ“FriedlanderвЂ“Iwaniec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        5
  4.2 X16: Shiu/AP Divisor Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        5
  4.3 Fixed Divisor-Function Second Moment . . . . . . . . . . . . . . . . . . . . . . . . .           5
  4.4 Standard Internal Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     5

5 Full Proof Body                                                                                       5
  5.1 Part 1. X10: DFI/X10 Kloosterman-fraction verification . . . . . . . . . . . . . . . .            6
  5.2 Part 2. X16: X16 divisor-sum/BRS verification . . . . . . . . . . . . . . . . . . . . .          17
  5.3 Part 3. G1a: CKP gcd splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
  5.4 Part 4. G2a: Smooth AP Fourier expansion . . . . . . . . . . . . . . . . . . . . . . .           24
  5.5 Part 5. G3a: CKP-to-DFI conversion . . . . . . . . . . . . . . . . . . . . . . . . . . .         29
  5.6 Part 6. CKPD: CKP/X10 smooth-weight derivative appendix . . . . . . . . . . . . .                33
  5.7 Part 7. G4a: DFI matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        38
  5.8 Part 8. G8a: CKP branch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .          47
  5.9 Part 9. X16BRS: BRS carrier-slice reduction . . . . . . . . . . . . . . . . . . . . . .          53
  5.10 Part 10. X16C: X16-Core Shiu/AP proof . . . . . . . . . . . . . . . . . . . . . . . .           56

6 Package Dependency Ledger and Synchronization Notes                                                  64
  6.1 Compact Package Dependency Graph . . . . . . . . . . . . . . . . . . . . . . . . . .             64
  6.2 Local Ledger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     65
  6.3 Synchronization Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       65

7 Bibliography                                                                                         65



                                                   1
8 References       65




               2
1     CKP/X10/X16 Analytic Independent Theorem Package
1.1     Abstract
This package gives the self-contained analytic brick used by the modular Goldbach proof. It
proves the CKP nonzero-frequency estimate through the DFI/X10 Kloosterman-fraction input and
proves the X16/Shiu carrier-slice estimate used by the BRS part of the TC1 route. The package
is independent as an analytic theorem package, while importing the common routing and local
projection interfaces from the other theorem packages.


2     Introduction and Statement of Results
2.1     Purpose of the Package
The package has two analytic responsibilities.
    First, it proves the CKP branch estimate. After CKP gcd splitting and smooth AP Fourier
expansion, the zero-frequency part is passed to the H4 local algebra package, while every central
nonzero-frequency term is matched to the DFI/X10 bilinear Kloosterman-fraction estimate with
the actual two-variable smooth weight verified by CKPD. The noncentral CKP ranges are not sent
to X10; they are routed to Edge or local branches by X10ER as recorded inside G4a/G8a/CKPD.
    Second, it proves the X16 carrier-slice estimate. X16BRS reduces the BRS carrier types to the
product-carrier model, and X16C proves the product carrier estimate by Shiu/AP divisor averages,
CauchyвЂ“Schwarz, and the fixed divisor-function second moment.

2.2     Package Theorem
2.2.1    Theorem CKP-X10-X16
Assume the common decomposition and routing setup supplied by the final modular assembly
package, the LPI local projection interface assembled by the H4 local algebra package, and the
global parameter hierarchy recorded in PAR/GEB. Then the following two conclusions hold.

    1. The CKP terminal contribution satisfies


                                  рќ‘…CKP (рќ‘Ѓ ) = рќ‘ЂCKP
                                               H4
                                                   (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ),
            LPI (рќ‘Ѓ ) is the tagged LPI-admitted local projection. The zero-frequency part is local,
   where рќ‘ЂCKP
and the central nonzero frequencies are рќ‘њ(рќ‘Ѓ ) by G1a, G2a, G3a, CKPD, G4a, and X10.

    1. Every active BRS carrier-slice input required by the TC1 no-rogue-short

    interval package is reduced by X16BRS to the product-carrier model and satisfies the X16C
Shiu/AP estimate, with the complementary carrier types routed to Edge, quotient-tag, local, or
already handled branches.
    Consequently this package exports exactly the analytic inputs needed by I1 and by the TNG/TC1
package:

             CKP = H4 local + рќ‘њ(рќ‘Ѓ )       and        BRS carrier slice в‡’ X16C control.



                                                 3
2.3      What This Package Imports
This package does not reprove the global routing theorem, the H4 local algebra, or the TC1 near-
global-or-routed theorem. It imports only the following named interfaces.

    1. PAR/GEB: the global parameter hierarchy and terminal summability.

    2. C1P/C1A/C1: Edge admission and strict Edge estimates for excluded CKP and BRS ranges.

    3. LPI/B1LD/H4: the tagged local projection interface for CKP zero frequency.

    4. BRS/TNG: the statement that the X16BRS output is used only inside the active B1-origin
       TC1 carrier-slice route.


3      Parameters, Notation, and Error Bookkeeping
Throughout рќ‘Ѓ is an even integer tending to infinity and рќђї = log рќ‘Ѓ . The HeathвЂ“Brown depth рќђЅ0 ,
the small structural exponents рќњ‚, рќњѓ, and the routing multiplicity constants are fixed by PAR. This
package uses only the following local parameters from that hierarchy.


         Parameter                                         Local use in this package
         рќђµHF                                               CKP high-frequency cutoff for Fourier modes в„Ћ
         рќђ¶DFI                                              smooth-weight derivative loss in CKPD/X10
         рќђ¶16                                               logarithmic loss in the X16C carrier estimate
         рќњЊ16                                               power-saving exponent in the X16C remainder
         рќђµ16                                               logarithmic floor exponent used before applying X16C




    The local error bookkeeping is as follows.


      Source                          Output                                  Routed or summed by
      CKP zero frequency              LPI local projection                    LPI/B1LD/H4
      CKP central nonzero frequency   рќ‘њ(рќ‘Ѓ )                                   CKPD + G4a + X10 + GEB
      CKP high-frequency range        рќ‘њ(рќ‘Ѓ ) or Edge                           G2a decay and C1P/C1A/C1
      CKP noncentral ranges           Edge/local/excluded before X10          X10ER in G4a/G8a/CKPD
      X16 product carrier             main controlled term plus power/log     X16C
                                      saving
      X16 complementary carriers      product-carrier, Edge, quotient-tag,    X16BRS
                                      local, or handled




   No parameter is chosen inside this package in a way that changes the global decomposition.
The X16 constants are exported to the TC1/TTH package, where the later near-global exponent
рќђµрќњ… is chosen after рќђµ16 , рќђ¶16 , and рќњЊ16 are fixed.


4      External Analytic Inputs
This package uses the following external or standard analytic inputs.




                                                       4
4.1    X10: DukeвЂ“FriedlanderвЂ“Iwaniec
The CKP nonzero-frequency central range uses the bilinear form estimate with Kloosterman frac-
tions of Duke, Friedlander, and Iwaniec. The active application is always made after G1aвЂ“G3a put
the phase in the form рќ‘’(рќ‘џрќ‘Ћ/рќ‘ћ), with coprimality (рќ‘Ћ, рќ‘ћ) = 1, and after CKPD proves the required
derivative estimates for the actual two-variable smooth weight.

4.2    X16: Shiu/AP Divisor Averages
The BRS carrier-slice estimate uses ShiuвЂ™s BrunвЂ“Titchmarsh theorem for multiplicative functions
in arithmetic progressions. X16C records the precise product-carrier form needed in the proof.

4.3    Fixed Divisor-Function Second Moment
The proof of X16C also uses the standard fixed-рќђѕ second moment
                                                                2
                                        рќњЏрќђѕ (рќ‘ў)2 в‰Єрќђѕ рќ‘€ (log 2рќ‘€ )рќђѕ в€’1 .
                                   в€‘пёЃ

                                  рќ‘ўв‰Ќрќ‘€

    The active reference is Tenenbaum, Ch. II.5, Theorem 5.

4.4    Standard Internal Inputs
The package also uses standard smooth Fourier decay, gcd splitting, and finite smooth partition
arguments, as recorded in the logical IDs X11, X14, and X15. These are bookkeeping-standard
inputs rather than separate active external analytic theorems.


5     Full Proof Body
The following proof-source files are included in full.

    1. External/x_10_verification_ltx.md вЂ“ DFI/X10 Kloosterman-fraction verification

    2. External/x_16_divisor_sum_brs_verification_ltx.md вЂ“ X16 divisor-sum/BRS verifica-
       tion

    3. Lemmas/g_1_a_ltx.md вЂ“ CKP gcd splitting

    4. Lemmas/g_2_a_ltx.md вЂ“ Smooth AP Fourier expansion

    5. Lemmas/g_3_a_ltx.md вЂ“ CKP-to-DFI conversion

    6. Lemmas/ckp_x10_smooth_weight_derivative_appendix_ltx.md вЂ“ CKP/X10 smooth-weight
       derivative appendix

    7. Lemmas/g_4_a_ltx.md вЂ“ DFI matching

    8. Lemmas/g_8_a_ltx.md вЂ“ CKP branch theorem

    9. Lemmas/x16_brs_carrier_slice_ltx.md вЂ“ BRS carrier-slice reduction

 10. Lemmas/x16_core_shiu_ap_proof_ltx.md вЂ“ X16-Core Shiu/AP proof



                                                   5
5.1    Part 1. X10: DFI/X10 Kloosterman-fraction verification
Source file: External/x_10_verification_ltx.md.

X10. DFI Kloosterman Fraction Input

X10.0. Role Logical ID: X10.
    Used by: G4a, G8a, CKPD. I1 uses X10 only through the CKP branch.
    Uses: G1a, G2a, G3a, G4a, CKPD, X10ER, C1A, C1, and the DukeвЂ“FriedlanderвЂ“Iwaniec
bilinear Kloosterman-fraction theorem.
    This document states and verifies the external black-box X10 used in the CKP branch:

                       рќђє3рќ‘Ћ + рќђє4рќ‘Ћ + рќђ¶рќђѕрќ‘ѓ рќђ· + рќ‘‹10рќђёрќ‘… + рќђ¶1рќђґ/рќђ¶1 =в‡’ рќђє8рќ‘Ћ.
    The external input is DukeвЂ“FriedlanderвЂ“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 рќђє8рќ‘Ћ?

   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 рќ‘‹10-рќђёрќ‘… and Lemmas C1P/C1A/C1 as stated.

   вЂ”

X10.1. Required CKP form              After Lemmas G1a, G2a, and G3a, a nonzero-frequency CKP
contribution has the form

                                                                                 в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                            (пё‚       )пё‚
                         рќ’Єрќ‘”,в„Ћ =                рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘”,в„Ћ (рќ‘ћ)рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)рќ‘’
                                     в€‘пёЃ
                                                                                       ,
                                  рќ‘Ћв€јрќђґрќ‘” ,рќ‘ћв€јрќ‘„рќ‘”
                                                                                  рќ‘ћ
                                    (рќ‘Ћ,рќ‘ћ)=1

   where

                                                 рќ‘Ѓ
                                          рќ‘Ѓрќ‘” =     ,       рќ‘ = в„Ћрќ‘Ѓрќ‘” ,
                                                 рќ‘”
   and

                                                     1           в„Ћ
                                                               (пё‚ )пё‚
                                        рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) = рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ
                                                     рќ‘ћ           рќ‘ћ
   is the smooth Fourier weight from Lemma G2a.
   In the balanced CKP range,

                                                       6
                                                                 рќ‘Ѓ 1/2+рќ‘‚(рќњ‚)
                               рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘” ,             рќ‘†рќ‘” =              .
                                                                      рќ‘”
   The Fourier-weight bound is

                             |рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)| в‰Єрќђґ (log рќ‘Ѓ )рќђ¶ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ .
   The central nonzero-frequency range is restricted to

                                          |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ .
   The complementary high-frequency range is already Edge by C1P/C1A/C1.
   It remains to prove:

                                                   рќ’Єрќ‘”,в„Ћ = рќ‘њ(рќ‘Ѓ ),
                                        в€‘пёЃ в€‘пёЃ

                                        рќ‘”|рќ‘Ѓ в„ЋМё=0
   after excluding C1-routed large-g, high-frequency, small-conductor, and boundary layers.
   вЂ”

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), 23вЂ“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

                                                                          рќ‘џрќ‘љ
                                                                     (пё‚     )пё‚
                              рќђµрќ‘џ (рќ‘Ђ, рќ‘„) =
                                                   в€‘пёЃ
                                                           рќ›јрќ‘љ рќ›Ѕрќ‘› рќ‘’           ,
                                             рќ‘Ђ <рќ‘љв‰¤2рќ‘Ђ
                                                                           рќ‘›
                                             рќ‘„<рќ‘›в‰¤2рќ‘„
                                              (рќ‘љ,рќ‘›)=1
    where рќ‘џ is a positive integer and рќ›јрќ‘љ , рќ›Ѕрќ‘› are arbitrary complex coefficients. DFI Theorem 2
gives

                       рќђµрќ‘џ (рќ‘Ђ, рќ‘„) в‰ЄрќњЂ вЂ–рќ›јвЂ–2 вЂ–рќ›ЅвЂ–2 (рќ‘џ + рќ‘Ђ рќ‘„)3/8 (рќ‘Ђ + рќ‘„)11/48+рќњЂ .
   DFI also allows a smooth weight, supported on the same dyadic box and normalized by |рќђ№ | в‰¤ 1,

                                               рќђ№ (рќ‘љ, рќ‘›)
   provided its derivatives satisfy controlled bounds

                          рќђ№ (рќ‘—,рќ‘) (рќ‘љ, рќ‘›) в‰Є рќњ‚ рќ‘—+рќ‘ рќ‘љв€’рќ‘— рќ‘›в€’рќ‘ ,           0 в‰¤ рќ‘—, рќ‘ в‰¤ 2,
   at the cost of multiplying the right-hand side by a harmless factor

                                                    рќњ‚2.
    For our use, рќњ‚ is at most a fixed power of log рќ‘Ѓ , so this is absorbed into the polylogarithmic
loss.
    вЂ”

                                                     7
X10.3. Parameter and hypothesis matching

Parameter dictionary

 DFI object                CKP object                    Source
 рќ‘љ                         CKP inverse variable рќ‘Ћ        G1a/G3a
 рќ‘›                         CKP modulus variable          G1a/G3a
                           рќ‘ћ
 dyadic length рќ‘Ђ           рќђґрќ‘”                            G8a central layer
 dyadic length рќ‘„           рќ‘„рќ‘”                            G8a central layer
 external integer рќ‘џ        \{}(r=                        h                            N_g\{})                  G2a/G3a
 coprimality (рќ‘љ, рќ‘›) = 1    (рќ‘Ћ, рќ‘ћ) = 1                    G1a
 coefficient рќ›јрќ‘љ            рќ›јрќ‘” (рќ‘Ћ)                        B1 finite-convolution
                                                         inheritance
 coefficient рќ›Ѕрќ‘›            рќ›ѕрќ‘”,в„Ћ (рќ‘ћ)                      G2a/G3a
 smooth         weight     normalized Fourier fi-        CKPD
 рќђ№ (рќ‘љ, рќ‘›)                  bre рќ‘ЉМѓпёЂ рќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)
 phase рќ‘’(рќ‘џрќ‘љ/рќ‘›)             рќ‘’(в„Ћрќ‘Ѓрќ‘” рќ‘Ћ/рќ‘ћ)                    G3a




    Thus the formal phase and coprimality conditions match exactly.
    Negative в„Ћ causes no problem: the corresponding phase is the complex conjugate/sign variant
of the same estimate. The case в„Ћ = 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 H4.

Hypothesis-by-hypothesis check

 DFI hypothesis            CKP verification              Routing if it fails
 Dyadic support рќ‘љ в€ј        The tagged CKP layer          Boundary or short-
 рќ‘Ђ, рќ‘› в€ј рќ‘„                  has рќ‘Ћ в€ј рќђґрќ‘” , рќ‘ћ в€ј рќ‘„рќ‘”           volume failures are
                           after G1a/G8a.                C1P/C1A/C1 Edge inputs.
 Coprimality of in-        G1a imposes (рќ‘Ћ, рќ‘ћ) =          Non-coprime pre-split
 verted variable and       1.                            layers are not sent to
 modulus                                                 X10; they are resolved
                                                         in the gcd split.
 Arbitrary complex co-     B1 finite-convolution         Coefficient-size     fail-
 efficients allowed with   coefficients        satisfy   ures are Edge/large-
 рќђї2 -norms                 divisor-type рќђї2 bounds        content inputs through
                           recorded in G3a/G4a.          C1P/C1A/C1.
 Smooth two-variable       CKPD proves this for          Noncentral       balance
 weight with derivatives   the actual nonsepa-           failures are routed by
 up to order two           rated рќ‘ЉМѓпёЂ рќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ).         X10ER and C1P/C1A/C1
                                                         before X10.
 Positive external inte-   Use \{}(r=                    h                            N_g\{}); the sign of в„Ћ   в„Ћ = 0 is the local
 ger рќ‘џ                                                                                is handled by conjuga-   term and is handled by
                                                                                      tion.                    G8a/LPI, then assem-
                                                                                                               bled by H4.
 Uniformity in рќ‘џ with      In     the      central       h                            g\{}le(\{}log            High-frequency layers
 loss (рќ‘џ + рќ‘Ђ рќ‘„)3/8         frequency        range                                     N)Л†B\{}).                are C1P/C1A/C1 Edge in-
                           \{}(r/(MQ)\{}asymp                                                                  puts.
 Central       balanced    рќђґрќ‘”     в‰Ќ      рќ‘„рќ‘”     в‰Ќ        Unbalanced and large-
 lengths                   рќ‘Ѓ 1/2+рќ‘‚(рќњ‚) /рќ‘”.                рќ‘” layers are X10ER
                                                         and C1P/C1A/C1 inputs.




   вЂ”



                                                                  8
X10.4. Coefficient admissibility DFI allows arbitrary complex coefficient sequences. Our
sequences satisfy the stronger bounds

                                 рќ‘” (log рќ‘Ѓ ) ,                           рќ‘” (log рќ‘Ѓ ) .
                                           рќђ¶                                      рќђ¶
                       вЂ–рќ›јрќ‘” вЂ–2 в‰Є рќђґ1/2                        вЂ–рќ›ѕрќ‘”,в„Ћ вЂ–2 в‰Є рќ‘„1/2
   These follow from the finite-convolution/divisor-bounded structure inherited from B1 and from
the CKP routing.
   Therefore the coefficient condition passes.
   вЂ”

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              в„Ћ
                                                                   (пё‚ )пё‚
                                         рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) = рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ          .
                                                      рќ‘ћ              рќ‘ћ
   By Lemma G2a, in the central CKP range it satisfies

                                 рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Єрќђґ (log рќ‘Ѓ )рќђ¶ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ .
                                                                   МѓпёЃрќ‘”,в„Ћ в‰Є 1 and has smooth deriva-
    Moreover, after normalizing by its supremum size, it satisfies рќ‘Љ
tive bounds of the DFI weighted-corollary type:

                               МѓпёЃрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Є (log рќ‘Ѓ )рќђ¶ рќ‘Ћв€’рќ‘— рќ‘ћ в€’рќ‘ ,
                       рќњ•рќ‘Ћрќ‘— рќњ•рќ‘ћрќ‘ рќ‘Љ                                           1 в‰¤ рќ‘— + рќ‘ в‰¤ 2,
   provided

                                              |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ .
    The nonseparated dependence on both рќ‘Ћ and рќ‘ћ is intentional. The weight is not absorbed into
рќ›ѕрќ‘”,в„Ћ (рќ‘ћ) alone. The chain-rule terms are the ones proved in Lemma CKPD: on the central CKP
support (рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ в‰Ќ рќ‘Њ вЂІ , so differentiating рќ‘Љрќ‘Њ вЂІ ((рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ) in рќ‘ћ gives a factor

                                         рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                 В· (рќ‘Њ вЂІ )в€’1 в‰Є рќ‘„в€’1
                                                               рќ‘” ,
                                           рќ‘ћ2
and differentiating in рќ‘Ћ gives
                                                  рќ‘¦
                                                    В· (рќ‘Њ вЂІ )в€’1 ,
                                                  рќ‘ћ
which is admissible in the central balanced range рќ‘Њ в‰Ќ рќ‘Њ вЂІ , рќђґрќ‘” в‰Ќ рќ‘„рќ‘” . 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 |в„Ћ|рќ‘” > (log рќ‘Ѓ )рќђµ , 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 DFIвЂ™s weighted
formulation is invoked in the stated form.
   вЂ”




                                                        9
X10.6. Uniformity in рќ‘ = в„Ћрќ‘Ѓрќ‘” DFI Theorem 2 is uniform in the positive integer external
parameter рќ‘џ, with right-hand side depending on рќ‘џ в‰Ў рќ‘ through

                                                (рќ‘џ + рќ‘Ђ рќ‘„)3/8 .
   In our central range,

                                                                    рќ‘Ѓ
                                         рќ‘Ђ рќ‘„ в‰Ќ рќђґрќ‘” рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘”2 в‰Ќ           ,
                                                                    рќ‘”2
   while

                                                             |в„Ћ|рќ‘Ѓ
                                            |рќ‘| = |в„Ћ|рќ‘Ѓрќ‘” =         .
                                                               рќ‘”
   Therefore

                                                 |рќ‘|
                                                     в‰Ќ |в„Ћ|рќ‘”.
                                                 рќ‘Ђрќ‘„
   On the central frequency range

                                              |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ ,
   we have

                                        |рќ‘| + рќ‘Ђ рќ‘„ в‰Є рќ‘Ђ рќ‘„(log рќ‘Ѓ )рќђµ .
   Thus the dependence on рќ‘ costs only a polylogarithmic factor. This is harmless.
   The case of small conductor рќ‘ћ/(рќ‘ћ, рќ‘) в‰¤ (log рќ‘Ѓ )рќђµ is already routed through C1A to Lemma C1,
Edge predicate E5. DFI itself does not require (рќ‘, рќ‘ћ) = 1, since its theorem is stated for arbitrary
positive integer external parameter. Therefore the gcd (рќ‘, рќ‘ћ) creates no additional obstruction for
X10.
   вЂ”

X10.7. Loss accounting for one (рќ‘”, в„Ћ)-layer                Let

                                                                  рќ‘Ѓ 1/2+рќ‘‚(рќњ‚)
                                  рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘” ,           рќ‘†рќ‘” =              .
                                                                       рќ‘”
    For simplicity write рќ‘Ђ = рќ‘„ = рќ‘†рќ‘” . DFI gives, with the normalized smooth Fourier weight
included,

               |рќ’Єрќ‘”,в„Ћ | в‰ЄрќњЂ (log рќ‘Ѓ )рќђ¶ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ вЂ–рќ›јрќ‘” вЂ–2 вЂ–рќ›ѕрќ‘”,в„Ћ вЂ–2 (|рќ‘| + рќ‘†рќ‘”2 )3/8 (2рќ‘†рќ‘” )11/48+рќњЂ .
   The prefactor рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ in this display is precisely the unnormalized amplitude рќ’њрќ‘”,в„Ћ,рќ‘…
from CKPD.7, after absorbing fixed powers of log рќ‘Ѓ and choosing рќђґ smaller than рќ‘… by a fixed
margin. Thus the displayed bound already includes the amplitude accounting for the normalization
              МѓпёЃрќ‘”,в„Ћ .
рќ’Ірќ‘”,в„Ћ = рќ’њрќ‘”,в„Ћ,рќ‘… рќ‘Љ
   Using

                                       вЂ–рќ›јрќ‘” вЂ–2 вЂ–рќ›ѕрќ‘”,в„Ћ вЂ–2 в‰Є рќ‘†рќ‘” (log рќ‘Ѓ )рќђ¶ ,
   and

                                                      10
                                         |рќ‘| + рќ‘†рќ‘”2 в‰Є рќ‘†рќ‘”2 (log рќ‘Ѓ )рќђµ ,
   we get

                         |рќ’Єрќ‘”,в„Ћ | в‰ЄрќњЂ (log рќ‘Ѓ )рќђ¶ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ рќ‘†рќ‘” рќ‘†рќ‘”3/4 рќ‘†рќ‘”11/48+рќњЂ .
   Since
                                                 3 11  95
                                            1+    +   = ,
                                                 4 48  48
   this becomes

                             |рќ’Єрќ‘”,в„Ћ | в‰ЄрќњЂ (log рќ‘Ѓ )рќђ¶ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ рќ‘†рќ‘”95/48+рќњЂ .
   Substituting рќ‘†рќ‘” = рќ‘Ѓ 1/2+рќ‘‚(рќњ‚) /рќ‘” в‰Ў рќ‘Ѓ 1/2 /рќ‘” at the exponent level,

                     |рќ’Єрќ‘”,в„Ћ | в‰ЄрќњЂ рќ‘Ѓ 95/96+рќњЂ+рќ‘‚(рќњ‚) (log рќ‘Ѓ )рќђ¶ рќ‘” в€’47/48в€’рќњЂ (1 + |в„Ћ|рќ‘”)в€’рќђґ .
   Thus one central CKP layer has a power saving over рќ‘Ѓ в‰Ў рќ‘Ѓ 1 , namely approximately

                                             рќ‘Ѓ в€’1/96+рќ‘‚(рќњЂ+рќњ‚) .
   Choosing рќњЂ, рќњ‚ sufficiently small preserves a fixed power saving.
   вЂ”

X10.8. Summation over в„Ћ and рќ‘”             The frequency sum is harmless because for large рќђґ,

                                               (1 + |в„Ћ|рќ‘”)в€’рќђґ в‰Є 1.
                                          в€‘пёЃ

                                          в„ЋМё=0

   More precise bounds give an additional рќ‘” в€’1 when useful, but this is not needed.
   The gcd parameter satisfies

                                                   рќ‘”|рќ‘Ѓ
   by G1a. Hence the number of possible рќ‘”-layers is divisor-bounded:

                                          #{рќ‘” : рќ‘” | рќ‘Ѓ } в‰ЄрќњЂ рќ‘Ѓ рќњЂ .
   Equivalently, this contributes only an рќ‘Ѓ рќ‘њ(1) or polylogarithmic/divisor loss in the ledger-level
asymptotic accounting.
   Thus

                                      |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќ‘њ(1)+рќ‘‚(рќњ‚)+рќњЂ = рќ‘њ(рќ‘Ѓ ),
                           в€‘пёЃ в€‘пёЃ

                           рќ‘”|рќ‘Ѓ в„ЋМё=0

   provided рќњ‚ > 0 and the DFI рќњЂ > 0 are fixed so small that
                                                        1
                                         рќ‘‚(рќњ‚) + рќњЂ + рќ‘њ(1) <.
                                                       96
   This leaves a fixed power saving over рќ‘Ѓ . All noncentral ranges are already routed to X10ER
and C1P/C1A/C1 before this summation is used.
   вЂ”

                                                     11
X10.9. Excluded-range routing

Lemma 5.1 (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:


                                        |в„Ћ|рќ‘” > (log рќ‘Ѓ )рќђµ .
   Routed to the Edge admission ledger C1A and then Lemma C1, Edge predicate E4.

  1. Small conductor:


                                      рќ‘ћ/(рќ‘ћ, рќ‘) в‰¤ (log рќ‘Ѓ )рќђµ .
   Routed to the Edge admission ledger C1A and then Lemma C1, Edge predicate E5.

  1. Large gcd/content:


                                            рќ‘” > рќ‘Ѓрќњ‚
    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.
    вЂ”

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:


       CKPв„ЋМё=0 = CentralDFI вЉ” HighFreq вЉ” SmallConductor вЉ” LargeG вЉ” Boundary/Short.

   Then the routing is:

                                      CentralDFI в†’ рќ‘‹10,


                                     HighFreq в†’ рќђ¶1рќђґ/рќђ¶1,

                                               12
                                           SmallConductor в†’ рќђ¶1рќђґ/рќђ¶1,


                                         LargeG в†’ рќ‘‹10рќђёрќ‘… в†’ рќђ¶1рќђґ/рќђ¶1,


                                           Boundary/Short в†’ рќђ¶1рќђґ/рќђ¶1.
   Thus the fact that X10 is only used on the central range is correct and necessary.
   вЂ”

X10.11. Restriction-by-restriction routing check

 X10 restriction          Required      proof-tree   Current source of sup-
                          support                    port
 Only central balanced    B3/F3/G8a must iso-        Lemmas B3, F3, F3T,
 CKP is sent to DFI       late central CKP and       G8a
                          avoid sending noncen-
                          tral atoms to X10
 High-frequency layers    Fourier decay must         h                        g>(\{}log N)Л†B\{})   Lemmas G2a, X10ER,
 are excluded             make \{}(                                           an Edge tail         C1A, C1 E4
 Small-conductor layers   Small conductor must       Lemmas C1A, C1 E5
 are excluded             be Edge only in CKP-
                          normalized oscillatory
                          scale
 Large-рќ‘” layers are ex-   GCD splitting gives        Lemmas G1a, G8a,
 cluded                   volume saving рќ‘Ѓ/рќ‘” 2        X10ER, C1A, C1 E3
 Boundary/short-          Boundary and short-        Lemmas   C1A,   C1
 volume    layers   are   volume atoms must          E1/E6/E7
 excluded                 satisfy strict Edge
                          predicates
 Smooth weighted fibre    AP expansion must          Lemmas G2a, G8a
 expansion is required    use full tagged fi-
                          bre weight, not bare
                          рќ‘Љрќ‘Њ (рќ‘¦) only
 GCD (рќ‘, рќ‘ћ) > 1 may       Small conductor cases      Lemmas C1A, C1 E5,
 occur                    are removed; DFI itself    and the X10 theorem
                          is uniform in external     statement
                          рќ‘џ=рќ‘
 Summation over      рќ‘”    G1a gives рќ‘” | рќ‘Ѓ , hence    Lemmas G1a, G8a
 must be harmless         divisor-bounded num-
                          ber of рќ‘”-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.
   вЂ”

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-рќ‘” terms did not have volume saving;


                                                            13
  4. boundary/short-volume terms did not satisfy strict C1 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:

                                      рќђ¶1рќђґ/рќђ¶1        closes Edge tails,

                                рќђє2рќ‘Ћ    closes weighted AP expansion,

                          рќђє8рќ‘Ћ    closes CKP normalization and routing,

                   рќђµ3 + рќђ№ 3 + рќђ№ 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 в†’ рќ‘‹10,          noncentral CKP residuals в†’ рќ‘‹10рќђёрќ‘… в†’ рќђ¶1рќђґ/рќђ¶1.
   вЂ”

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-рќ‘”, 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


                                           |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ ;
  1. рќ‘”-summation uses the fact that G1a gives рќ‘” | рќ‘Ѓ .
   Under these conditions,

                                                    рќ’Єрќ‘”,в„Ћ = рќ‘њ(рќ‘Ѓ ).
                                         в€‘пёЃ в€‘пёЃ

                                         рќ‘”|рќ‘Ѓ в„ЋМё=0


                                                     14
Output for the CKP Branch The external X10 input discharges the DFI applicability condi-
tion in G4a/G8a.
    Consequently, X10 is verified with restrictions. DFI Theorem 2 and its smooth-weight corollary
apply to the central CKP nonzero-frequency sums with рќ‘Ђ = рќђґрќ‘” , рќ‘„ = рќ‘„рќ‘” , and positive external
integer parameter рќ‘џ = |в„Ћ|рќ‘Ѓрќ‘” . For в„Ћ < 0, the same estimate is applied to the conjugate phase.
The resulting saving is рќ‘Ѓ в€’1/96+рќ‘њ(1) in the balanced range, sufficient after summation over в„Ћ and
divisor-bounded рќ‘”-layers. Boundary, high-frequency, small-conductor and large-рќ‘” 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.
    вЂ”

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), 23вЂ“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 рќђ№ (рќ‘љ, рќ‘›) 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 рќ‘Ђ, рќ‘„ в‰Ґ 1, рќ‘џ в‰Ґ 1, and let рќ›јрќ‘љ , рќ›Ѕрќ‘ћ be arbitrary complex sequences
supported on рќ‘љ в€ј рќ‘Ђ , рќ‘ћ в€ј рќ‘„. Let рќђ№ (рќ‘љ, рќ‘ћ) be a smooth weight supported in the same dyadic box,
with |рќђ№ (рќ‘љ, рќ‘ћ)| в‰¤ 1, and satisfying, for 0 в‰¤ рќ‘–, рќ‘— в‰¤ 2,

                                               рќњ•рќ‘ћ рќђ№ (рќ‘љ, рќ‘ћ) в‰Є рќњ‚ рќ‘–+рќ‘— рќ‘Ђ в€’рќ‘– рќ‘„в€’рќ‘— .
                                             рќ‘– рќ‘—
                                            рќњ•рќ‘љ
   Then, for every рќњЂ > 0,

                                       рќ‘џрќ‘љ
                                  (пё‚         )пё‚
                рќ›јрќ‘љ рќ›Ѕрќ‘ћ рќђ№ (рќ‘љ, рќ‘ћ)рќ‘’                   в‰ЄрќњЂ рќњ‚ 2 вЂ–рќ›јвЂ–2 вЂ–рќ›ЅвЂ–2 (рќ‘џ + рќ‘Ђ рќ‘„)3/8 (рќ‘Ђ + рќ‘„)11/48+рќњЂ .       (DFI-X10)
         в€‘пёЃ

     рќ‘љв€јрќ‘Ђ, рќ‘ћв€јрќ‘„
                                        рќ‘ћ
      (рќ‘љ,рќ‘ћ)=1

    In the CKP application, рќњ‚ в‰¤ (log рќ‘Ѓ )рќђ¶ , so the рќњ‚ 2 factor is absorbed into the existing polyloga-
rithmic loss.

Exact substitution      The central CKP nonzero-frequency sum has

                   рќ‘љ = рќ‘Ћ,         рќ‘ћ = рќ‘ћ,              рќ‘Ђ = рќђґрќ‘” ,       рќ‘„ = рќ‘„рќ‘” ,       рќ‘џ = |в„Ћ|рќ‘Ѓрќ‘” .
   The coprimality (рќ‘Ћ, рќ‘ћ) = 1 is supplied by Lemma G1a. The smooth weight is the normalized
Fourier fibre weight

                                                              МѓпёЃрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)
                                                   рќђ№ (рќ‘љ, рќ‘ћ) = рќ‘Љ
   from G8a.3/G3a.2. Explicitly, CKPD.4 defines

                                  1                                         рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦     в„Ћрќ‘¦
                                       в€«пёЃ                              (пё‚         )пё‚ (пё‚      )пё‚
                 рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) =              рќњ”рќђґ (рќ‘Ћ)рќњ”рќ‘„ (рќ‘ћ)рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ                  рќ‘’ в€’           рќ‘‘рќ‘¦
                                  рќ‘ћ                                            рќ‘ћ         рќ‘ћ

                                                             15
    and рќђ№ = рќ‘Љ  МѓпёЃрќ‘”,в„Ћ = рќ’њв€’1 рќ’Ірќ‘”,в„Ћ , with рќ’њрќ‘”,в„Ћ,рќ‘… accounted for in X10.7. Its derivative bounds,
                          рќ‘”,в„Ћ,рќ‘…
including the chain-rule dependence of рќђ№рќ‘Ћ,рќ‘ћ on рќ‘Ћ and рќ‘ћ, are proved in Lemma CKPD. They use the
central frequency condition |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ and the central balance restrictions.
    The coefficient norms are

                                рќ‘” (log рќ‘Ѓ ) ,                       рќ‘” (log рќ‘Ѓ ) .
                                          рќђ¶                                  рќђ¶
                      вЂ–рќ›јрќ‘” вЂ–2 в‰Є рќђґ1/2                    вЂ–рќ›ѕрќ‘”,в„Ћ вЂ–2 в‰Є рќ‘„1/2
   In the central balanced range

                              рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘” ,        рќ‘†рќ‘” = рќ‘Ѓ 1/2+рќ‘‚(рќњ‚0 ) /рќ‘”,
   and

                                     |в„Ћ|рќ‘Ѓрќ‘”
                                           в‰Ќ |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµ .
                                     рќђґрќ‘” рќ‘„рќ‘”
   Thus DFI-X10 gives

                      |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќ‘‚(рќњ‚0 )+рќњЂ (log рќ‘Ѓ )рќђ¶ рќ‘” в€’47/48 (1 + |в„Ћ|рќ‘”)в€’рќђґ .
   After summing over в„Ћ Мё= 0 and divisor-bounded рќ‘” | рќ‘Ѓ , this is рќ‘њ(рќ‘Ѓ ) once рќ‘‚(рќњ‚0 ) + рќњЂ + рќ‘њ(1) < 1/96.

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. |в„Ћ|рќ‘” > (log рќ‘Ѓ )рќђµ : high Fourier tail, routed by Lemmas G2a, X10ER, and C1;

  2. рќ‘ћ/(рќ‘ћ, в„Ћрќ‘Ѓрќ‘” ) в‰¤ (log рќ‘Ѓ )рќђµ : small conductor, routed by C1P/C1A/C1;

  3. large рќ‘”: routed by Lemmas G1a, G8a, X10ER, and C1;

  4. boundary/short-volume ranges: routed by C1P/C1A/C1;

  5. в„Ћ = 0: local/main term handled by Lemma G8a through LPI and then assembled by H4.

    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.
    вЂ”

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;


                                                  16
  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: DukeвЂ“FriedlanderвЂ“Iwaniec bilinear Kloosterman-fraction estimate in the
form stated in X10.2/X10.14.
   Internal dependencies: G1a, G2a, G3a, G4a, CKPD, X10ER, C1A, C1, G8a.
   Children served: G4a, G8a, CKPD, I1.

5.2   Part 2. X16: X16 divisor-sum/BRS verification
Source file: External/x_16_divisor_sum_brs_verification_ltx.md.

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 C1 Edge unless it already carries a routing tag. The only exter-
nal/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 d

   Logical dependencies are X16BRS, X16C, BRS, TTH, F4, and ShiuвЂ™s arithmetic-progression
BrunвЂ“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 BrunвЂ“Titchmarsh theorem for multiplicative functions, Journal fuer die reine und
     angewandte Mathematik 313 (1980), 161вЂ“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.
  вЂ”




                                               17
X16.1. Statement Fix the HeathвЂ“Brown depth рќђЅ0 . Let в„¬ be a B1 typed dyadic block and let рќђ¶
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 рќ‘Ѓ в€’ рќ‘ѓ ;

  3. quotient carrier рќ‘  from a recorded equation рќђї = рќ‘‘рќ‘ ;

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

   Let рќ‘‹рќђ¶ be the dyadic height of рќђ¶, and let рќђј be an additive interval. Put

                                      рќ‘Њ16 := max{|рќђј в€© Z|, рќ‘‹рќђ¶ (log рќ‘Ѓ )в€’рќђµ16 }.
   Then

                                                       рќ‘Њ16
                   Massв„¬ (рќђ¶ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16           + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 ,               (X16-BRS)
                                                       рќ‘‹рќђ¶
    where рќђ¶16 , рќњЊ16 > 0 depend only on рќђЅ0 , 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.
    вЂ”

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

                 рќњЏрќђѕ1 (рќ‘ќ)         рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 в‰Є рќ‘Њ16 рќ‘€ (log рќ‘Ѓ )рќђ¶ + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶ ,
           в€‘пёЃ              в€‘пёЃ

           рќ‘ќв€€рќђј             рќ‘ўв‰Ќрќ‘€

    for fixed рќђѕ1 , рќђѕ2 , рќђѕ3 , рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ , after dyadic localization and with the harmless polylogarithmic
losses coming from the B1 coefficient types рќњ‡, 1, log.
    The floor рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16 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
strict C1 saving once рќђµ16 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 рќ‘ѓ , same-side complement рќ‘€ , and opposite-side product рќ‘„ =
     рќ‘Ѓ в€’ рќ‘ѓрќ‘€;
                                                                рќђґ , applied to рќ‘„ = рќ‘Ѓ в€’ рќ‘ќрќ‘ў after
  3. ShiuвЂ™s arithmetic-progression BrunвЂ“Titchmarsh theorem for рќњЏрќђѕ
     fixing рќ‘ќ or рќ‘ў;

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

  5. partial summation for smooth dyadic weights;

                                                        18
  6. divisor-sum stability under fixed divisor quotients and polylogarithmic CRT restrictions.

   No prime distribution theorem is used in X16.
   вЂ”

X16.3. Proof Outline for X16-BRS The following is the reduction outline; the full proof of
the analytic correlation estimate, including the two CauchyвЂ“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 HeathвЂ“Brown depth
рќђЅ0 is fixed, every coefficient sequence produced by B1, B3, F3/F4 tags and E5-clean transports is
bounded by (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) рќњЏрќђѕ (В·) for a fixed рќђѕ = рќђѕ(рќђЅ0 ), after dyadic localization and after absorbing
smooth cutoffs by partial summation.
    For a grouped product carrier рќђ¶ = рќ‘ѓ , fixing рќ‘ѓ = рќ‘ќ leaves a fixed-depth number of factorizations
of рќ‘ќ, but it also leaves a genuine complementary correlation рќ‘„ = рќ‘Ѓ в€’ рќ‘ќрќ‘ў. Thus

        Massв„¬ (рќђ¶ в€€ рќђј) в‰Є (log рќ‘Ѓ )рќђ¶                                рќњЏрќђѕ1 (рќ‘ќ)         рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 ,
                                                  в€‘пёЃ                       в€‘пёЃ

                                           рќ‘ќв€€рќђј16 в€©[рќ‘‹рќ‘ѓ /2,3рќ‘‹рќ‘ѓ ]             рќ‘ўв‰Ќрќ‘€

    where рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ . This is not bounded by averaging only рќњЏрќђѕ1 (рќ‘ќ). Instead, for fixed рќ‘ќ, the values
рќ‘Ѓ в€’рќ‘ќрќ‘ў lie in one arithmetic progression modulo рќ‘ќ; for fixed рќ‘ў, they lie in one arithmetic progression
modulo рќ‘ў.
    Lemma X16C applies ShiuвЂ™s AP theorem, combined with CauchyвЂ“Schwarz and second moments
for fixed divisor functions, and proves

                  рќњЏрќђѕ1 (рќ‘ќ)         рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 в‰Є рќ‘Њ16 рќ‘€ (log рќ‘Ѓ )рќђ¶ + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶ .
           в€‘пёЃ               в€‘пёЃ

          рќ‘ќв€€рќђј16             рќ‘ўв‰Ќрќ‘€

   Since рќ‘€ в‰Ќ рќ‘Ѓ/рќ‘‹рќ‘ѓ , this gives exactly

                                                       рќ‘Њ16
                                         рќ‘Ѓ (log рќ‘Ѓ )рќђ¶       + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶ .
                                                       рќ‘‹рќђ¶
    The complementary carrier рќ‘Ѓ в€’ рќ‘ѓ is identical after replacing рќђј by рќ‘Ѓ в€’ рќђј. A quotient carrier рќ‘ 
from рќђї = рќ‘‘рќ‘  is reduced to the grouped product case for рќ‘‘рќ‘ ; the factor рќ‘‘ changes both the interval
length and dyadic scale by the same controlled amount, so the ratio рќ‘Њ16 /рќ‘‹рќђ¶ is preserved up to
polylogarithmic losses. Controlled CRT restrictions split the interval into рќ‘‚((log рќ‘Ѓ )рќђ¶ ) residue
subintervals, and full-rank affine transports change lattice index and derivatives by рќ‘‚((log рќ‘Ѓ )рќђ¶ ).
These losses are absorbed in рќђ¶16 .
    This reduction outline is deliberately standard rather than deep, but it is not the rejected one-
variable shortcut: the рќ‘Ѓ в€’ рќ‘ќрќ‘ў correlation is retained and estimated by AP divisor averages. No
prime distribution theorem and no cancellation of О› is used in X16.
    вЂ”

X16.4. Match to BRS.1

Grouped product carrier Fixing рќђ¶ = рќ‘› leaves boundedly many factorizations of рќ‘› and bound-
edly many remaining parent variables, all of fixed depth рќ‘‚(рќђЅ0 ). Summing over рќ‘› в€€ рќђј gives a
fixed-order divisor correlation. X16-BRS gives the relative factor рќ‘Њ16 /рќ‘‹рќђ¶ , with only polylogarith-
mic loss.



                                                           19
Complementary carrier        If рќђ¶ = рќ‘Ѓ в€’ рќ‘ѓ , then рќђ¶ в€€ рќђј is equivalent to рќ‘ѓ в€€ рќ‘Ѓ в€’ рќђј. The previous
case applies to рќ‘ѓ .

Quotient carrier If рќђї = рќ‘‘рќ‘  and рќђ¶ = рќ‘ , then рќ‘  в€€ рќђј restricts рќ‘‘рќ‘  to total length рќ‘‚(рќђ·рќ‘Њ ) inside
dyadic scale рќђ·рќ‘‹рќђ¶ . Applying the grouped-product carrier estimate to рќ‘‘рќ‘  gives

                                              рќђ·рќ‘Њ
                              рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16       + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 ,
                                              рќђ·рќ‘‹рќђ¶
   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 coordi-
nate changes alter indices and lengths by at most polylogarithmic factors in the BRS route. Those
losses are absorbed by рќђ¶16 . Tagged rank drops do not enter BRS as untagged B1-origin carriers;
they are handled by ROC/BRS case 3 or by E10M.
    вЂ”

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

                                    |рќђїрќ‘љ (О©)| < рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђµ
   and choosing рќђµ larger than the fixed C1 and X16 losses gives strict C1 Edge:

                  Mass(рќђїрќ‘љ (О©)) в‰Є рќ‘Ѓ (log рќ‘Ѓ )в€’рќђ¶0 в€’10 + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 = рќ‘њ(рќ‘Ѓ ).
   Therefore a TC1 coarea test that remains after ROC/BRS must satisfy the near-global lower
bound used by TTH:

                                        рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµрќњ… .
   This is the precise reason the TC1 proof does not require a low-рќњѓ X9L theorem.
   вЂ”

Parameter check 5.2 (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 рќ‘ќ- and рќ‘ў-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 рќ‘›в€€рќђј рќњЏрќ‘ (рќ‘›) after fixing the carrier value is still not
                                             в€‘пёЂ

used, because the remaining variables impose a divisor correlation along рќ‘Ѓ в€’ рќ‘›рќ‘Ј. The new proof
controls that correlation directly.

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

5.3   Part 3. G1a: CKP gcd splitting
Source file: Lemmas/g_1_a_ltx.md.

                                                   20
G1a. CKP GCD Splitting Lemma

G1a.0. Role Logical ID: G1a.
   Used by: G2a, G3a, G4a, 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

                                                  рќ‘ўрќ‘¦ + рќ‘ўвЂІ рќ‘¦ вЂІ = рќ‘Ѓ
   into the coprime form required for the later Fourier/AP expansion and for the application of
Kloosterman-fraction estimates.
   The main output is

                       рќ‘” = gcd(рќ‘ў, рќ‘ўвЂІ ),          рќ‘ў = рќ‘”рќ‘Ћ,        рќ‘ўвЂІ = рќ‘”рќ‘ћ,          (рќ‘Ћ, рќ‘ћ) = 1.
   If рќ‘” в€¤ рќ‘Ѓ , there are no solutions. If рќ‘” | рќ‘Ѓ , the equation becomes

                                                                            рќ‘Ѓ
                                         рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,       рќ‘Ѓрќ‘” =          .
                                                                            рќ‘”
   вЂ”

G1a.1. Initial CKP block          Consider a balanced CKP atom of the form

                                                                          рќ‘ў рќ‘ўвЂІ рќ‘¦ рќ‘¦ вЂІ
                                                                     (пё‚                )пё‚
             в„›(рќ‘Ѓ ) =                   рќ›ј(рќ‘ў)рќ›јвЂІ (рќ‘ўвЂІ )рќ›Ѕ(рќ‘¦)рќ›Ѕ вЂІ (рќ‘¦ вЂІ )рќ‘Љ
                           в€‘пёЃ
                                                                           , , ,     1рќ‘ўрќ‘¦+рќ‘ўвЂІ рќ‘¦вЂІ =рќ‘Ѓ ,
                                                                          рќ‘€ рќ‘€вЂІ рќ‘Њ рќ‘Њ вЂІ
                       рќ‘ўв€јрќ‘€, рќ‘ўвЂІ в€јрќ‘€ вЂІ
                       рќ‘¦в€јрќ‘Њ, рќ‘¦ вЂІ в€јрќ‘Њ вЂІ

   where рќ‘Љ is a smooth compactly supported weight and the coefficients are finite-convolution/divisor-
bounded:

                            |рќ›ј(рќ‘ў)|, |рќ›јвЂІ (рќ‘ўвЂІ )|, |рќ›Ѕ(рќ‘¦)|, |рќ›Ѕ вЂІ (рќ‘¦ вЂІ )| в‰Є (log рќ‘Ѓ )рќђ¶(рќђЅ0 ) .
   The balanced CKP range means that, after grouping variables,

                           рќ‘€ в‰Ќ рќ‘€ вЂІ в‰Ќ рќ‘Ѓ 1/2+рќ‘‚(рќњ…) ,             рќ‘Њ в‰Ќ рќ‘Њ вЂІ в‰Ќ рќ‘Ѓ 1/2+рќ‘‚(рќњ…) .
   The exact shape of the ranges is not important for G1a. The only point needed here is that рќ‘ў
and рќ‘ўвЂІ are the two grouped convolution variables to which gcd splitting is applied.
   вЂ”

G1a.2. GCD splitting         For each pair (рќ‘ў, рќ‘ўвЂІ ), set

                                                  рќ‘” = gcd(рќ‘ў, рќ‘ўвЂІ ).
   Then there are unique positive integers рќ‘Ћ, рќ‘ћ such that

                                  рќ‘ў = рќ‘”рќ‘Ћ,           рќ‘ўвЂІ = рќ‘”рќ‘ћ,         (рќ‘Ћ, рќ‘ћ) = 1.
   Substituting into

                                                 рќ‘ўрќ‘¦ + рќ‘ўвЂІ рќ‘¦ вЂІ = рќ‘Ѓ,

                                                         21
   gives

                                               рќ‘”рќ‘Ћрќ‘¦ + рќ‘”рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓ.
   If

                                                      рќ‘” в€¤ рќ‘Ѓ,
   there are no solutions. If

                                                      рќ‘” | рќ‘Ѓ,
   then, writing

                                                               рќ‘Ѓ
                                                     рќ‘Ѓрќ‘” =        ,
                                                               рќ‘”
   we obtain the reduced equation

                                     рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,                (рќ‘Ћ, рќ‘ћ) = 1.
   вЂ”

G1a.3. Exact reparametrization of the block                       The decomposition by рќ‘” is exact:

                                             в„›(рќ‘Ѓ ) =            в„›рќ‘” (рќ‘Ѓ ),
                                                          в€‘пёЃ

                                                          рќ‘”|рќ‘Ѓ

   where

              в„›рќ‘” (рќ‘Ѓ ) =                     рќ›ј(рќ‘”рќ‘Ћ)рќ›јвЂІ (рќ‘”рќ‘ћ)                   рќ›Ѕ(рќ‘¦)рќ›Ѕ вЂІ (рќ‘¦ вЂІ )рќ‘Љрќ‘” (рќ‘Ћ, рќ‘ћ, рќ‘¦, рќ‘¦ вЂІ ),
                                в€‘пёЃ                               в€‘пёЃ

                          рќ‘Ћв€јрќ‘€/рќ‘”, рќ‘ћв€јрќ‘€ вЂІ /рќ‘”                  рќ‘¦в€јрќ‘Њ, рќ‘¦ вЂІ в€јрќ‘Њ вЂІ
                             (рќ‘Ћ,рќ‘ћ)=1                       рќ‘Ћрќ‘¦+рќ‘ћрќ‘¦ вЂІ =рќ‘Ѓрќ‘”

   and

                                                                рќ‘”рќ‘Ћ рќ‘”рќ‘ћ рќ‘¦ рќ‘¦ вЂІ
                                                           (пё‚                      )пё‚
                                                вЂІ
                                 рќ‘Љрќ‘” (рќ‘Ћ, рќ‘ћ, рќ‘¦, рќ‘¦ ) = рќ‘Љ             , , ,     .
                                                                рќ‘€ рќ‘€вЂІ рќ‘Њ рќ‘Њ вЂІ
   If рќ‘” в€¤ рќ‘Ѓ , the corresponding layer is empty. Therefore the sum is only over рќ‘” | рќ‘Ѓ .
   вЂ”

G1a.4. Ranges after splitting          Define

                                                    рќ‘€                   рќ‘€вЂІ
                                            рќђґрќ‘” =      ,         рќ‘„рќ‘” =       .
                                                    рќ‘”                   рќ‘”
   Then

                                             рќ‘Ћ в€ј рќђґрќ‘” ,           рќ‘ћ в€ј рќ‘„рќ‘” .
   In the balanced symmetric case рќ‘€ в‰Ќ рќ‘€ вЂІ в‰Ќ рќ‘Ѓ 1/2 , this gives

                                                                рќ‘Ѓ 1/2
                                              рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ               .
                                                                 рќ‘”

                                                          22
   This is the form used later in G3a/G4a:

                                                        рќ‘Ѓ 1/2
                                               рќ‘†рќ‘” :=          .
                                                         рќ‘”
   вЂ”

G1a.5. Coefficient preservation          Define the new coefficients

                                рќ›јрќ‘” (рќ‘Ћ) = рќ›ј(рќ‘”рќ‘Ћ),            рќ›ѕрќ‘” (рќ‘ћ) = рќ›јвЂІ (рќ‘”рќ‘ћ).
   If the original coefficients are divisor-bounded, then

                                    |рќ›јрќ‘” (рќ‘Ћ)|, |рќ›ѕрќ‘” (рќ‘ћ)| в‰Є (log рќ‘Ѓ )рќђ¶(рќђЅ0 ) .
   Moreover, on dyadic intervals,

                                                рќ‘” (log рќ‘Ѓ )
                                                           рќђ¶(рќђЅ0 )
                                      вЂ–рќ›јрќ‘” вЂ–2 в‰Є рќђґ1/2               ,


                                                рќ‘” (log рќ‘Ѓ )
                                                           рќђ¶(рќђЅ0 )
                                      вЂ–рќ›ѕрќ‘” вЂ–2 в‰Є рќ‘„1/2               .
   These estimates are needed for the later DFI/Kloosterman-fraction matching.
   вЂ”

G1a.6. Local meaning of the condition (рќ‘Ћ, рќ‘ћ) = 1                  The condition

                                                   (рќ‘Ћ, рќ‘ћ) = 1
    is not an additional restriction; it is part of the exact gcd parametrization. It guarantees the
existence of the inverse class

                                               рќ‘Ћ     (mod рќ‘ћ),
   which appears when solving the congruence

                                          рќ‘Ћрќ‘¦ в‰Ў рќ‘Ѓрќ‘”        (mod рќ‘ћ).
   This condition matches the coprimality condition in the DFI Kloosterman-fraction estimate
used by X10.
   вЂ”

G1a.7. Lemma G1a

Lemma 5.3 (Lemma G1a). Suppose a CKP atom contains the equation

                                             рќ‘ўрќ‘¦ + рќ‘ўвЂІ рќ‘¦ вЂІ = рќ‘Ѓ.
   Then exact gcd splitting gives a disjoint decomposition by

                                             рќ‘” = gcd(рќ‘ў, рќ‘ўвЂІ ),
   and on every nonzero layer рќ‘” | рќ‘Ѓ the equation becomes


                                                      23
                                                                   рќ‘Ѓ
                           рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,             рќ‘Ѓрќ‘” =         ,          (рќ‘Ћ, рќ‘ћ) = 1.
                                                                   рќ‘”
   The coefficients remain finite-convolution/divisor-bounded, and the new dyadic ranges are

                                      рќ‘Ћ в€ј рќ‘€/рќ‘”,                    рќ‘ћ в€ј рќ‘€ вЂІ /рќ‘”.
   In the balanced range this gives

                                                             рќ‘Ѓ 1/2
                                                    рќ‘Ћ, рќ‘ћ в‰Ќ         .
                                                              рќ‘”
Proof. All assertions follow from the uniqueness of the decomposition

                               рќ‘ў = рќ‘”рќ‘Ћ,               рќ‘ўвЂІ = рќ‘”рќ‘ћ,            (рќ‘Ћ, рќ‘ћ) = 1,
   where рќ‘” = gcd(рќ‘ў, рќ‘ўвЂІ ), and from substitution into the original equation. If рќ‘” в€¤ рќ‘Ѓ , the equation

                                                  рќ‘”(рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ ) = рќ‘Ѓ
    is impossible. If рќ‘” | рќ‘Ѓ , division by рќ‘” gives the reduced equation. The coefficient and range
statements follow immediately from dyadic support and divisor-boundedness.
    The lemma follows.
    вЂ”


Remark 5.4 (G1a.8. Output).

                             G1a gives the exact CKP gcd splitting.

    Nonzero layers require рќ‘” | рќ‘Ѓ , and every such layer has reduced equation рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓ/рќ‘” with
(рќ‘Ћ, рќ‘ћ) = 1.

G1a.9. Logical Dependencies Internal dependencies: B3, F3, F4, and the CKP terminal
predicate.
   Children served: G2a, G3a, G4a, G8a, X10.

5.4   Part 4. G2a: Smooth AP Fourier expansion
Source file: Lemmas/g_2_a_ltx.md.

G2a. Weighted Smooth AP Fourier Expansion for CKP

G2a.0. Role Logical ID: G2a.
  Used by: G3a, G4a, G8a, X10, C1A, C1.
  Uses: G1a, C1A, C1, G8a, 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 рќ‘Љрќ‘Њ (рќ‘¦), but the full tagged fibre weight:

                                                  рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦                             рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                             (пё‚              )пё‚                  (пё‚           )пё‚
                                         вЂІ
                      рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦) = рќ›Ѕ(рќ‘¦)рќ›Ѕ                    рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ                          .
                                                     рќ‘ћ                                   рќ‘ћ

                                                         24
  Lemma G2a turns

                                   рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,                  (рќ‘Ћ, рќ‘ћ) = 1
  into a smooth AP Fourier expansion in which:

  вЂў в„Ћ = 0 gives the local/main zero-frequency term;

  вЂў в„Ћ Мё= 0 gives the oscillatory Kloosterman-fraction input for Lemma G3a;

  вЂў Fourier weights satisfy rapid decay sufficient for the high-frequency Edge routing in C1P/C1A/C1
    and for the CKP assembly in G8a.

  вЂ”

G2a.1. Reduced CKP equation           On a fixed рќ‘”-layer after Lemma G1a, we have

                         рќ‘ў = рќ‘”рќ‘Ћ,        рќ‘ўвЂІ = рќ‘”рќ‘ћ,                (рќ‘Ћ, рќ‘ћ) = 1,         рќ‘” | рќ‘Ѓ.
  Set

                                                               рќ‘Ѓ
                                                    рќ‘Ѓрќ‘” =         .
                                                               рќ‘”
  Then the CKP equation becomes

                                                 рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” .
  Eliminate рќ‘¦ вЂІ :

                                                        рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                 рќ‘¦вЂІ =           .
                                                           рќ‘ћ
  The condition рќ‘¦ вЂІ в€€ Z is equivalent to

                                            рќ‘Ћрќ‘¦ в‰Ў рќ‘Ѓрќ‘”         (mod рќ‘ћ).
  Since (рќ‘Ћ, рќ‘ћ) = 1, this is equivalent to the congruence

                                            рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ        (mod рќ‘ћ).
  вЂ”

G2a.2. Tagged weighted fibre        For fixed (рќ‘”, рќ‘Ћ, рќ‘ћ), define the tagged fibre contribution

                                   рќ’®рќ‘Ћ,рќ‘ћ =                             рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦),
                                                        в€‘пёЃ

                                                 рќ‘¦в‰Ўрќ‘Ѓрќ‘” рќ‘Ћ    (mod рќ‘ћ)

  where

                                                 рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦                           рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                            (пё‚             )пё‚                 (пё‚             )пё‚
                                        вЂІ
                     рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦) = рќ›Ѕ(рќ‘¦)рќ›Ѕ                    рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ                        .
                                                    рќ‘ћ                                 рќ‘ћ
  Here:


                                                          25
  вЂў рќ‘Љрќ‘Њ , рќ‘Љрќ‘Њ вЂІ are smooth dyadic weights inherited from the fixed tag (в„¬, рќњЏ );

  вЂў \{}(\{}beta,

   \{}betaвЂ™\{}) are divisor-bounded finite-convolution coefficient weights;

  вЂў the summand is defined as zero unless (рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ в€€ Z and lies in the tagged dyadic support.

     For Fourier expansion, the smooth part is expanded directly. If some finite-convolution coef-
ficient 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.
     вЂ”

G2a.3. Smooth AP Fourier identity Let рќђ№ be a smooth compactly supported tagged fibre
weight on Z. For a residue class рќ‘џ (mod рќ‘ћ),

                                                    1 в€‘пёЃ М‚пёЂ          в„Ћ   в„Ћрќ‘џ
                                                                    (пё‚ )пё‚ (пё‚    )пё‚
                                            рќђ№ (рќ‘¦) =
                                   в€‘пёЃ
                                                          рќђ№            рќ‘’    ,
                             рќ‘¦в‰Ўрќ‘џ    (mod рќ‘ћ)
                                                    рќ‘ћ в„Ћв€€Z            рќ‘ћ   рќ‘ћ

   where, in discrete normalization,

                                        рќђ№М‚пёЂ (рќњ‰) =         рќђ№ (рќ‘¦)рќ‘’(в€’рќ‘¦рќњ‰).
                                                    в€‘пёЃ

                                                    рќ‘¦в€€Z

   Applying this with

                                            рќ‘џ = рќ‘Ѓрќ‘” рќ‘Ћ       (mod рќ‘ћ),
   we get

                                            1 в€‘пёЃ М‚пёЂ           в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                          (пё‚ )пё‚ (пё‚         )пё‚
                                   рќ’®рќ‘Ћ,рќ‘ћ =         рќђ№рќ‘Ћ,рќ‘ћ          рќ‘’       .
                                            рќ‘ћ в„Ћв€€Z             рќ‘ћ    рќ‘ћ
    This identity is exact for the tagged smooth fibre after the standard smooth extension conven-
tion. Boundary errors caused by compact support truncation are C1A-admitted C1 boundary/short-
volume errors.
    вЂ”

G2a.4. Zero-frequency term           The zero-frequency term is
                                          1             1 в€‘пёЃ
                                     (0)
                                    рќ’®рќ‘Ћ,рќ‘ћ = рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ (0) =      рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦).
                                          рќ‘ћ             рќ‘ћ рќ‘¦
   It has no oscillatory phase

                                                  в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                    (пё‚         )пё‚
                                                рќ‘’
                                                   рќ‘ћ
    with в„Ћ Мё= 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 H4:



                                                         26
                               (0)
                             рќ‘ЂCKP,в„¬,рќњЏ (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                   CKP
                                                       (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
   Thus Lemma G2a supplies the AP/Fourier identity, while Lemma G8a supplies the LPI nor-
malization check for H4.
   вЂ”

G2a.5. Nonzero-frequency oscillatory terms                 The nonzero-frequency contribution is

                                                     1            в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                 (пё‚ )пё‚ (пё‚       )пё‚
                      рќ’Єрќ‘” =               рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ
                             в€‘пёЃ       в€‘пёЃ
                                                                    рќ‘’       .
                           в„ЋМё=0 рќ‘Ћв€јрќђґ ,рќ‘ћв€јрќ‘„
                                                     рќ‘ћ            рќ‘ћ    рќ‘ћ
                                       рќ‘”       рќ‘”
                                     (рќ‘Ћ,рќ‘ћ)=1

   This is the precise input for Lemma G3a: a weighted bilinear Kloosterman fraction sum with
parameters

                               рќ‘Ђ = рќђґрќ‘” ,            рќ‘„ = рќ‘„рќ‘” ,     рќ‘ = в„Ћрќ‘Ѓрќ‘” .
   The finite-convolution coefficients and the tagged fibre transform are absorbed into divisor-
bounded weighted coefficient sequences, with polylogarithmic losses only.
   вЂ”

G2a.6. Fourier-weight decay          Assume the CKP balanced range:

                                                                 рќ‘Ѓ 1/2+рќ‘‚(рќњ‚)
                            рќ‘Њ в‰Ќ рќ‘Ѓ 1/2+рќ‘‚(рќњ‚) ,          рќ‘ћ в‰Ќ рќ‘„рќ‘” в‰Ќ              .
                                                                      рќ‘”
   For the smooth fibre weight we have, for every рќђґ > 0,
                                                            )пё‚в€’рќђґ
                             вѓ’1
                             вѓ’ рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ в„Ћ вѓ’ в‰Єрќђґ рќђїрќђ¶рќђ№ рќ‘Њ 1 + |в„Ћ|рќ‘Њ
                             вѓ’       (пё‚ )пё‚вѓ’        (пё‚
                                          вѓ’
                             вѓ’рќ‘ћ                                  .
                                       рќ‘ћ вѓ’       рќ‘ћ       рќ‘ћ
   Since

                                                    рќ‘Њ
                                                      в‰Ќрќ‘”
                                                    рќ‘ћ
   up to fixed dyadic constants, this gives
                               вѓ’1
                               вѓ’
                               вѓ’ рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ в„Ћ вѓ’ в‰Єрќђґ рќђїрќђ¶рќђ№ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ .
                                       (пё‚ )пё‚вѓ’
                                            вѓ’
                               вѓ’рќ‘ћ        рќ‘ћ вѓ’
    The polylogarithmic factor рќђїрќђ¶рќђ№ records derivative bounds and finite-convolution coefficient
losses. It is harmless in C1P/C1A/C1 and G8a because all those estimates have arbitrary polylogarithmic
saving margins.
    вЂ”




                                                      27
G2a.7. Lemma G2a
Lemma 5.5 (Lemma G2a). For each fixed balanced CKP рќ‘”-layer after G1a, the reduced equation

                                      рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,                (рќ‘Ћ, рќ‘ћ) = 1,
   is equivalent, after eliminating рќ‘¦ вЂІ , to

                                                рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ       (mod рќ‘ћ).
   For the tagged weighted fibre

                                       рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦                                      рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                (пё‚            )пё‚               (пё‚          )пё‚
                                            вЂІ
                      рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦) = рќ›Ѕ(рќ‘¦)рќ›Ѕ         рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ                                   ,
                                          рќ‘ћ                                            рќ‘ћ
   we have the exact smooth AP expansion

                                                      1 в€‘пёЃ М‚пёЂ               в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                          (пё‚ )пё‚ (пё‚        )пё‚
                                           рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦) =
                               в€‘пёЃ
                                                            рќђ№рќ‘Ћ,рќ‘ћ              рќ‘’       .
                         рќ‘¦в‰Ўрќ‘Ѓрќ‘” рќ‘Ћ    (mod рќ‘ћ)
                                                      рќ‘ћ в„Ћв€€Z                 рќ‘ћ    рќ‘ћ

   The zero-frequency term
                                                         1 М‚пёЂ
                                                           рќђ№рќ‘Ћ,рќ‘ћ (0)
                                                         рќ‘ћ
   is the CKP local/main term, and the nonzero frequencies produce the DFI/Kloosterman input

                                                       1                     в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                           (пё‚ )пё‚ (пё‚        )пё‚
                        рќ’Єрќ‘” =               рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ
                               в€‘пёЃ      в€‘пёЃ
                                                                               рќ‘’       .
                             в„ЋМё=0 рќ‘Ћв€јрќђґ ,рќ‘ћв€јрќ‘„
                                                       рќ‘ћ                     рќ‘ћ    рќ‘ћ
                                        рќ‘”            рќ‘”
                                      (рќ‘Ћ,рќ‘ћ)=1

   Moreover, in the balanced range,
                                   вѓ’1
                                   вѓ’
                                   вѓ’ рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ в„Ћ вѓ’ в‰Єрќђґ рќђїрќђ¶рќђ№ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ .
                                           (пё‚ )пё‚вѓ’
                                                вѓ’
                                   вѓ’рќ‘ћ        рќ‘ћ вѓ’
Proof. The congruence follows from

                                  рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” в‡ђв‡’ рќ‘Ћрќ‘¦ в‰Ў рќ‘Ѓрќ‘”                 (mod рќ‘ћ),
    and from (рќ‘Ћ, рќ‘ћ) = 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
follows by separating в„Ћ = 0 from в„Ћ Мё= 0. The decay estimate follows from rapid decay of the Fourier
transform of the tagged smooth fibre weight and from \{}(Y/q \{}asymp g\{}) in the balanced
CKP range. Lemma proved.
    вЂ”


Remark 5.6 (G2a.8. Output).

                  G2a gives the weighted smooth AP Fourier expansion for CKP.

    After G1a, the reduced equation is converted to the congruence рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ (mod рќ‘ћ). 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.

                                                            28
G2a.9. Logical Dependencies Internal dependencies: G1a, C1A, C1, G8a, and the CKP terminal
predicate.
   Children served: G3a, G4a, G8a, X10, C1A, C1.

5.5    Part 5. G3a: CKP-to-DFI conversion
Source file: Lemmas/g_3_a_ltx.md.

G3a. CKP to Kloosterman-Fraction Reduction

G3a.0. Role Logical ID: G3a.
    Used by: G4a, G8a, X10.
    Uses: G1a, G2a, CKPD, X10.
    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.
    The target is:

                                                            1 М‚пёЃ               в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                             (пё‚ )пё‚ (пё‚          )пё‚
                                                рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ
                                 в€‘пёЃ         в€‘пёЃ
                   рќ’Єрќ‘”      в‡ќ                                    рќ‘Њ                рќ‘’       ,
                                 в„ЋМё=0 рќ‘Ћв€јрќђґ , рќ‘ћв€јрќ‘„
                                                            рќ‘ћ                  рќ‘ћ    рќ‘ћ
                                            рќ‘”        рќ‘”
                                           (рќ‘Ћ,рќ‘ћ)=1

   that is, to a Kloosterman-fraction sum with parameters

                                 рќ‘Ђ = рќђґрќ‘” ,             рќ‘„ = рќ‘„рќ‘” ,            рќ‘ = в„Ћрќ‘Ѓрќ‘” .
   In the balanced CKP case,

                                                                рќ‘Ѓ 1/2
                                                рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ             .
                                                                 рќ‘”
   вЂ”

G3a.1. Input from G1a and G2a                   After G1a we have the exact gcd splitting

                                                                                        рќ‘Ѓ
                          рќ‘ў = рќ‘”рќ‘Ћ,          рќ‘ўвЂІ = рќ‘”рќ‘ћ,         (рќ‘Ћ, рќ‘ћ) = 1,         рќ‘Ѓрќ‘” =      .
                                                                                        рќ‘”
   After G2a, the reduced equation

                                                  рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘”
   gives the congruence

                                                рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ    (mod рќ‘ћ).
   The smooth AP Fourier expansion gives the nonzero-frequency part

                                                          1 М‚пёЃ              в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                          (пё‚ )пё‚ (пё‚        )пё‚
                     рќ’Єрќ‘” =                     рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ
                               в€‘пёЃ      в€‘пёЃ
                                                              рќ‘Њ               рќ‘’       ,
                               в„ЋМё=0 рќ‘Ћв€јрќђґ , рќ‘ћв€јрќ‘„
                                                          рќ‘ћ                 рќ‘ћ    рќ‘ћ
                                       рќ‘”         рќ‘”
                                      (рќ‘Ћ,рќ‘ћ)=1

   up to harmless smooth weights in рќ‘Ћ, рќ‘ћ inherited from the dyadic decomposition.
   вЂ”


                                                           29
G3a.2. Incorporating smooth dyadic weights                     The coefficients after gcd splitting and dyadic
localization can be written as

                           рќ›Ѕрќ‘” (рќ‘Ћ) = рќ›ј(рќ‘”рќ‘Ћ)рќњ”рќђґ (рќ‘Ћ),              рќ›ѕрќ‘” (рќ‘ћ) = рќ›јвЂІ (рќ‘”рќ‘ћ)рќњ”рќ‘„ (рќ‘ћ),
   where рќњ”рќђґ , рќњ”рќ‘„ are smooth dyadic cutoffs supported on

                                             рќ‘Ћ в‰Ќ рќђґрќ‘” ,         рќ‘ћ в‰Ќ рќ‘„рќ‘” .
    All smooth weights depending only on рќ‘Ћ or only on рќ‘ћ may be absorbed into рќ›Ѕрќ‘” or рќ›ѕрќ‘” . The CKP
fibre weight from G8a is slightly more general: after eliminating рќ‘¦ вЂІ , the factor

                                                    рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ (в„Ћ/рќ‘ћ)

depends smoothly on both рќ‘Ћ and рќ‘ћ. This two-variable weight is not replaced by a separated
surrogate. It is kept as a normalized smooth DFI weight рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ). The derivative admissibility
of рќ‘Љрќ‘”,в„Ћ , including the chain-rule terms from рќ›Ѕ вЂІ ((рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ) and рќ‘Љрќ‘Њ вЂІ ((рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ), 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 рќ›Ѕрќ‘” and рќ›ѕрќ‘” .
    Thus it is enough to treat sums of the form

                                                                   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                                    (пё‚         )пё‚
                         рќ’Єрќ‘”,в„Ћ =           рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ)рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)рќ‘’
                                        в€‘пёЃ
                                                                         .
                                рќ‘Ћв€јрќђґ , рќ‘ћв€јрќ‘„
                                                                    рќ‘ћ
                                        рќ‘”       рќ‘”
                                      (рќ‘Ћ,рќ‘ћ)=1
   вЂ”

G3a.3. Weighted DFI form              In the separated model one may define the weighted coefficient

                                                           1 М‚пёЃ          в„Ћ
                                                                    (пё‚ )пё‚
                                        рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ) = рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ  рќ‘Њ           .
                                                           рќ‘ћ             рќ‘ћ
   Then

                                                                               в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                          (пё‚              )пё‚
                              рќ’Єрќ‘”,в„Ћ =                  рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ)рќ‘’
                                             в€‘пёЃ
                                                                                     .
                                        рќ‘Ћв€јрќђґрќ‘” , рќ‘ћв€јрќ‘„рќ‘”
                                                                                рќ‘ћ
                                          (рќ‘Ћ,рќ‘ћ)=1

   For the actual CKP fibre, the equivalent DFI form is

                                                                        в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                   (пё‚          )пё‚
               рќ’Єрќ‘”,в„Ћ =                 рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ)рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)рќ‘’                                      (G3a-DFI-weight)
                           в€‘пёЃ
                                                                              ,
                        рќ‘Ћв€јрќђґрќ‘” , рќ‘ћв€јрќ‘„рќ‘”
                                                                         рќ‘ћ
                          (рќ‘Ћ,рќ‘ћ)=1

   where рќ‘Љрќ‘”,в„Ћ 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

                                                             рќ‘рќ‘љ
                                                                          (пё‚         )пё‚
                                рќђµрќ‘ (рќ‘Ђ, рќ‘„) =
                                                      в€‘пёЃ
                                                     рќ›јрќ‘љ рќ›Ѕрќ‘› рќ‘’    ,
                                            рќ‘љв€јрќ‘Ђ, рќ‘›в€јрќ‘„
                                                              рќ‘›
                                                    (рќ‘љ,рќ‘›)=1


                                                        30
   with the dictionary

                   рќ‘љ = рќ‘Ћ,       рќ‘› = рќ‘ћ,        рќ‘Ђ = рќђґрќ‘” ,        рќ‘„ = рќ‘„рќ‘” ,      рќ‘ = в„Ћрќ‘Ѓрќ‘” .
   вЂ”

G3a.4. Coefficient norms        From G1a coefficient preservation, finite-convolution divisor-boundedness
gives

                                               рќ‘” (log рќ‘Ѓ )
                                                          рќђ¶(рќђЅ0 )
                                     вЂ–рќ›Ѕрќ‘” вЂ–2 в‰Є рќђґ1/2               ,


                                               рќ‘” (log рќ‘Ѓ )
                                                          рќђ¶(рќђЅ0 )
                                     вЂ–рќ›ѕрќ‘” вЂ–2 в‰Є рќ‘„1/2               .
    The exponent рќђ¶(рќђЅ0 ) is uniform in рќ‘”. Indeed, the B1 elementary coefficients are bounded by
fixed powers of log рќ‘Ѓ on every dyadic block; after the exact substitution рќ‘ў = рќ‘”рќ‘Ћ, рќ‘ўвЂІ = рќ‘”рќ‘ћ, the
bounds become |рќ›Ѕрќ‘” (рќ‘Ћ)|, |рќ›ѕрќ‘” (рќ‘ћ)| в‰Є (log рќ‘Ѓ )рќђ¶(рќђЅ0 ) on supports of lengths рќђґрќ‘” and рќ‘„рќ‘” . Thus

                               рќ‘” (log рќ‘Ѓ )                           рќ‘” (log рќ‘Ѓ )
                                          рќђ¶(рќђЅ0 )                               рќђ¶(рќђЅ0 )
                     вЂ–рќ›Ѕрќ‘” вЂ–2 в‰Є рќђґ1/2               ,        вЂ–рќ›ѕрќ‘” вЂ–2 в‰Є рќ‘„1/2

with the same structural exponent for every admissible рќ‘”-layer. Summing over рќ‘” | рќ‘Ѓ later uses the рќ‘”-
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:
                                                           )пёѓв€’рќђґ
                                вѓ’1
                                                 (пёѓ
                                                      |в„Ћ|рќ‘Њ
                                вѓ’
                                   М‚пёЃрќ‘Њ в„Ћ вѓ’ в‰Єрќђґ рќ‘Њ
                                      (пё‚ )пё‚вѓ’
                                                   1+
                                           вѓ’
                                вѓ’ рќ‘Љ                             ,
                                вѓ’рќ‘ћ      рќ‘ћ вѓ’   рќ‘„рќ‘”       рќ‘„рќ‘”
   and in balanced range рќ‘Њ /рќ‘„рќ‘” в‰Ќ рќ‘”, we obtain

                            вЂ–рќ›ѕМѓпёЂрќ‘”,в„Ћ вЂ–2 в‰Єрќђґ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ рќ‘„1/2
                                                         рќ‘” (log рќ‘Ѓ )
                                                                    рќђ¶(рќђЅ0 )
                                                                           .
   For the nonseparated weighted form (G3a-DFI-weight), the same coefficient norms are used,
while the supremum and derivative losses of рќ‘Љрќ‘”,в„Ћ are charged to the smooth-weight parameter in
X10. These are precisely the hypotheses used in G4a/X10.
   вЂ”

G3a.5. Balanced parameter matching              In balanced CKP range,

                                                                  рќ‘Ѓ 1/2
                                   рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘” ,          рќ‘†рќ‘” =         .
                                                                   рќ‘”
   The DFI external parameter is

                                                          в„Ћрќ‘Ѓ
                                           рќ‘ = в„Ћрќ‘Ѓрќ‘” =         .
                                                           рќ‘”
   Then

                                              |в„Ћ|рќ‘Ѓ  рќ‘Ѓ   рќ‘Ѓ
                              |рќ‘| + рќђґрќ‘” рќ‘„рќ‘” в‰Ќ        + 2 = 2 (1 + |в„Ћ|рќ‘”).
                                                рќ‘”   рќ‘”   рќ‘”
   This is exactly the expression used in G4a:


                                                     31
                             (|рќ‘| + рќђґрќ‘” рќ‘„рќ‘” )3/8 = рќ‘Ѓ 3/8 рќ‘” в€’3/4 (1 + |в„Ћ|рќ‘”)3/8 .
   вЂ”

G3a.6. Lemma G3a

Lemma 5.7 (Lemma G3a). The nonzero-frequency contribution produced by G2a on each CKP
gcd layer рќ‘” is a finite sum of weighted bilinear Kloosterman fraction sums

                                                                               в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                          (пё‚       )пё‚
                              рќ’Єрќ‘”,в„Ћ =                 рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ)рќ‘’
                                           в€‘пёЃ
                                                                                     ,
                                       рќ‘Ћв€јрќђґрќ‘” , рќ‘ћв€јрќ‘„рќ‘”
                                                                                рќ‘ћ
                                         (рќ‘Ћ,рќ‘ћ)=1

   where

                                                          1 М‚пёЃ           в„Ћ
                                                                    (пё‚ )пё‚
                                       рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ) = рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ  рќ‘Њ            .
                                                           рќ‘ћ             рќ‘ћ
   This matches the DFI Kloosterman-fraction form with

                                рќ‘Ђ = рќђґрќ‘” ,         рќ‘„ = рќ‘„рќ‘” ,                рќ‘ = в„Ћрќ‘Ѓрќ‘” .
   In balanced range,

                                                               рќ‘Ѓ 1/2
                                            рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ                ,
                                                                рќ‘”
   and the coefficient norms satisfy

                                                   рќ‘” (log рќ‘Ѓ ) ,
                                                             рќђ¶
                                         вЂ–рќ›Ѕрќ‘” вЂ–2 в‰Є рќђґ1/2


                               вЂ–рќ›ѕМѓпёЂрќ‘”,в„Ћ вЂ–2 в‰Єрќђґ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ рќ‘„1/2
                                                            рќ‘” (log рќ‘Ѓ ) .
                                                                      рќђ¶


Proof. The congruence and Fourier expansion are G2a. Absorbing all one-variable smooth dyadic
weights into рќ›Ѕрќ‘” and рќ›ѕрќ‘” , and then absorbing the Fourier factor into рќ›ѕМѓпёЂрќ‘”,в„Ћ , gives exactly the displayed
bilinear Kloosterman fraction sum. The dictionary \{}((m,n,k)=(a,q,hN_g) \{}) is immediate.
The coefficient norm estimates follow from finite-convolution divisor-boundedness and Fourier decay
from G2a. Lemma proved.
    вЂ”


Remark 5.8 (G3a.7. Output).

         G3a converts CKP nonzero frequencies to weighted Kloosterman-fraction sums.

   The parameters are рќ‘Ђ = рќђґрќ‘” , рќ‘„ = рќ‘„рќ‘” , рќ‘ = в„Ћрќ‘Ѓрќ‘” , and the coefficient norms match the hypotheses
used by G4a and CKPD.

G3a.8. Logical Dependencies Internal dependencies: G1a, G2a, CKPD, and X10.
  Children served: G4a, G8a, and X10.



                                                      32
5.6    Part 6. CKPD: CKP/X10 smooth-weight derivative appendix
Source file: Lemmas/ckp_x10_smooth_weight_derivative_appendix_ltx.md.

CKPD. CKP/X10 Smooth-Weight Derivative Check

CKPD.0. Role Logical ID: CKPD.
  Used by: G3a, G4a, G8a, X10, GEB, I1.
  Uses: G1a, G2a, G3a, G8a, X10, X10ER, C1A, C1, and DFI Theorem 2.
  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 and the X10 external input, which invoke DFI Theorem 2.

   The conclusion is:


the CKP nonzero-frequency weight is DFI-admissible with only polylogarithmic derivative parameter.

   вЂ”

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), 23вЂ“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 рќ‘Ђ, рќ‘„ в‰Ґ 1, рќ‘џ в‰Ґ 1, and let рќ›јрќ‘љ , рќ›Ѕрќ‘ћ be arbitrary
complex sequences supported on рќ‘љ в‰Ќ рќ‘Ђ , рќ‘ћ в‰Ќ рќ‘„. Let рќђ№ (рќ‘љ, рќ‘ћ) be a smooth weight supported on
the same dyadic box and satisfying

           |рќђ№ (рќ‘љ, рќ‘ћ)| в‰¤ 1,        рќњ•рќ‘ћ рќђ№ (рќ‘љ, рќ‘ћ) в‰Є рќњ‚ рќ‘–+рќ‘— рќ‘Ђ в€’рќ‘– рќ‘„в€’рќ‘— ,
                                рќ‘– рќ‘—
                               рќњ•рќ‘љ                                     0 в‰¤ рќ‘–, рќ‘— в‰¤ 2.        (DFI-wt)
   Then, for every рќњЂ > 0,


                               рќ‘џрќ‘љ
                              (пё‚    )пё‚
               рќ›јрќ‘љ рќ›Ѕрќ‘ћ рќђ№ (рќ‘љ, рќ‘ћ)рќ‘’           в‰ЄрќњЂ рќњ‚ 2 вЂ–рќ›јвЂ–2 вЂ–рќ›ЅвЂ–2 (рќ‘џ + рќ‘Ђ рќ‘„)3/8 (рќ‘Ђ + рќ‘„)11/48+рќњЂ .   (DFI-X10)
         в€‘пёЃ

      рќ‘љв‰Ќрќ‘Ђ, рќ‘ћв‰Ќрќ‘„
                                рќ‘ћ
       (рќ‘љ,рќ‘ћ)=1


   In the CKP application, рќњ‚ is a fixed power of log рќ‘Ѓ . Thus the рќњ‚ 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:


                                                    33
 DFI quantity               CKP quantity                  Verified in
 рќ‘љв€јрќ‘Ђ                        рќ‘Ћ в€ј рќђґрќ‘”                        G1a/G8a
 рќ‘ћв€јрќ‘„                        рќ‘ћ в€ј рќ‘„рќ‘”                        G1a/G8a
 (рќ‘љ, рќ‘ћ) = 1                 (рќ‘Ћ, рќ‘ћ) = 1                    G1a
 рќ‘џв‰Ґ1                        \{}(r=                        h                         N_g\{}), в„Ћ Мё= 0     G2a/G3a
 рќ›јрќ‘љ , рќ›Ѕ рќ‘ћ                   finite-convolution coef-      G3a/G4a
                            ficient sequences
 рќђ№ (рќ‘љ, рќ‘ћ)                   normalized рќ‘Љ МѓпёЂ рќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)    CKPD.3вЂ“CKPD.6




    All noncentral, high-frequency, small-conductor, large-рќ‘”, and boundary ranges are excluded
before this table is used; they are routed through C1P/C1A/C1 through the excluded-range routing
statement X10ER.
    вЂ”

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      CKP realization            Verification locus   Verification type
 parameter
 dyadic     support     рќ‘Ћ в‰Ќ рќђґрќ‘” , рќ‘ћ в‰Ќ рќ‘„рќ‘” af-        G1a/G8a              internal
 рќ‘љ в‰Ќ рќ‘Ђ, рќ‘ћ в‰Ќ рќ‘„           ter the рќ‘”-split
 coprimality            (рќ‘Ћ, рќ‘ћ) = 1 after the       G1a                  internal
 (рќ‘љ, рќ‘ћ) = 1             CKP gcd normal-
                        ization
 integer parameter      \{}(r=                     h                    N_g\{}) with в„Ћ Мё=     G2a/G3a        internal
 рќ‘џв‰Ґ1                                                                    0
 arbitrary в„“2 coeffi-   finite-convolution         G3a/G4a and X10      internal
 cient sequences        CKP          coefficient
                        sequences
 smooth         two-    рќ‘ЉМѓпёЂ рќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)             CKPD.3вЂ“CKPD.6        internal
 variable weight
 derivative order re-   all рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— , 0 в‰¤ рќ‘–, рќ‘— в‰¤   CKPD.4вЂ“CKPD.6        internal
 quired by DFI          2
 derivative parame-     (log рќ‘Ѓ )рќђ¶DFI               CKPD.6               internal,   charged
 ter рќњ‚                                                                  to    the   polylog
                                                                        budget
 excluded в„Ћ = 0         local CKP contri-          G8a/LPI, then H4     internal
 term                   bution                     assembly
 high     frequency,    not    sent   to           X10ER, C1P/C1A/C1        internal
 noncentral, bound-     DFI/X10
 ary, small conduc-
 tor
 exact    agreement     the     displayed          external DFI paper   external    theorem
 with DFI Theorem       dyadic statement           / X10                check
 2 and formulas         (DFI-X10)
 (1.7)вЂ“(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.
    вЂ”



                                                                   34
CKPD.3. Setup: central CKP notation Fix one tagged central balanced CKP layer after
the G1a gcd split. We use the notation of G8a:

                                     рќ‘Ѓ
                           рќ‘Ѓрќ‘” =        ,        рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,                 (рќ‘Ћ, рќ‘ћ) = 1,
                                     рќ‘”
   with

                           рќ‘Ћ в‰Ќ рќђґрќ‘” ,           рќ‘ћ в‰Ќ рќ‘„рќ‘” ,           рќ‘¦ в‰Ќ рќ‘Њ,            рќ‘¦вЂІ в‰Ќ рќ‘Њ вЂІ,
   and central balance

                                                                            рќ‘Њ
                                рќђґрќ‘” в‰Ќ рќ‘„рќ‘” ,                рќ‘Њ в‰Ќ рќ‘Њ вЂІ,              в‰Ќ рќ‘”.            (CB)
                                                                            рќ‘„рќ‘”
   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

                                                                  рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                       рќ‘§ = рќ‘§(рќ‘Ћ, рќ‘ћ, рќ‘¦) :=                  .
                                                                     рќ‘ћ
    On this support, рќ‘§ в‰Ќ рќ‘Њ вЂІ . Let рќњ”рќђґ , рќњ”рќ‘„ , рќ‘Љрќ‘Њ , рќ‘Љрќ‘Њ вЂІ be the smooth dyadic cutoffs belonging to the
fixed tag. They satisfy, for every fixed рќ‘џ в‰Ґ 0,
                                (рќ‘џ)                                (рќ‘џ)
                               рќњ”рќђґ (рќ‘Ћ) в‰Єрќ‘џ рќђґв€’рќ‘џ
                                          рќ‘” ,                    рќњ”рќ‘„ (рќ‘ћ) в‰Єрќ‘џ рќ‘„в€’рќ‘џ
                                                                            рќ‘” ,


                               (рќ‘џ)                                (рќ‘џ)
                            рќ‘Љрќ‘Њ (рќ‘¦) в‰Єрќ‘џ рќ‘Њ в€’рќ‘џ ,                    рќ‘Љрќ‘Њ вЂІ (рќ‘§) в‰Єрќ‘џ (рќ‘Њ вЂІ )в€’рќ‘џ .           (S)
   Any nonsmooth finite-convolution coefficient inherited from B1 is kept in the outer coefficient
sequences рќ›јрќ‘” (рќ‘Ћ), рќ›ѕрќ‘” (рќ‘ћ). Thus the smooth object differentiated below is

                           О¦рќ‘Ћ,рќ‘ћ (рќ‘¦) = рќњ”рќђґ (рќ‘Ћ)рќњ”рќ‘„ (рќ‘ћ)рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ (рќ‘§(рќ‘Ћ, рќ‘ћ, рќ‘¦)).                     (Phi)
   If one chooses to include a smooth coefficient cutoff inside О¦, it satisfies the same derivative
bounds and only changes the final logarithmic constant.
   вЂ”

CKPD.4. Exact formula for the DFI weight                          For в„Ћ Мё= 0, define

                                                         1 М‚пёЂ     в„Ћ
                                                                         (пё‚ )пё‚
                                           рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) := О¦   рќ‘Ћ,рќ‘ћ   ,
                                                         рќ‘ћ        рќ‘ћ
   where
                                                    в€«пёЃ
                                     М‚пёЂ рќ‘Ћ,рќ‘ћ (рќњ‰) =
                                     О¦                    О¦рќ‘Ћ,рќ‘ћ (рќ‘¦)рќ‘’(в€’рќ‘¦рќњ‰) рќ‘‘рќ‘¦.                   (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




                                                           35
                                                                                    в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                               (пё‚       )пё‚
                        рќ’Єрќ‘”,в„Ћ =                 рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ)рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)рќ‘’                       (CKP-X10)
                                     в€‘пёЃ
                                                                                          .
                                 рќ‘Ћв€јрќђґрќ‘” , рќ‘ћв€јрќ‘„рќ‘”
                                                                                     рќ‘ћ
                                   (рќ‘Ћ,рќ‘ћ)=1

   For DFI, set

                                 рќ’њрќ‘”,в„Ћ,рќ‘… := (log рќ‘Ѓ )рќђ¶* рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќ‘… ,                             (Agh)
    where рќђ¶* is a fixed constant large enough to dominate the dyadic smoothness constants in the
estimates below.
    The normalized DFI weight is

                                   МѓпёЃрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) := рќ’њв€’1 рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ),
                                   рќ‘Љ                                                            (Wtilde)
                                                    рќ‘”,в„Ћ,рќ‘…

    with рќ‘… 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 рќ’њрќ‘”,в„Ћ,рќ‘… 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 рќ‘”, в„Ћ-
summation.
    вЂ”

CKPD.5. Elementary chain-rule bounds                     On the central support,
                                              рќ‘¦                    рќ‘§
                                      рќњ•рќ‘Ћ рќ‘§ = в€’ ,           рќњ•рќ‘ћ рќ‘§ = в€’ .
                                              рќ‘ћ                    рќ‘ћ
   Using (CB), (S), and рќ‘§ в‰Ќ рќ‘Њ вЂІ , we get

                                                          рќ‘¦              рќ‘Њ
                                                    (пё‚        )пё‚
                           рќњ•рќ‘Ћ рќ‘Љрќ‘Њ вЂІ (рќ‘§) = рќ‘Љрќ‘ЊвЂІ вЂІ (рќ‘§) в€’               в‰Є           в‰Є рќђґв€’1
                                                                                  рќ‘” ,              (A1)
                                                          рќ‘ћ             рќ‘„рќ‘” рќ‘Њ вЂІ
   because рќђґрќ‘” в‰Ќ рќ‘„рќ‘” and рќ‘Њ в‰Ќ рќ‘Њ вЂІ . Similarly,

                                                                  рќ‘§
                                                           (пё‚          )пё‚
                                 рќњ•рќ‘ћ рќ‘Љрќ‘Њ вЂІ (рќ‘§) = рќ‘Љрќ‘ЊвЂІ вЂІ (рќ‘§)        в€’           в‰Є рќ‘„в€’1
                                                                               рќ‘” .                 (Q1)
                                                                  рќ‘ћ
   Repeated derivatives are no worse. More precisely, for 0 в‰¤ рќ‘–, рќ‘— в‰¤ рќ‘…0 with fixed рќ‘…0 ,

                                 рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— рќ‘Љрќ‘Њ вЂІ (рќ‘§(рќ‘Ћ, рќ‘ћ, рќ‘¦)) в‰Єрќ‘…0 рќђґв€’рќ‘– в€’рќ‘—
                                                                рќ‘” рќ‘„рќ‘” .                            (Zder)
    The proof is by induction and Faa di Bruno. Every рќ‘Ћ-derivative of рќ‘§ contributes рќ‘‚(рќ‘Њ /рќ‘„рќ‘” ),
                                             (рќ‘џ)
                                                          рќ‘” ). Every рќ‘ћ-derivative of рќ‘§ is рќ‘‚(рќ‘Њ рќ‘„рќ‘” )
and after division by one рќ‘Њ вЂІ -scale from рќ‘Љрќ‘Њ вЂІ this is рќ‘‚(рќђґв€’1                                 вЂІ в€’рќ‘—

at order рќ‘—, and the corresponding derivative of рќ‘Љрќ‘Њ вЂІ contributes (рќ‘Њ ) for each рќ‘§-factor, giving
                                                                     вЂІ в€’1

рќ‘‚(рќ‘„в€’рќ‘—рќ‘” ). Mixed derivatives combine the two estimates.
    Including the explicit рќњ”рќђґ , рќњ”рќ‘„ derivatives gives

                                   рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— О¦рќ‘Ћ,рќ‘ћ (рќ‘¦) в‰Єрќ‘…0 рќђґв€’рќ‘– в€’рќ‘—
                                                         рќ‘” рќ‘„рќ‘” 1рќ‘¦в‰Ќрќ‘Њ                              (Phider)
    for 0 в‰¤ рќ‘–, рќ‘— в‰¤ рќ‘…0 , with the same statement after applying any bounded number of рќ‘¦-derivatives,
at the cost of the expected powers of рќ‘Њ в€’1 .
    вЂ”



                                                     36
CKPD.6. Fourier decay with parameter derivatives                         For every fixed рќђµ, рќ‘–, рќ‘— в‰Ґ 0,
                                                                            (пёѓ           )пёѓв€’рќђµ
                                                               в€’рќ‘— рќ‘Њ               |в„Ћ|рќ‘Њ
                  рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Єрќђµ,рќ‘–,рќ‘— (1 + |в„Ћ|рќ‘”)рќ‘–+рќ‘— рќђґв€’рќ‘–
                                                            рќ‘” рќ‘„рќ‘”               1+               .
                                                                 рќ‘„рќ‘”                рќ‘„рќ‘”
   Equivalently, after increasing the constant and using рќ‘Њ /рќ‘„рќ‘” в‰Ќ рќ‘”,

                          рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Єрќђµ,рќ‘–,рќ‘— рќђґв€’рќ‘– в€’рќ‘—
                                                      рќ‘” рќ‘„рќ‘” рќ‘”(1 + |в„Ћ|рќ‘”)
                                                                      в€’рќђµ+рќ‘–+рќ‘—
                                                                             .                      (Wder-raw)

Proof. Write

                                              1                       в„Ћрќ‘¦
                                                     в€«пёЃ           (пё‚      )пё‚
                                рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) =             О¦рќ‘Ћ,рќ‘ћ (рќ‘¦)рќ‘’ в€’    рќ‘‘рќ‘¦.
                                              рќ‘ћ                        рќ‘ћ
   When |в„Ћ|рќ‘” в‰¤ 1, no oscillatory integration is needed. The trivial bound

                                     1                           рќ‘Њ
                                         в€«пёЃ
                                              |О¦рќ‘Ћ,рќ‘ћ (рќ‘¦)| рќ‘‘рќ‘¦ в‰Є       в‰Ќрќ‘”
                                     рќ‘ћ                           рќ‘„рќ‘”
    gives the рќ‘– = рќ‘— = 0 case, and the differentiated version follows from (Phider), together with the
harmless derivatives of рќ‘ћ в€’1 and of the phase. Since 1 + |в„Ћ|рќ‘” в‰Ќ 1 in this range, this gives precisely
the right-hand side of (Wder-raw).
    It remains to consider |в„Ћ|рќ‘” > 1, where oscillation is available. First ignore рќ‘Ћ, рќ‘ћ-derivatives.
Integrating by parts рќђµ times in рќ‘¦, using that every рќ‘¦-derivative of О¦ costs рќ‘Њ в€’1 , gives
                                                (пёѓ             )пёѓв€’рќђµ
                                       рќ‘Њ            |в„Ћ|рќ‘Њ
                      |рќ’Ірќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ)| в‰Єрќђµ           1+                   в‰Єрќђµ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђµ .                  (FD)
                                       рќ‘„рќ‘”            рќ‘„рќ‘”
    Now differentiate in рќ‘Ћ, рќ‘ћ. Derivatives falling on О¦ are controlled by (Phider), giving the expected
         рќ‘” рќ‘„рќ‘” . Derivatives falling on рќ‘ћ
factors рќђґв€’рќ‘–  в€’рќ‘—                           в€’1 also give powers of рќ‘„в€’1 . Derivatives falling on the phase
                                                                    рќ‘”
contribute powers of

                                                 |в„Ћ|рќ‘Њ   |в„Ћ|рќ‘”
                                                    2
                                                      в‰Ќ      ,
                                                  рќ‘„рќ‘”     рќ‘„рќ‘”

               рќ‘” (1 + |в„Ћ|рќ‘”). Thus рќ‘– + рќ‘— total рќ‘Ћ, рќ‘ћ-derivatives can lose at most (1 + |в„Ћ|рќ‘”)
   which are рќ‘„в€’1                                                                           рќ‘–+рќ‘— from the

Fourier-decay exponent. Repeating the integration-by-parts argument after these differentiations
proves (Wder-raw).
   вЂ”


Parameter check 5.9 (CKPD.7. Parameter check: DFI-admissibility in the X10 range). Let
рќ‘…DFI = 2, matching the derivative order required in X10. Choose рќ‘… в‰Ґ рќ‘…DFI + 10 in the Fourier
decay step above. In the central X10 range,

                                              |в„Ћ|рќ‘” в‰¤ (log рќ‘Ѓ )рќђµHF .                                        (HF)
   Combining (Wder-raw), (Agh), and (HF), we obtain

                                               МѓпёЃрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Є 1,
                                               рќ‘Љ                                                        (DFI-0)
   and, for 1 в‰¤ рќ‘– + рќ‘— в‰¤ 2,



                                                          37
                                     МѓпёЃрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ) в‰Є (log рќ‘Ѓ )рќђ¶DFI рќђґв€’рќ‘– рќ‘„в€’рќ‘— .
                             рќњ•рќ‘Ћрќ‘– рќњ•рќ‘ћрќ‘— рќ‘Љ                                                 (DFI-der)
                                                                  рќ‘”   рќ‘”

    Also рќ‘Љ МѓпёЃрќ‘”,в„Ћ is supported on the same dyadic box рќ‘Ћ в‰Ќ рќђґрќ‘” , рќ‘ћ в‰Ќ рќ‘„рќ‘” , because of рќњ”рќђґ рќњ”рќ‘„ . Thus it
satisfies the smooth-weight hypotheses of the DFI-X10 statement with

                                         рќњ‚ = (log рќ‘Ѓ )рќђ¶DFI .
   The unnormalized factor рќ’њрќ‘”,в„Ћ,рќ‘… is not lost. It is kept outside the normalized DFI weight and
charged to the в„Ћ-summation:

                                рќ’њрќ‘”,в„Ћ,рќ‘… = (log рќ‘Ѓ )рќђ¶* рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќ‘… .                      (A-loss)
   Choosing рќ‘… larger than the DFI derivative loss, the 3/8 growth from (|в„Ћ|рќ‘Ѓрќ‘” + рќђґрќ‘” рќ‘„рќ‘” )3/8 , and
the fixed divisor summation losses leaves an absolutely summable (1 + |в„Ћ|рќ‘”)в€’2 -type tail. This is
the decay used in G4a/G8a.
   вЂ”

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

                              рќ‘Ђ = рќђґрќ‘” ,       рќ‘„ = рќ‘„рќ‘” ,         рќ‘џ = |в„Ћ|рќ‘Ѓрќ‘” .
                                                     МѓпёЃрќ‘”,в„Ћ is DFI-admissible with polylogarithmic
    By (DFI-der), the normalized two-variable weight рќ‘Љ
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. в„Ћ = 0 is the CKP local term handled by G8a/LPI and then assembled by H4;

  2. |в„Ћ|рќ‘” > (log рќ‘Ѓ )рќђµHF 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/C1 as recorded in X10.

   Thus the CKP/X10 smooth-weight derivative obligation is discharged.
   вЂ”
Remark 5.10 (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, G8a, X10ER, C1A, C1.
  Children served: X10, G3a, G4a, G8a, GEB, I1.

5.7   Part 7. G4a: DFI matching
Source file: Lemmas/g_4_a_ltx.md.

                                                  38
G4a. Exact Kloosterman Black-Box Matching

G4a.0. Role Logical ID: G4a.
   Used by: G8a, I1.
   Uses: G1a, G2a, G3a, CKPD, X10, X10ER, C1A, C1, G8a.
   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 DukeвЂ“FriedlanderвЂ“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

                                                 рќ‘њ(рќ‘Ѓ )
   after summing over рќ‘” and в„Ћ.
   вЂ”

G4a.1. External analytic theorem         We use an external estimate for bilinear forms with Kloost-
erman fractions. Let

                                                                         рќ‘рќ‘љ
                                                                    (пё‚     )пё‚
                               рќђµрќ‘ (рќ‘Ђ, рќ‘„) =
                                                в€‘пёЃ
                                                         рќ›јрќ‘љ рќ›Ѕрќ‘ћ рќ‘’            ,
                                             рќ‘љв€јрќ‘Ђ, рќ‘ћв€јрќ‘„
                                                                          рќ‘ћ
                                              (рќ‘љ,рќ‘ћ)=1

   where рќ‘љ is the inverse of рќ‘љ modulo рќ‘ћ, and

                                             рќ‘’(рќ‘Ґ) = рќ‘’2рќњ‹рќ‘–рќ‘Ґ .
   The working form needed here is

                      рќђµрќ‘ (рќ‘Ђ, рќ‘„) в‰ЄрќњЂ вЂ–рќ›јвЂ–2 вЂ–рќ›ЅвЂ–2 (|рќ‘| + рќ‘Ђ рќ‘„)3/8 (рќ‘Ђ + рќ‘„)11/48+рќњЂ .
    Here рќ›ј and рќ›Ѕ are arbitrary complex coefficients. This is important because our coefficients are
finite-convolution coefficients built from рќњ‡, 1, and log, and are controlled through their рќђї2 -norms.
    вЂ”

G4a.2. Exact external theorem and formulation check For G4a we fix the concrete external
theorem.
    DFI Theorem 2. In DukeвЂ“FriedlanderвЂ“Iwaniec, *Bilinear forms with Kloosterman fractions*,
Invent. Math. 128 (1997), 23вЂ“43, DOI 10.1007/s002220050135, Theorem 2 states that, for the
bilinear form

                                                                         рќ‘Ћрќ‘љ
                                                                    (пё‚     )пё‚
                               рќђµ(рќ‘Ђ, рќ‘Ѓ ) =
                                                в€‘пёЃ
                                                         рќ›јрќ‘љ рќ›Ѕ рќ‘› рќ‘’           ,
                                             рќ‘Ђ <рќ‘љв‰¤2рќ‘Ђ
                                                                          рќ‘›
                                             рќ‘Ѓ <рќ‘›в‰¤2рќ‘Ѓ
                                              (рќ‘љ,рќ‘›)=1

   where рќ‘љ is the inverse of рќ‘љ modulo рќ‘›, and рќ›јрќ‘љ , рќ›Ѕрќ‘› are arbitrary complex coefficients, one has

                       рќђµ(рќ‘Ђ, рќ‘Ѓ ) в‰ЄрќњЂ вЂ–рќ›јвЂ–2 вЂ–рќ›ЅвЂ–2 (рќ‘Ћ + рќ‘Ђ рќ‘Ѓ )3/8 (рќ‘Ђ + рќ‘Ѓ )11/48+рќњЂ .
   This formulation is compatible with G4a for the following reasons.
   1. The phase matches. In DFI the phase has the form

                                                  39
                                                        рќ‘Ћрќ‘љ
                                                   (пё‚        )пё‚
                                               рќ‘’           .
                                                         рќ‘›
   In our CKP sum the phase has the form

                                               в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                               (пё‚              )пё‚
                                             рќ‘’       .
                                                рќ‘ћ
   The parameter correspondence is

                            рќ‘љ в†” рќ‘Ћ,       рќ‘› в†” рќ‘ћ,               рќ‘ЋDFI в†” |рќ‘| = |в„Ћ|рќ‘Ѓрќ‘” .
    For в„Ћ < 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 |в„Ћ|рќ‘Ѓрќ‘” .
    2. The coprimality condition matches. DFI requires

                                              (рќ‘љ, рќ‘›) = 1.
   In our sum this is exactly

                                              (рќ‘Ћ, рќ‘ћ) = 1,
    which is required for the existence of рќ‘Ћ (mod рќ‘ћ).
    3. The coefficients are admissible. DFI allows arbitrary complex coefficients рќ›јрќ‘љ , рќ›Ѕрќ‘› and
estimates the sum in terms of their рќђї2 -norms. Our рќ›Ѕрќ‘” (рќ‘Ћ) and рќ›ѕрќ‘” (рќ‘ћ) are finite-convolution coefficients
built from рќњ‡, 1, and log, with smooth dyadic weights. They are therefore admissible once their
рќђї2 -norms are estimated.
    4. The ranges are dyadic. DFI works on dyadic intervals. After gcd splitting we have

                                         рќ‘Ћ в€ј рќ‘†рќ‘” ,             рќ‘ћ в€ј рќ‘†рќ‘” ,
   so

                                            рќ‘Ђ = рќ‘„ = рќ‘†рќ‘” .
   5. A large external parameter is allowed. The DFI bound contains the factor

                                            (рќ‘Ћ + рќ‘Ђ рќ‘Ѓ )3/8 ,
   so the external parameter may be large. In our problem

                                                                  в„Ћрќ‘Ѓ
                                           рќ‘ = в„Ћрќ‘Ѓрќ‘” =
                                                                   рќ‘”
   may exceed

                                                             рќ‘Ѓ
                                               рќ‘†рќ‘”2 =            .
                                                             рќ‘”2
   This is why the computation produces the factor

                                             (1 + |в„Ћ|рќ‘”)3/8 ,
   which is then compensated by the smooth Fourier weight.


                                                        40
    6. The weighted coefficient is admissible. In the separated model we apply DFI not to
рќ›ѕрќ‘” (рќ‘ћ), but to

                                                           1 М‚пёЃ     в„Ћ
                                                                   (пё‚ )пё‚
                                        рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ) = рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ  рќ‘Њ      .
                                                           рќ‘ћ        рќ‘ћ
    This is still an arbitrary complex sequence in рќ‘ћ, so DFI applies after estimating its рќђї2 -norm.
In the CKP interface, the actual weight may be the more general nonseparated weight рќ‘Љрќ‘”,в„Ћ (рќ‘Ћ, рќ‘ћ).
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.
    вЂ”

G4a.3. CKP sum after gcd splitting                   A balanced CKP block has the base form

                                                 рќ‘ўрќ‘¦ + рќ‘ўвЂІ рќ‘¦ вЂІ = рќ‘Ѓ.
   Write

                       рќ‘” = gcd(рќ‘ў, рќ‘ўвЂІ ),          рќ‘ў = рќ‘”рќ‘Ћ,       рќ‘ўвЂІ = рќ‘”рќ‘ћ,        (рќ‘Ћ, рќ‘ћ) = 1.
   If рќ‘” в€¤ рќ‘Ѓ , there are no solutions. If рќ‘” | рќ‘Ѓ , write

                                                             рќ‘Ѓ
                                                     рќ‘Ѓрќ‘” =      .
                                                             рќ‘”
   Then the equation becomes

                                                 рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” .
   Solve the congruence in рќ‘¦:

                                             рќ‘Ћрќ‘¦ в‰Ў рќ‘Ѓрќ‘”        (mod рќ‘ћ).
   Since (рќ‘Ћ, рќ‘ћ) = 1, this gives

                                             рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ       (mod рќ‘ћ).
   For the smooth weight рќ‘Љрќ‘Њ , use the Fourier expansion for counting points in an arithmetic
progression:

                                                        1 в€‘пёЃ М‚пёЃ        в„Ћ   в„Ћрќ‘џ
                                                                    (пё‚ )пё‚ (пё‚      )пё‚
                                            рќ‘Љрќ‘Њ (рќ‘¦) =
                                  в€‘пёЃ
                                                              рќ‘Љрќ‘Њ         рќ‘’    .
                             рќ‘¦в‰Ўрќ‘џ (mod рќ‘ћ)
                                                        рќ‘ћ в„Ћв€€Z          рќ‘ћ   рќ‘ћ

   The term в„Ћ = 0 gives the local/main term. The terms в„Ћ Мё= 0 give the oscillatory contribution.
   For fixed рќ‘” this gives

                                                        1 М‚пёЃ            в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                       (пё‚ )пё‚ (пё‚        )пё‚
                        рќ’Єрќ‘” =                рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ
                              в€‘пёЃ        в€‘пёЃ
                                                            рќ‘Њ             рќ‘’       .
                             в„ЋМё=0 рќ‘Ћв€јрќђґ , рќ‘ћв€јрќ‘„
                                                        рќ‘ћ               рќ‘ћ    рќ‘ћ
                                        рќ‘”        рќ‘”
                                       (рќ‘Ћ,рќ‘ћ)=1

   In the balanced range,


                                                        41
                                                                            рќ‘Ѓ 1/2
                                 рќђґрќ‘” в‰Ќ рќ‘„рќ‘” в‰Ќ рќ‘†рќ‘” ,             рќ‘†рќ‘” =                  .
                                                                             рќ‘”
   вЂ”

G4a.4. Why the Fourier weight must be included in the coefficient                            The external phase
parameter is

                                                           в„Ћрќ‘Ѓ
                                           рќ‘ = в„Ћрќ‘Ѓрќ‘” =          .
                                                            рќ‘”
   It can be larger than

                                                          рќ‘Ѓ
                                              рќђґрќ‘” рќ‘„рќ‘” в‰Ќ        .
                                                          рќ‘”2
   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 рќ‘ћ-variable by

                                                        1 М‚пёЃ            в„Ћ
                                                                   (пё‚ )пё‚
                                     рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ) = рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ  рќ‘Њ             .
                                                        рќ‘ћ               рќ‘ћ
   Then

                                                                                 в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                            (пё‚        )пё‚
                            рќ’Єрќ‘”,в„Ћ =                 рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕМѓпёЂрќ‘”,в„Ћ (рќ‘ћ)рќ‘’
                                         в€‘пёЃ
                                                                                       .
                                     рќ‘Ћв€јрќ‘†рќ‘” , рќ‘ћв€јрќ‘†рќ‘”
                                                                                  рќ‘ћ
                                       (рќ‘Ћ,рќ‘ћ)=1

   This is exactly the form рќђµрќ‘ (рќ‘Ђ, рќ‘„) with parameters

                                                                                      в„Ћрќ‘Ѓ
                           рќ‘Ђ = рќ‘†рќ‘” ,        рќ‘„ = рќ‘†рќ‘” ,            рќ‘ = в„Ћрќ‘Ѓрќ‘” =                 .
                                                                                       рќ‘”
   вЂ”

G4a.5. рќђї2 -norms of the coefficients          Finite-convolution coefficients built from рќњ‡, 1, and log
are divisor-bounded. Hence

                                       вЂ–рќ›Ѕрќ‘” вЂ–2 в‰Є рќ‘†рќ‘”1/2 (log рќ‘Ѓ )рќђ¶ ,


                                       вЂ–рќ›ѕрќ‘” вЂ–2 в‰Є рќ‘†рќ‘”1/2 (log рќ‘Ѓ )рќђ¶ .
   Now estimate the Fourier weight. Let

                                                               рќ‘¦
                                                          (пё‚       )пё‚
                                         рќ‘Љрќ‘Њ (рќ‘¦) = рќ‘Љ                     ,
                                                               рќ‘Њ
   where рќ‘Љ в€€ рќђ¶рќ‘ђв€ћ . Then

                                                     М‚пёЃ (рќ‘Њ рќњ‰).
                                         М‚пёЃрќ‘Њ (рќњ‰) = рќ‘Њ рќ‘Љ
                                         рќ‘Љ
   Therefore


                                                     42
                                     вѓ’1
                                     вѓ’              вѓ’ (пё‚
                                        М‚пёЃрќ‘Њ в„Ћ вѓ’ = рќ‘Њ вѓ’рќ‘ЉМ‚пёЃ в„Ћрќ‘Њ вѓ’ .
                                           (пё‚ )пё‚вѓ’          )пё‚вѓ’
                                                вѓ’   вѓ’        вѓ’
                                     вѓ’ рќ‘Љ
                                     вѓ’рќ‘ћ      рќ‘ћ  вѓ’ рќ‘ћ вѓ’     рќ‘ћ вѓ’
  On a balanced рќ‘”-layer,

                                                 рќ‘Ѓ 1/2
                                    рќ‘ћ в‰Ќ рќ‘†рќ‘” =           ,              рќ‘Њ в‰Ќ рќ‘Ѓ 1/2 ,
                                                  рќ‘”
  and therefore

                                                      рќ‘Њ
                                                        в‰Ќ рќ‘”.
                                                      рќ‘ћ
                     М‚пёЃ gives, for every рќ‘… > 0,
  The rapid decay of рќ‘Љ
                                    вѓ’1
                                    вѓ’
                                       М‚пёЃрќ‘Њ в„Ћ вѓ’ в‰Єрќ‘… рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќ‘… .
                                          (пё‚ )пё‚вѓ’
                                               вѓ’
                                    вѓ’ рќ‘Љ
                                    вѓ’рќ‘ћ      рќ‘ћ вѓ’
  Hence

                                вЂ–рќ›ѕМѓпёЂрќ‘”,в„Ћ вЂ–2 в‰Єрќ‘… рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќ‘… рќ‘†рќ‘”1/2 (log рќ‘Ѓ )рќђ¶ .
  вЂ”

G4a.6. Application of the DFI theorem                     By the external DFI estimate,
                                                                         )пё‚3/8
                                                          |в„Ћ|рќ‘Ѓ
                                                     (пё‚
                      |рќ’Єрќ‘”,в„Ћ | в‰Є вЂ–рќ›Ѕрќ‘” вЂ–2 вЂ–рќ›ѕМѓпёЂрќ‘”,в„Ћ вЂ–2              + рќ‘†рќ‘”2             (2рќ‘†рќ‘” )11/48+рќњЂ .
                                                            рќ‘”
  Substitute the coefficient norms:

                            вЂ–рќ›Ѕрќ‘” вЂ–2 вЂ–рќ›ѕМѓпёЂрќ‘”,в„Ћ вЂ–2 в‰Є рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќ‘… рќ‘†рќ‘” (log рќ‘Ѓ )рќђ¶ .
  Since

                                                           рќ‘Ѓ 1/2
                                                 рќ‘†рќ‘” =            ,
                                                            рќ‘”
  we have

                                                рќ‘”рќ‘†рќ‘” = рќ‘Ѓ 1/2 .
  Moreover,

                                                               рќ‘Ѓ
                                                     рќ‘†рќ‘”2 =        ,
                                                               рќ‘”2
  and

                                       |в„Ћ|рќ‘Ѓ        рќ‘Ѓ
                                            + рќ‘†рќ‘”2 = 2 (1 + |в„Ћ|рќ‘”).
                                         рќ‘”         рќ‘”
  Therefore
                                             )пё‚3/8
                                |в„Ћ|рќ‘Ѓ
                           (пё‚
                                     + рќ‘†рќ‘”2           = рќ‘Ѓ 3/8 рќ‘” в€’3/4 (1 + |в„Ћ|рќ‘”)3/8 .
                                  рќ‘”

                                                          43
  Also

                                         рќ‘†рќ‘”11/48 = рќ‘Ѓ 11/96 рќ‘” в€’11/48 .
  Thus

               |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 1/2 рќ‘Ѓ 3/8 рќ‘Ѓ 11/96 рќ‘” в€’3/4 рќ‘” в€’11/48 (1 + |в„Ћ|рќ‘”)в€’рќ‘…+3/8 (log рќ‘Ѓ )рќђ¶ .
  The exponents of рќ‘Ѓ add to
                               1 3 11   48 36 11  95
                                + +   =   +  +   = .
                               2 8 96   96 96 96  96
  The exponents of рќ‘” add to
                                   3 11   36 11   47
                                  в€’ в€’   =в€’ в€’    =в€’ .
                                   4 48   48 48   48
  We obtain

                           |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќњЂ рќ‘” в€’47/48 (1 + |в„Ћ|рќ‘”)в€’рќ‘…+3/8 .
  The logarithmic factors are absorbed into рќ‘Ѓ рќњЂ .
  вЂ”

G4a.7. Summation over в„Ћ         Take

                                                    рќ‘… = 2.
  Then
                                                     3   13
                                             в€’рќ‘… +      =в€’ .
                                                     8   8
  Also

                                           (1 + |в„Ћ|рќ‘”)в€’13/8 в‰Є рќ‘” в€’13/8 .
                                      в€‘пёЃ

                                      в„ЋМё=0

  Therefore

                                      |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќњЂ рќ‘” в€’47/48 рќ‘” в€’13/8 .
                               в€‘пёЃ

                               в„ЋМё=0

  Since
                                                   13  78
                                                      = ,
                                                    8  48
  we get
                                                 47 78    125
                                             в€’     в€’   =в€’     .
                                                 48 48     48
  Thus

                                         |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќњЂ рќ‘” в€’125/48 .
                                  в€‘пёЃ

                                  в„ЋМё=0
  вЂ”

                                                      44
G4a.8. Summation over рќ‘”         Since
                                                  125
                                                      > 1,
                                                  48
   the series

                                                       рќ‘” в€’125/48
                                                 в€‘пёЃ

                                                 рќ‘”в‰Ґ1

   converges. Hence
                                    в€‘пёЃ в€‘пёЃ
                                                 |рќ’Єрќ‘”,в„Ћ | в‰Є рќ‘Ѓ 95/96+рќњЂ .
                                        рќ‘” в„ЋМё=0

   Choose
                                                          1
                                                   рќњЂ<        .
                                                          96
   Then

                                           рќ‘Ѓ 95/96+рќњЂ = рќ‘њ(рќ‘Ѓ ).
   Consequently,

                                                  рќ’Єрќ‘” = рќ‘њ(рќ‘Ѓ ).
                                            в€‘пёЃ

                                             рќ‘”
   вЂ”

G4a.9. The term в„Ћ = 0 The term в„Ћ = 0 is not an error. It is the zero Fourier frequency and
gives the CKP local/main contribution:

                                    в„Ћ=0           =в‡’         рќ‘ЂCKP (рќ‘Ѓ ).
   All terms with в„Ћ Мё= 0 contribute рќ‘њ(рќ‘Ѓ ). Thus, at the oscillatory analysis level,

                                    CKP = рќ‘ЂCKP (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
   вЂ”

G4a.10. Coprimality and conductor issue                   The condition

                                                  (рќ‘Ћ, рќ‘ћ) = 1
    is present in our sum and is required to define рќ‘Ћ (mod рќ‘ћ). It matches the coprimality condition
in the external DFI Kloosterman-fraction estimate.
    The external numerator

                                                  рќ‘ = в„Ћрќ‘Ѓрќ‘”
    may have a common divisor with рќ‘ћ. This does not break the matching, because the DFI theorem
estimates phases of the form



                                                        45
                                                               рќ‘рќ‘Ћ
                                                          (пё‚        )пё‚
                                                      рќ‘’
                                                                рќ‘ћ
     with arbitrary integer рќ‘; coprimality is required between the inverted variable рќ‘Ћ and the modulus
рќ‘ћ.
     If one further decomposes by conductor
                                                             рќ‘ћ
                                                 рќ‘ћ1 =               ,
                                                          gcd(рќ‘ћ, рќ‘)
    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.
    вЂ”

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 М‚пёЃ               в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                             (пё‚ )пё‚ (пё‚   )пё‚
                        рќ’Є=                     рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќ‘Љ
                             в€‘пёЃ в€‘пёЃ      в€‘пёЃ
                                                               рќ‘Њ                рќ‘’       ,
                              рќ‘” в„ЋМё=0 рќ‘Ћв€јрќ‘† , рќ‘ћв€јрќ‘†             рќ‘ћ                  рќ‘ћ    рќ‘ћ
                                        рќ‘”        рќ‘”
                                       (рќ‘Ћ,рќ‘ћ)=1

     where

                                                 рќ‘Ѓ 1/2                      рќ‘Ѓ
                                       рќ‘†рќ‘” =            ,             рќ‘Ѓрќ‘” =     ,
                                                  рќ‘”                         рќ‘”

                                     вЂ–рќ›Ѕрќ‘” вЂ–2 , вЂ–рќ›ѕрќ‘” вЂ–2 в‰Є рќ‘†рќ‘”1/2 (log рќ‘Ѓ )рќђ¶ ,
     and рќ‘Љ в€€ рќђ¶рќ‘ђв€ћ . Then, using the DFI theorem for bilinear Kloosterman fractions,

                                                     рќ’Є = рќ‘њ(рќ‘Ѓ ).
     More precisely,

                                         |рќ’Є| в‰Є рќ‘Ѓ 95/96+рќњЂ = рќ‘њ(рќ‘Ѓ )
     for every sufficiently small fixed рќњЂ > 0.
     вЂ”
Remark 5.11 (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 рќђї2 -norm bounds;

     3. the large parameter в„Ћрќ‘Ѓ/рќ‘” is correctly compensated by the smooth Fourier weight;

     4. summation over в„Ћ Мё= 0 and рќ‘” gives рќ‘њ(рќ‘Ѓ );


                                                           46
  5. в„Ћ = 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, G2a, and G8a.

G4a.13. Logical Dependencies External dependency: X10 / DFI.
  Internal dependencies: G1a, G2a, G3a, CKPD, X10ER, C1A, C1, and G8a.
  Children served: G8a and the CKP branch closure.

5.8   Part 8. G8a: CKP branch theorem
Source file: Lemmas/g_8_a_ltx.md.

G8a. CKP Theorem and Zero-Frequency Normalization

G8a.0. Role Logical ID: G8a.
   Used by: H4, I1, CKP branch closure.
   Uses: G1a, G2a, G3a, G4a, CKPD, X10, X10ER, 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 H4, the schematic formulation

                                 рќ‘…CKP (рќ‘Ѓ ) = рќ‘ЂCKP (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ )
   is no longer sufficient. One must prove the sharper statement

                             рќ‘ЂCKP,в„¬,рќњЏ (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ )

   for every tagged CKP atom \{}((\{}mathcal B, \{}tau) \{}).
   Otherwise the local/main assembly is not entitled to accept the CKP zero-frequency term.
   Thus G8a proves two things:

  1. zero-frequency normalization:


                             в„Ћ=0     =в‡’        Locрќ‘„ рќ‘…в„¬,рќњЏ (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ );

  1. nonzero-frequency cancellation:


                                               рќ’Єрќ‘”,в„Ћ = рќ‘њ(рќ‘Ѓ )
                                        в€‘пёЃ

                                        в„ЋМё=0

   after summing over all relevant CKP layers, using G3a, G4a, X10, X10ER, and C1P/C1A/C1.
   вЂ”




                                                  47
G8a.1. Tagged CKP atom                Let \{}((\{}mathcal B, \{}tau) \{}) be a tagged CKP atom
produced by

                                              рќђµ1 в†’ рќђµ3 в†’ рќђ№ 3/рќђ№ 4.
   It has the schematic form

                  (рќ‘Ѓ ) =                  рќ›ј(рќ‘ў)рќ›јвЂІ (рќ‘ўвЂІ )рќ›Ѕ(рќ‘¦)рќ›Ѕ вЂІ (рќ‘¦ вЂІ )рќ‘Љрќ‘€ (рќ‘ў)рќ‘Љрќ‘€ вЂІ (рќ‘ўвЂІ )рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ (рќ‘¦ вЂІ ),
                               в€‘пёЃ
              CKP
             рќ‘…в„¬,рќњЏ
                           рќ‘ўрќ‘¦+рќ‘ўвЂІ рќ‘¦ вЂІ =рќ‘Ѓ

   where:

  вЂў рќ‘ў, рќ‘ўвЂІ are balanced finite-convolution grouped variables;

  вЂў рќ‘¦, рќ‘¦ вЂІ are complementary variables;

  вЂў coefficients are divisor-bounded finite-convolution sequences inherited from B1;

  вЂў all weights and ranges are tagged by \{}((\{}mathcal B,

   \{}tau) \{});

  вЂў the CKP balance regime gives


                           рќ‘€ в‰Ќ рќ‘€ вЂІ в‰Ќ рќ‘Ѓ 1/2+рќ‘‚(рќњ‚) ,              рќ‘Њ в‰Ќ рќ‘Њ вЂІ в‰Ќ рќ‘Ѓ 1/2+рќ‘‚(рќњ‚) .
   The tag \{}((\{}mathcal B, \{}tau) \{}) is fixed throughout. This ensures compatibility with
the LPI local projection interface.
   вЂ”

G8a.2. GCD splitting         By Lemma G1a, write

                                  рќ‘ў = рќ‘”рќ‘Ћ,           рќ‘ўвЂІ = рќ‘”рќ‘ћ,           (рќ‘Ћ, рќ‘ћ) = 1.
   Then a necessary condition for a nonempty layer is

                                                       рќ‘” | рќ‘Ѓ,
   and after putting

                                                              рќ‘Ѓ
                                                     рќ‘Ѓрќ‘” =       ,
                                                              рќ‘”
   we obtain the reduced equation

                                      рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” ,              (рќ‘Ћ, рќ‘ћ) = 1.
   Thus

                                               (рќ‘Ѓ ) =                  (рќ‘Ѓ ).
                                                         в€‘пёЃ
                                           CKP                   CKP
                                          рќ‘…в„¬,рќњЏ                  рќ‘…в„¬,рќњЏ,рќ‘”
                                                         рќ‘”|рќ‘Ѓ

  If a gcd layer with рќ‘” в€¤ рќ‘Ѓ appears during the formal gcd split, its equation рќ‘”рќ‘Ћ рќ‘¦ + рќ‘”рќ‘ћ рќ‘¦ вЂІ = рќ‘Ѓ has
empty support. The layer is not silently discarded: it carries the inherited tag (в„¬, рќњЏ, рќ‘”), contributes

                                                         48
zero, and is terminal Edge of zero effective volume. Thus the B3 CKP predicate remains a scale-
structural predicate; divisibility by рќ‘Ѓ is handled inside the exact G1a/G8a gcd decomposition.
    Large рќ‘”-layers outside the balanced CKP range are routed by X10ER to C1P/C1A/C1 and contribute
рќ‘њ(рќ‘Ѓ ). Hence it suffices to treat the balanced range.
    вЂ”

G8a.3. Weighted smooth AP expansion                                    For fixed рќ‘”, рќ‘Ћ, рќ‘ћ, the reduced equation is

                                                           рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘” .
   Eliminate \{}(yвЂ™ \{}):

                                                                      рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                           рќ‘¦вЂІ =               ,
                                                                         рќ‘ћ
   and impose

                                                      рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ           (mod рќ‘ћ).
   The weighted fibre is

                                                                     рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦                       рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                                (пё‚           )пё‚              (пё‚            )пё‚
                                                            вЂІ
                    рќ’®рќ‘Ћ,рќ‘ћ =                                                   рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ
                                       в€‘пёЃ
                                             рќ›Ѕ(рќ‘¦)рќ›Ѕ                                                         .
                             рќ‘¦в‰Ўрќ‘Ѓрќ‘” рќ‘Ћ mod рќ‘ћ
                                                                        рќ‘ћ                             рќ‘ћ

   Define the smooth tagged fibre weight

                                                           рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦                          рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                                      (пё‚                )пё‚             (пё‚             )пё‚
                                                 вЂІ
                       рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦) = рќ›Ѕ(рќ‘¦)рќ›Ѕ                            рќ‘Љрќ‘Њ (рќ‘¦)рќ‘Љрќ‘Њ вЂІ                       ,
                                                              рќ‘ћ                                рќ‘ћ
    with the convention that the summand is zero unless (рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ в€€ Z and lies in the dyadic
support.
    The dependence of рќђ№рќ‘Ћ,рќ‘ћ on both рќ‘Ћ and рќ‘ћ 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 (рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦)/рќ‘ћ в‰Ќ рќ‘Њ вЂІ ; hence
                                       рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦           рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦ вЂІ рќ‘Ѓрќ‘” в€’ рќ‘Ћрќ‘¦
                                  (пё‚             )пё‚                               (пё‚          )пё‚
                      рќњ•рќ‘ћ рќ‘Љ   рќ‘ЊвЂІ                       =в€’        рќ‘Љрќ‘Њ вЂІ                               в‰Є рќ‘„в€’1
                                                                                                      рќ‘” ,
                                          рќ‘ћ                рќ‘ћ2         рќ‘ћ
              (1)
after using рќ‘Љрќ‘Њ вЂІ в‰Є (рќ‘Њ вЂІ )в€’1 and рќ‘ћ в‰Ќ рќ‘„рќ‘” . Similarly, рќњ•рќ‘Ћ produces (рќ‘¦/рќ‘ћ)рќ‘Љрќ‘ЊвЂІ вЂІ , which is admissible in
the central balanced CKP range рќ‘Њ в‰Ќ рќ‘Њ вЂІ , рќђґрќ‘” в‰Ќ рќ‘„рќ‘” . 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 в€‘пёЃ М‚пёЂ               в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                        (пё‚ )пё‚ (пё‚       )пё‚
                                         рќ’®рќ‘Ћ,рќ‘ћ =             рќђ№рќ‘Ћ,рќ‘ћ              рќ‘’       ,
                                                      рќ‘ћ в„Ћв€€Z                 рќ‘ћ    рќ‘ћ
   where the smooth Fourier transform satisfies rapid decay inherited from the dyadic weights.
Any nonsmooth bounded finite-convolution coefficient that cannot be included into рќђ№рќ‘Ћ,рќ‘ћ 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 рќ‘Љрќ‘Њ (рќ‘¦) only.
   вЂ”

                                                                       49
G8a.4. Zero-frequency term         The zero-frequency contribution is
                                        1             1 в€‘пёЃ
                                   (0)
                                  рќ’®рќ‘Ћ,рќ‘ћ = рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ (0) =      рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦).
                                        рќ‘ћ             рќ‘ћ рќ‘¦
   Therefore the tagged CKP zero-frequency contribution is

                           CKP,0                                       1
                                 (рќ‘Ѓ ) =                    рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ) рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ (0),
                                           в€‘пёЃ в€‘пёЃ
                          рќ‘Ђв„¬,рќњЏ
                                           рќ‘”|рќ‘Ѓ     рќ‘Ћ,рќ‘ћ                 рќ‘ћ
                                                 (рќ‘Ћ,рќ‘ћ)=1

   with all dyadic weights and tags inherited from \{}((\{}mathcal B, \{}tau) \{}).
   This expression is local because it contains no oscillatory phase

                                                в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                 (пё‚          )пё‚
                                              рќ‘’
                                                 рќ‘ћ
   with в„Ћ Мё= 0.
   However, LPI-admission requires more: this local term must equal the canonical tagged local
projection that H4 later assembles.
   вЂ”

G8a.5. CKP zero-frequency equals the LPI tagged local projection

Lemma 5.12 (Lemma G8a.1). For every tagged CKP atom \{}((\{}mathcal B, \{}tau) \{}),
                              CKP,0
                             рќ‘Ђв„¬,рќњЏ   (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                 CKP
                                                     (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).

Proof. The LPI tagged local projection \{}(\{}operatorname{Loc}_Q R_{\{}mathcal B, \{}tau}Л†{\{}mathrm{C
\{}) 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

                                              рќ‘„=
                                                       в€ЏпёЃ
                                                             рќ‘ќ.
                                                       рќ‘ќв‰¤рќ‘¤

   In the CKP tagged atom, after gcd splitting and local residue decomposition modulo рќ‘„, all
congruence restrictions are local. The smooth variables remain distributed over the same tagged
dyadic cells. The fibre part for fixed (рќ‘”, рќ‘Ћ, рќ‘ћ) is exactly

                                     рќ’®рќ‘Ћ,рќ‘ћ =                       рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦).
                                                      в€‘пёЃ

                                              рќ‘¦в‰Ўрќ‘Ѓрќ‘” рќ‘Ћ mod рќ‘ћ

   The finite AP identity gives

                                     1 в€‘пёЃ М‚пёЂ        в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                           (пё‚ )пё‚ (пё‚           )пё‚
                              рќ’®рќ‘Ћ,рќ‘ћ =           рќђ№рќ‘Ћ,рќ‘ћ   рќ‘’       ,
                                     рќ‘ћ в„Ћ mod рќ‘ћ      рќ‘ћ    рќ‘ћ

   up to the already routed endpoint smoothing error. Its в„Ћ = 0 term is
                                      1 М‚пёЂ         1 в€‘пёЃ
                                        рќђ№рќ‘Ћ,рќ‘ћ (0) =      рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦).
                                      рќ‘ћ            рќ‘ћ рќ‘¦
   Therefore the full zero-frequency CKP term is the explicitly tagged sum


                                                      50
                           CKP,0                                            1 в€‘пёЃ
                                 (рќ‘Ѓ ) =                      рќ›јрќ‘” (рќ‘Ћ)рќ›ѕрќ‘” (рќ‘ћ)        рќђ№рќ‘Ћ,рќ‘ћ (рќ‘¦),
                                             в€‘пёЃ в€‘пёЃ
                          рќ‘Ђв„¬,рќњЏ
                                             рќ‘”|рќ‘Ѓ     рќ‘Ћ,рќ‘ћ                    рќ‘ћ рќ‘¦
                                                   (рќ‘Ћ,рќ‘ћ)=1

    with the same tag (в„¬, рќњЏ ).
    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 в„Ћ = 0 term are exactly the coefficient
factors used by Locрќ‘„ рќ‘…в„¬,рќњЏCKP (рќ‘Ѓ ).

    The equation

                                                 рќ‘Ћрќ‘¦ + рќ‘ћрќ‘¦ вЂІ = рќ‘Ѓрќ‘”
   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 в„Ћ = 0 is precisely the average over the residue class

                                             рќ‘¦ в‰Ў рќ‘Ѓрќ‘” рќ‘Ћ        (mod рќ‘ћ),
    with density factor 1/рќ‘ћ. 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 admis-
sion and C1 Edge predicate E1/E6 and therefore contribute \{}(o(N) \{}). The parent tag
\{}((\{}mathcal B, \{}tau) \{}) is preserved throughout the gcd splitting, AP expansion and
zero-frequency extraction. Therefore no local term is moved between different tags.
    Hence
                                  CKP,0
                                 рќ‘Ђв„¬,рќњЏ   (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                     CKP
                                                         (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
   Lemma proved.
   The zero-frequency term is therefore not merely local-looking; it is explicitly identified with the
LPI tagged О›рќ‘„ -projection of the same CKP cell.
   вЂ”


G8a.6. Nonzero frequencies and DFI reduction                        For в„Ћ Мё= 0, the contribution is

                                                        1               в„Ћ   в„Ћрќ‘Ѓрќ‘” рќ‘Ћ
                                                                      (пё‚ )пё‚ (пё‚           )пё‚
                        рќ’Єрќ‘”,в„Ћ =            рќ›Ѕрќ‘” (рќ‘Ћ)рќ›ѕрќ‘”,в„Ћ (рќ‘ћ) рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ
                                    в€‘пёЃ
                                                                          рќ‘’       .
                                 рќ‘Ћв€јрќђґ ,рќ‘ћв€јрќ‘„
                                                        рќ‘ћ               рќ‘ћ    рќ‘ћ
                                     рќ‘”       рќ‘”
                                   (рќ‘Ћ,рќ‘ћ)=1

    By Lemma G3a, this is reduced to a weighted bilinear Kloosterman fraction sum with param-
eters

                                 рќ‘Ђ = рќђґрќ‘” ,           рќ‘„ = рќ‘„рќ‘” ,         рќ‘ = |в„Ћ|рќ‘Ѓрќ‘” .
    For negative в„Ћ, the same DFI estimate is applied to the conjugate phase.
    By Lemma G4a, the external DFI theorem applies in the balanced range and gives the required
saving. The DFI smooth-weight derivative hypotheses for the actual nonseparated CKP fibre are
supplied by CKPD. The Fourier decay from the weighted G2a step gives, for every рќђґ > 0,

                                                        51
                                 вѓ’1
                                 вѓ’
                                 вѓ’ рќђ№М‚пёЂрќ‘Ћ,рќ‘ћ в„Ћ вѓ’ в‰Єрќђґ рќ‘”(1 + |в„Ћ|рќ‘”)в€’рќђґ рќђїрќђ¶ .
                                         (пё‚ )пё‚вѓ’
                                              вѓ’
                                 вѓ’рќ‘ћ        рќ‘ћ вѓ’
   The extra factor рќђїрќђ¶ absorbs the finite-convolution coefficient losses and derivatives of the tagged
smooth fibre weight.
   Thus the nonzero-frequency contribution satisfies

                                                  рќ’Єрќ‘”,в„Ћ = рќ‘њ(рќ‘Ѓ ).
                                        в€‘пёЃ в€‘пёЃ

                                         рќ‘” в„ЋМё=0

  Large-рќ‘”, high-frequency and small-conductor boundary ranges are excluded from the central
DFI range and are routed through X10ER and C1P/C1A/C1.
  вЂ”

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 рќ‘” в†’ рќђё3,


                                           high в„Ћ в†’ рќђё4,


                                       small conductor в†’ рќђё5,


                                 boundary/short volume в†’ рќђё1/рќђё6.
   Therefore they contribute \{}(o(N) \{}).
   вЂ”

G8a.8. CKP theorem

Theorem 5.13 (Theorem G8a). For every tagged CKP atom \{}((\{}mathcal B, \{}tau) \{}),

                                  CKP
                                 рќ‘…в„¬,рќњЏ (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                   CKP
                                                       (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
   Consequently, summing over all CKP tags,

                                   рќ‘…CKP (рќ‘Ѓ ) = рќ‘ЂCKP (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ),
   where

                                                   52
                              рќ‘ЂCKP (рќ‘Ѓ ) =               рќ‘ђв„¬ Locрќ‘„ рќ‘…в„¬,рќњЏ (рќ‘Ѓ ).
                                              в€‘пёЃ
                                                                 CKP

                                            в„¬,рќњЏ в€€CKP

   Thus the CKP main term is LPI-admissible and can be assembled by Lemma H4.

Proof. Apply G1a to split gcd layers:

                                рќ‘ў = рќ‘”рќ‘Ћ,      рќ‘ўвЂІ = рќ‘”рќ‘ћ,         (рќ‘Ћ, рќ‘ћ) = 1.
    For each balanced layer, apply the weighted smooth AP expansion. Separating the frequency
в„Ћ = 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 в„Ћ Мё= 0 are reduced by G3a to DFI/Kloosterman fraction sums. G4a
supplies the DFI saving in the central range, while X10ER and C1P/C1A/C1 handle high-frequency,
small-conductor, large-рќ‘”, and boundary layers. Therefore the total nonzero-frequency contribution
is рќ‘њ(рќ‘Ѓ ).
    Hence

                                 CKP
                                рќ‘…в„¬,рќњЏ (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                  CKP
                                                      (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
    Summing over the finite tagged CKP family gives the theorem. The number of tags is polylog-
arithmic and all error estimates have sufficient savings to survive this summation. The theorem is
proved.
    вЂ”


Remark 5.14 (G8a.9. Output).

           Every tagged CKP atom equals its LPI canonical local projection plus рќ‘њ(рќ‘Ѓ ).

   The LPI-admissible statement is:

                              рќ‘ЂCKP,в„¬,рќњЏ (рќ‘Ѓ ) = Locрќ‘„ рќ‘…в„¬,рќњЏ
                                                    CKP
                                                        (рќ‘Ѓ ) + рќ‘њ(рќ‘Ѓ ).
    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 sepa-
rated from Edge boundary ranges; and the dependence on DFI remains explicit through G4a/X10.
    вЂ”

G8a.10. Logical Dependencies External dependency: X10 / DFI through G4a.
  Internal dependencies: G1a, G2a, G3a, G4a, CKPD, X10ER, C1A, C1, B1LD, and LPI.
  Children served: H4, I1, and the CKP branch closure.

5.9   Part 9. X16BRS: BRS carrier-slice reduction
Source file: Lemmas/x16_brs_carrier_slice_ltx.md.

X16BRS. Carrier-Slice Divisor Estimate for BRS




                                                   53
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 ShiuвЂ™s arithmetic-progression
BrunвЂ“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.
    вЂ”

X16BRS.1. Statement: B1 Carrier-Slice Estimate Let в„¬ be a typed B1 dyadic block of
fixed depth рќђЅ0 . Let Massв„¬ (рќђ¶ в€€ рќђј) denote the sum of absolute values of the B1 coefficient weights
over tuples in в„¬ for which the carrier рќђ¶ lies in an additive interval рќђј of length рќ‘Њ . The carrier height
is рќ‘‹рќђ¶ .
    Fix the X16 slice-floor exponent рќђµ16 from the parameter register, and put

                                рќ‘Њ16 := max{|рќђј в€© Z|, рќ‘‹рќђ¶ (log рќ‘Ѓ )в€’рќђµ16 }.
   The BRS carrier estimate needed by TTH is

                                                 рќ‘Њ16
                 Massв„¬ (рќђ¶ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16       + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 ,                (X16-BRS)
                                                 рќ‘‹рќђ¶
   where рќђ¶16 = рќђ¶16 (рќђЅ0 ) and рќњЊ16 = рќњЊ16 (рќђЅ0 ) > 0.
   The allowed BRS carriers are:

  1. grouped product carriers;

  2. Goldbach complementary carriers рќ‘Ѓ в€’ рќ‘ѓ ;

  3. quotient carriers рќ‘  from a recorded equation рќђї = рќ‘‘рќ‘ ;

  4. controlled divisor quotients рќђї/рќ‘‘0 , with рќ‘‘0 в‰¤ (log рќ‘Ѓ )рќђ¶ .

   вЂ”

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 рќ‘ѓ of height рќ‘‹рќ‘ѓ , with рќ‘Њ16 = max{|рќђј в€© Z|, рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16 },

                                                 рќ‘Њ16
                 Massв„¬ (рќ‘ѓ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16       + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 .                (X16-Core)
                                                 рќ‘‹рќ‘ѓ
   This is the fixed-depth divisor-correlation estimate proved in Lemma X16C. The proof keeps the
genuine рќ‘Ѓ в€’ рќ‘ќрќ‘ў divisor correlation and controls it by Shiu-type averages in arithmetic progressions,
switching between the carrier and complementary variables according to the dyadic range.




                                                    54
X16BRS.2a. Excluded Shortcut: One-Variable Divisor Averaging                                The estimate cannot
be proved by fixing рќ‘ѓ = рќ‘› в€€ рќђј and bounding only

                                                           рќњЏрќ‘ (рќ‘›)
                                                     в€‘пёЃ

                                                     рќ‘›в€€рќђј
   by a standard average divisor estimate. After fixing рќ‘ѓ = рќ‘›, the remaining variables still satisfy
a Goldbach-complementary equation of the form

                                                   рќ‘› рќ‘Ј + рќ‘¤ = рќ‘Ѓ.
  Thus the relevant majorant is not merely рќњЏрќ‘ (рќ‘›); it is a fixed-depth divisor correlation along the
moving complementary values рќ‘Ѓ в€’ рќ‘›рќ‘Ј, for example schematically

                                         рќњЏрќ‘1 (рќ‘›)          рќњЏрќ‘2 (рќ‘Ј)рќњЏрќ‘3 (рќ‘Ѓ в€’ рќ‘›рќ‘Ј).
                                   в€‘пёЃ              в€‘пёЃ

                                   рќ‘›в€€рќђј             рќ‘Јв‰Ќрќ‘‰рќ‘›
   A bound for рќ‘›в€€рќђј рќњЏрќ‘ (рќ‘›) alone does not control the correlation with рќ‘Ѓ в€’ рќ‘›рќ‘Ј, especially when
                  в€‘пёЂ

the modulus/step рќ‘› 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.
   вЂ”

X16BRS.3. Proof: Product and Complementary Carriers For a grouped product carrier
рќђ¶ = рќ‘ѓ , (X16-BRS) is exactly X16-Core.
   For a complementary carrier рќђ¶ = рќ‘Ѓ в€’ рќ‘ѓ , the condition рќђ¶ в€€ рќђј is equivalent to рќ‘ѓ в€€ рќ‘Ѓ в€’ рќђј. Since
|рќ‘Ѓ в€’ рќђј| = |рќђј| and the dyadic height is unchanged up to fixed constants, X16-Core gives (X16-BRS).
   вЂ”

X16BRS.4. Proof: Quotient Carriers                   Let рќђ¶ = рќ‘  occur through a recorded quotient equation

                                   рќђї = рќ‘‘рќ‘ ,           рќ‘‘ в‰Ќ рќђ·,         рќ‘  в‰Ќ рќ‘‹рќђ¶ .
   If рќ‘‘ is fixed or controlled by a dyadic divisor tag, then рќ‘  в€€ рќђј implies
                       рќђї в€€ рќ‘‘рќђј,      |рќ‘‘рќђј в€© Z| в‰Є рќ‘‘|рќђј в€© Z| + рќ‘‚(рќ‘‘),                рќ‘‹рќђї в‰Ќ рќ‘‘рќ‘‹рќђ¶ .
Applying X16-Core to the carrier рќђї gives
                                                   рќ‘Њ16
                                 рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16         + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 ,
                                                   рќ‘‹рќђ¶
which is (X16-BRS).
    If the divisor рќ‘‘ is not fixed by a routing tag and summing over рќ‘‘ 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 |рќ›ј(рќ‘‘)| в‰Є рќњЏрќ‘‚рќђЅ0 (1) (рќ‘‘)(log рќ‘Ѓ )рќ‘‚(1) .
On a dyadic рќ‘‘-block,
                                     в€‘пёЃ рќњЏрќ‘‚рќђЅ (1) (рќ‘‘)
                                            0
                                                    в‰Є (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
                                    рќ‘‘в‰Ќрќђ·
                                              рќ‘‘
Thus the controlled sum over tagged рќ‘‘-layers preserves (X16-BRS), after enlarging рќђ¶16 by a constant
depending only on рќђЅ0 .
   вЂ”

                                                           55
X16BRS.5. Proof: Controlled Divisor Quotients Let рќђ¶ = рќђї/рќ‘‘0 , where рќ‘‘0 в‰¤ (log рќ‘Ѓ )рќђ¶ is
fixed or controlled. Then
                            рќђ¶ в€€ рќђј в‡ђв‡’ рќђї в€€ рќ‘‘0 рќђј.
The carrier height changes from рќ‘‹рќђ¶ to рќ‘‘0 рќ‘‹рќђ¶ , while the interval length changes from рќ‘Њ to рќ‘‘0 рќ‘Њ .
Their ratio is unchanged, and the polylogarithmic factor рќ‘‘0 is absorbed into рќђ¶16 . Hence X16-Core
again gives (X16-BRS).
   вЂ”
Remark 5.15 (X16BRS.6. Output). By X16-Core, all four BRS carrier types satisfy X16-BRS.
Therefore BRS may use the estimate with constants рќђ¶16 (рќђЅ0 ), рќњЊ16 (рќђЅ0 ) > 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.

5.10   Part 10. X16C: X16-Core Shiu/AP proof
Source file: Lemmas/x16_core_shiu_ap_proof_ltx.md.

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

                                                 рќњЏрќ‘ (рќ‘›).
                                           в€‘пёЃ

                                           рќ‘›в€€рќђј

   Instead it uses the arithmetic-progression form of ShiuвЂ™s BrunвЂ“Titchmarsh theorem for non-
negative multiplicative functions, applied to the moving complementary values рќ‘Ѓ в€’ рќ‘ђрќ‘ў. 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 ShiuвЂ™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), 161вЂ“170.
    Logical dependencies are X16BRS, BRS, F4, CKPD, the parameter register, and ShiuвЂ™s arithmetic-
progression BrunвЂ“Titchmarsh theorem for multiplicative functions. X16C is used by X16BRS, BRS,
TTH, and TNG.
    вЂ”



                                                 56
X16C.1. External Input: Shiu in Divisor-Function Form                           We use the following standard
consequence of ShiuвЂ™s theorem.

Lemma 5.16 (Lemma X16-SH. Divisor functions in AP intervals). Fix рќђѕ, рќђґ в‰Ґ 1 and 0 < рќ›ї < 1/10.
Let

                                                рќ‘“ (рќ‘›) = рќњЏрќђѕ (рќ‘›)рќђґ .
   Let рќђЅ вЉ‚ [1, рќ‘Ѓ ] be an interval of length рќђ», where рќ‘Ѓ рќ›ї в‰¤ рќђ» в‰¤ рќ‘Ѓ , and let рќ‘ћ в‰¤ рќђ» 1в€’рќ›ї . Then for
every residue class рќ‘Ћ mod рќ‘ћ,

                                                           рќђ»
                                                      (пё‚         )пё‚
                                     рќ‘“ (рќ‘›) в‰Єрќђѕ,рќђґ,рќ›ї            + 1 (log рќ‘Ѓ )рќђ¶SH (рќђѕ,рќђґ,рќ›ї) в„°рќ‘ћ,рќ‘Ћ ,             (SH)
                          в€‘пёЃ

                        рќ‘›в€€рќђЅ
                                                           рќ‘ћ
                     рќ‘›в‰Ўрќ‘Ћ (mod рќ‘ћ)

    where the possible non-coprime local factor is supported on primes dividing (рќ‘Ћ, рќ‘ћ). 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 5.17 (Lemma X16-LFA. Local factor averaging). Fix рќђѕ0 , рќђѕ, рќђґ в‰Ґ 1. Let в„°рќ‘ђ,рќ‘Ѓ denote any
local factor produced by applying X16-SH to the residue class рќ‘Ѓ mod рќ‘ђ, after extracting the common
                                                       #                     #
divisor (рќ‘ђ, рќ‘Ѓ ). Then, for every X16 carrier interval рќђј16 вЉ‚ [рќ‘‹/2, 3рќ‘‹] with |рќђј16 | в‰« рќ‘‹(log рќ‘Ѓ )в€’рќђµ16 ,
                                                1/2            #
                                    рќњЏрќђѕ0 (рќ‘ђ)рќђґ в„°рќ‘ђ,рќ‘Ѓ в‰Єрќђѕ0 ,рќђѕ,рќђґ,рќ›ї |рќђј16 |(log рќ‘Ѓ )рќђ¶loc .                    (SH-loc)
                            в€‘пёЃ

                               #
                            рќ‘ђв€€рќђј16

    The same bound holds for a full dyadic interval рќ‘ђ в‰Ќ рќ‘‹. Consequently it also applies in the
interchanged orientation of Case 2, where the averaging variable is the same-side complement рќ‘ў в‰Ќ
рќ‘€.

Proof of X16-LFA. ShiuвЂ™s theorem is stated for coprime residue classes. For a non-coprime class
write рќ‘” = (рќ‘Ћ, рќ‘ћ), рќ‘Ћ = рќ‘”рќ‘Ћ1 , рќ‘ћ = рќ‘”рќ‘ћ1 , with (рќ‘Ћ1 , рќ‘ћ1 ) = 1. The summand is рќ‘“ (рќ‘”рќ‘›1 ) on a coprime class
modulo рќ‘ћ1 . Since рќ‘“ = рќњЏрќђѕрќђґ is submultiplicative up to constants depending only on рќђѕ, рќђґ,


                                         рќ‘“ (рќ‘”рќ‘›1 ) в‰Єрќђѕ,рќђґ рќ‘“ (рќ‘”)рќ‘“ (рќ‘›1 ).
   The local cost is therefore bounded by a fixed divisor power of рќ‘” = (рќ‘Ћ, рќ‘ћ), together with the
harmless Euler factor рќ‘ќ|рќ‘ћ (1 + рќ‘‚рќђѕ,рќђґ (1/рќ‘ќ)). In our application рќ‘ћ = рќ‘ђ and рќ‘Ћ = рќ‘Ѓ , so рќ‘” = (рќ‘ђ, рќ‘Ѓ ).
                      в€ЏпёЂ
                               #
   Average this cost over рќ‘ђ в€€ рќђј16 . Equivalently, apply ShiuвЂ™s ordinary interval theorem to the
multiplicative function

                                      рќ‘”рќ‘Ѓ (рќ‘ђ) = рќњЏрќђѕ0 (рќ‘ђ)рќђґ рќњЏрќ‘Ђ ((рќ‘ђ, рќ‘Ѓ ))рќђµ ,
   where рќђµ is fixed large enough to dominate the local factor. This is a non-negative multiplicative
                                                                                              #
function of рќ‘ђ, uniformly divisor-bounded for fixed рќђѕ0 , рќђґ, рќ‘Ђ, рќђµ. If рќ‘‹ > (log рќ‘Ѓ )2рќђµ16 , then |рќђј16 |в‰«
рќ‘‹(log рќ‘Ѓ )в€’рќђµ 16 в‰« рќ‘‹ , and Shiu gives
                    1/2


                                                         #
                                              рќ‘”рќ‘Ѓ (рќ‘ђ) в‰Є |рќђј16 |(log рќ‘Ѓ )рќ‘‚(1) .
                                       в€‘пёЃ

                                         #
                                      рќ‘ђв€€рќђј16

   If рќ‘‹ в‰¤ (log рќ‘Ѓ )2рќђµ16 , the same bound is trivial after increasing the logarithmic exponent, because
every рќ‘ђ в€€ [рќ‘‹/2, 3рќ‘‹] is polylogarithmic. The full-dyadic-interval case is the ordinary mean-value


                                                           57
estimate for the same fixed divisor-bounded multiplicative function. The proof for the interchanged
variable is identical. Lemma proved.


Lemma 5.18 (Lemma X16-SH-class. Squared divisor functions are admissible). For every fixed
рќђѕ в‰Ґ 1, the function

                                               рќ‘“ (рќ‘›) = рќњЏрќђѕ (рќ‘›)2
   belongs to the divisor-bounded multiplicative class to which X16-SH applies, with constants de-
pending only on рќђѕ. Indeed, for prime powers,
                                                        )пёѓ2
                                        в„“+рќђѕ в€’1
                                       (пёѓ
                         рќњЏрќђѕ (рќ‘ќв„“ )2 =                          в‰Єрќђѕ (1 + в„“)2рќђѕв€’2 в‰¤ рќђґв„“рќђѕ ,
                                         рќђѕ в€’1
   after increasing рќђґрќђѕ . Also

                                        рќњЏрќђѕ (рќ‘›)2 в‰¤ рќњЏрќђѕ 2 (рќ‘›) в‰Єрќђѕ,рќњЂ рќ‘›рќњЂ
   for every рќњЂ > 0. Hence the applications of X16-SH below with рќ‘“ = рќњЏрќђѕ
                                                                     2 are legitimate.
                                                                      3
   вЂ”

X16C.2. Statement: X16-Core                 Let в„¬ be a B1 typed dyadic block of depth at most рќђЅ0 . Its
parent equation is
                                  рќ‘џ            рќ‘ 
                                       рќ‘Ћрќ‘– +         рќ‘Џрќ‘— = рќ‘Ѓ,         рќ‘џ, рќ‘  в‰¤ 2рќђЅ0 .                 (B1)
                                 в€ЏпёЃ           в€ЏпёЃ

                                 рќ‘–=1          рќ‘—=1

    Fix a slice-floor exponent рќђµ16 , chosen in the parameter register after рќђ¶16 and before рќђµрќњ… .
    Let рќ‘ѓ be a one-side grouped product carrier. Thus, after possibly interchanging the two sides
of (B1),
                                                    рќ‘џ
                                                        рќ‘Ћрќ‘– = рќ‘ѓ рќ‘€,
                                                   в€ЏпёЃ

                                                рќ‘–=1
   where рќ‘€ is the complementary product of the remaining variables on that side. Let рќ‘‹рќ‘ѓ be the
dyadic height of рќ‘ѓ , and let

                                рќђј # = рќђј в€© Z,             рќ‘Њ # = max(1, |рќђј # |).
   Define

                                 рќ‘Њ16 := max{рќ‘Њ # , рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16 }.                            (Y16)
   If рќ‘Њ # < рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16 , enlarge рќђј to an interval рќђј16 вЉ‚ [рќ‘‹рќ‘ѓ /2, 3рќ‘‹рќ‘ѓ ] with |рќђј16 в€© Z| в‰Ќ рќ‘Њ16 .
This only enlarges the mass. Hence the proof below is carried out for рќђј16 ; if \{}(YЛ†\{}#\{}ge
X_P(\{}log N)Л†{-B_{16}}\{}), take рќђј16 = рќђј.
   The BRS form of X16-Core is

                                                         рќ‘Њ16
               Massв„¬ (рќ‘ѓ в€€ рќђј) в‰ЄрќђЅ0 рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16               + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 .       (X16-Core)
                                                         рќ‘‹рќ‘ѓ
   One may take, after harmless enlargement,

                                                         58
                                                                 1
                       рќђ¶16 = 100рќђЅ02 + 100,            рќњЊ16 =             .                     (X16-constants)
                                                              106 рќђЅ04
   The рќ‘Њ # convention is the usual integer-lattice correction. The floor in рќ‘Њ16 is essential: a single
highly composite carrier value may carry a local divisor factor larger than any fixed power of log рќ‘Ѓ .
BRS does not need such a one-point estimate. If the actual marked image is shorter than the floor,
the monotone enlargement to рќђј16 still gives a strict C1 saving once рќђµ16 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

                                         рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16
                         рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16                   = рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16 в€’рќђµ16 ,
                                               рќ‘‹рќ‘ѓ
    up to the fixed C1/B1 coefficient losses. The parameter condition recorded in the parameter
register,

                                     рќђµ16 > рќђ¶0 + рќђ¶1 + рќђ¶16 + 20,
  makes this a strict C1 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.
  вЂ”

X16C.3. Setup: Reduction to a Bilinear Divisor Correlation The elementary B1 coeffi-
cients are of type рќњ‡ В· 1в‰¤рќ‘¦ , 1, and log. Hence, after dyadic localization and taking absolute values,
each coefficient product is bounded by

                                              (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
    If рќ‘ѓ = рќ‘ќ, the number of factorizations of рќ‘ќ by the carrier variables is в‰Є рќњЏрќђѕ1 (рќ‘ќ), with рќђѕ1 в‰¤ 2рќђЅ0 .
If the complementary product on the same side is рќ‘€ = рќ‘ў, the number of its factorizations is
в‰Є рќњЏрќђѕ2 (рќ‘ў), with рќђѕ2 в‰¤ 2рќђЅ0 . The opposite side is then forced to have product

                                              рќ‘„ = рќ‘Ѓ в€’ рќ‘ќрќ‘ў,
    and, on the positive support рќ‘Ѓ в€’ рќ‘ќрќ‘ў > 0, the number of its factorizations is в‰Є рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў),
with рќђѕ3 в‰¤ 2рќђЅ0 .
    Discarding dyadic restrictions on рќ‘„ only enlarges the count. The support condition рќ‘Ѓ в€’ рќ‘ќрќ‘ў > 0
is retained; terms with рќ‘Ѓ в€’ рќ‘ќрќ‘ў в‰¤ 0 contribute nothing. Therefore

            Massв„¬ (рќ‘ѓ в€€ рќђј) в‰Є (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1)           рќњЏрќђѕ1 (рќ‘ќ)         рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 ,       (1)
                                              в€‘пёЃ                в€‘пёЃ

                                                 #
                                              рќ‘ќв€€рќђј16             рќ‘ўв‰Ќрќ‘€

    where рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ 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 рќ‘† of B1 variables whose product is the carrier
рќ‘ѓ ; the complementary subset on the same side has product рќ‘€ . Every original B1 tuple maps to a
unique pair

                              рќ‘ќ = рќ‘ѓ (рќ‘Ћрќ‘– : рќ‘– в€€ рќ‘†),           рќ‘ў = рќ‘€ (рќ‘Ћрќ‘– : рќ‘– в€€
                                                                          / рќ‘†),



                                                       59
    and then the opposite side is forced to have product рќ‘„ = рќ‘Ѓ в€’ рќ‘ќрќ‘ў. Conversely, for fixed рќ‘ќ
and рќ‘ў, the number of compatible B1 factorizations is bounded by the displayed divisor factors
рќњЏрќђѕ1 (рќ‘ќ)рќњЏрќђѕ2 (рќ‘ў), and the number of opposite-side factorizations is bounded by рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў). 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 рќђ¶16 .
    It remains to prove that the double sum in (1) is

                           в‰ЄрќђЅ0 рќ‘Њ16 рќ‘€ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .                   (2)
   Since рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ , (2) is exactly (X16-Core).
   вЂ”
Parameter check 5.19 (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.

  вЂў Carrier fixing. For a fixed product carrier рќ‘ќ = рќ‘ѓ (рќ‘Ћрќ‘– : рќ‘– в€€ рќ‘†),

                                       #{(рќ‘Ћрќ‘– ) : рќ‘ѓ (рќ‘Ћрќ‘– ) = рќ‘ќ} в‰¤ рќњЏрќђѕ1 (рќ‘ќ).

      This loss is absorbed in рќђ¶16 .

  вЂў Same-side complement. For the complementary factor рќ‘ў = рќ‘€ (рќ‘Ћрќ‘— : рќ‘— в€€
                                                                     / рќ‘†),

                                       #{(рќ‘Ћрќ‘— ) : рќ‘€ (рќ‘Ћрќ‘— ) = рќ‘ў} в‰¤ рќњЏрќђѕ2 (рќ‘ў).

      This loss is absorbed in рќђ¶16 .

  вЂў Opposite side. The remaining Goldbach complement is рќ‘Ѓ в€’рќ‘ќрќ‘ў. The divisor weight рќњЏрќђѕ3 (рќ‘Ѓ в€’
    рќ‘ќрќ‘ў) remains inside the correlation; it is essential and is not averaged away.

  вЂў **Fixed рќ‘ќ arithmetic progression.** The expression рќ‘Ѓ в€’ рќ‘ќрќ‘ў, with рќ‘ў in a fixed residue class
    and a dyadic interval, is estimated by ShiuвЂ™s divisor estimate in an arithmetic progression.
    The modulus is рќ‘ќ in the non-small-volume range, and the loss is (log рќ‘Ѓ )рќ‘‚рќђѕ (1) .

  вЂў **Fixed рќ‘ў arithmetic progression.** The expression рќ‘Ѓ в€’ рќ‘ўрќ‘ќ, with рќ‘ќ in a fixed residue class
    and a dyadic interval, is estimated by the same Shiu/AP estimate with modulus рќ‘ў whenever
    this is the admissible orientation. The loss is again (log рќ‘Ѓ )рќ‘‚рќђѕ (1) .

  вЂў CauchyвЂ“Schwarz passage. Products of the fixed divisor weights are controlled by CauchyвЂ“
    Schwarz followed by Shiu/AP on the squared divisor weight. The recorded loss is square-
    rooted and polylogarithmic.

  вЂў Divisor second moment. The sums

                                                         рќњЏрќђѕ (рќ‘ў)2
                                                  в€‘пёЃ

                                                  рќ‘ўв€јрќ‘€

      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 рќ‘€ (log рќ‘€ )рќђѕ в€’1 .

                                                    60
  вЂў 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.

  вЂў 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.

  вЂў Residual small volume. If рќ‘Њ16 рќ‘€ в‰¤ рќ‘Ѓ 1в€’рќњЊ , or if the symmetric analogue holds, the trivial
    divisor bound with рќњЂ в‰Є рќњЊ gives a power saving.

   Thus the argument never replaces the carrier-complement correlation by the one-variable aver-
age рќ‘ќв€€рќђј рќњЏрќђѕ (рќ‘ќ). The complementary variable and the Goldbach expression рќ‘Ѓ в€’ рќ‘ќрќ‘ў remain present
    в€‘пёЂ

until the Shiu/AP estimate is applied on the correct fixed arithmetic progression.
   вЂ”

X16C.5. Proof: The Bilinear Correlation Estimate                             Let

                         рќ‘†=           рќњЏрќђѕ1 (рќ‘ќ)         рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 .
                              в€‘пёЃ                в€‘пёЃ

                                 #
                              рќ‘ќв€€рќђј16             рќ‘ўв‰Ќрќ‘€

   We prove (2).

Case 1. The complementary variable is not too small Assume рќ‘‹рќ‘ѓ в‰¤ рќ‘Ѓ 1в€’рќ›ї , with рќ›ї =
                      #
1/(20рќђЅ02 ). Fix рќ‘ќ в€€ рќђј16 . The values рќ‘Ѓ в€’ рќ‘ќрќ‘ў, as рќ‘ў в‰Ќ рќ‘€ , lie in an arithmetic progression modulo
рќ‘ќ, in an interval of length в‰Є рќ‘ќрќ‘€ . After intersecting with the positive support рќ‘Ѓ в€’ рќ‘ќрќ‘ў > 0, this
set is contained in an interval рќђЅрќ‘ќ вЉ‚ [1, рќ‘Ѓ ] of length рќђ»рќ‘ќ = рќ‘Ѓ ; this monotone enlargement can only
increase the AP divisor sum. Since рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ , the expected number of admissible residue-class
points is still

                                      рќђ»рќ‘ќ     рќ‘Ѓ
                                         +1в‰Ќ   + 1 в‰Ќ рќ‘€ + 1 в‰Ќ рќ‘€.
                                      рќ‘ќ      рќ‘ќ
                                                 1в€’рќ›ї/2
   The modulus satisfies рќ‘ќ в‰¤ рќ‘Ѓ 1в€’рќ›ї в‰¤ рќђ»рќ‘ќ     . Applying (SH) to рќ‘“ = рќњЏрќђѕ
                                                                    2 , and Cauchy-Schwarz
                                                                     3
together with the ordinary second moment bound for рќњЏрќђѕ2 , gives

                                            (пёѓ                    )пёѓ1/2 (пёѓ                                )пёѓ1/2
          рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 в‰¤               рќњЏрќђѕ2 (рќ‘ў)                     рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў) 1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0
     в€‘пёЃ                                          в€‘пёЃ                          в€‘пёЃ
                                                              2                              2
                                                                                                                  .
    рќ‘ўв‰Ќрќ‘€                                         рќ‘ўв‰Ќрќ‘€                      рќ‘ўв‰Ќрќ‘€

   The first factor is

                                           в‰Є рќ‘€ 1/2 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1)
    by the standard second moment for fixed divisor functions. For instance, the SelbergвЂ“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 рќђѕ
                                                                             2
                                         рќњЏрќђѕ (рќ‘ў)2 в‰Єрќђѕ рќ‘€ (log 2рќ‘€ )рќђѕ в€’1 .
                                   в€‘пёЃ

                                   рќ‘ўв‰Ќрќ‘€


                                                         61
   For the second factor, рќ‘Ѓ в€’ рќ‘ќрќ‘ў runs through the residue class рќ‘Ѓ mod рќ‘ќ in the enlarged interval
                                     2 gives
рќђЅрќ‘ќ вЉ‚ [1, рќ‘Ѓ ]. X16-SH applied to рќ‘“ = рќњЏрќђѕ 3


                              рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)2 1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 в‰Є рќ‘€ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в„°рќ‘ќ,рќ‘Ѓ .
                         в€‘пёЃ

                        рќ‘ўв‰Ќрќ‘€
   Multiplying the two square-root estimates yields
                                                                                  1/2
                            рќњЏрќђѕ2 (рќ‘ў)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ќрќ‘ў)1рќ‘Ѓ в€’рќ‘ќрќ‘ў>0 в‰Є рќ‘€ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в„°рќ‘ќ,рќ‘Ѓ .
                      в€‘пёЃ

                      рќ‘ўв‰Ќрќ‘€
                     #
   Summing over рќ‘ќ в€€ рќђј16 and using X16-LFA yields
                                                 1/2
                                         рќњЏрќђѕ1 (рќ‘ќ)в„°рќ‘ќ,рќ‘Ѓ в‰Є рќ‘Њ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
                                  в€‘пёЃ

                                    #
                                 рќ‘ќв€€рќђј16

   Therefore

                                          рќ‘† в‰Є рќ‘€ рќ‘Њ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .                            (3)
   This is the desired main term рќ‘Њ16 рќ‘€ .

Case 2. The carrier is very large Assume рќ‘‹рќ‘ѓ > рќ‘Ѓ 1в€’рќ›ї . Since рќ‘‹рќ‘ѓ рќ‘€ в‰Ќ рќ‘Ѓ , we have рќ‘€ в‰Є рќ‘Ѓ рќ›ї .
     If рќ‘Њ16 рќ‘€ в‰¤ рќ‘Ѓ 1в€’рќњЊ16 , the trivial divisor bound рќњЏрќђѕ (рќ‘›) в‰Єрќђѕ,рќњЂ рќ‘›рќњЂ , with рќњЂ chosen much smaller than
рќњЊ16 , gives the required power-saving term. Explicitly,

                       рќ‘† в‰Є рќ‘Ѓ рќњЂ рќ‘Њ16 рќ‘€ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в‰¤ рќ‘Ѓ 1в€’рќњЊ16 +рќњЂ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
   Taking
                                              1           1
                                           рќњЂ = рќњЊ16 =
                                              2      2 В· 106 рќђЅ04
   and enlarging рќђ¶16 absorbs the logarithmic factor. Equivalently, with
                                                       1
                                                 рќњЊвЂІ16 = рќњЊ16 ,
                                                       2
   we have
                                                                  вЂІ
                             рќ‘Ѓ 1в€’рќњЊ16 +рќњЂ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в‰Є рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 .
    After this point rename рќњЊвЂІ16 as рќњЊ16 . This is the harmless initial shrinkage of the displayed
positive constant in (X16-constants).
                                                                         #
    Assume now that рќ‘Њ16 рќ‘€ > рќ‘Ѓ 1в€’рќњЊ16 . We fix рќ‘ў instead of рќ‘ќ. As рќ‘ќ в€€ рќђј16    , the values рќ‘Ѓ в€’ рќ‘ўрќ‘ќ lie
in the residue class рќ‘Ѓ mod рќ‘ў, and the positive part is contained in an interval \{}(J_u\{}subset
[1,N]\{}) of length

                                                 рќђ»рќ‘ў в‰Ќ рќ‘ўрќ‘Њ16 .
   Since рќ‘ў в‰Ќ рќ‘€ , the non-small-volume assumption gives

                                           рќђ»рќ‘ў в‰« рќ‘€ рќ‘Њ16 > рќ‘Ѓ 1в€’рќњЊ16 .
   The Shiu modulus condition follows from the explicit parameter inequality

                                                       62
                                             рќ›ї < (1 в€’ рќњЊ16 )(1 в€’ рќ›ї/2).                            (4)
   Indeed, рќ‘ў в‰Ќ рќ‘€ в‰Є рќ‘Ѓ рќ›ї , while рќђ»рќ‘ў в‰« рќ‘Ѓ 1в€’рќњЊ16 ; hence, for large рќ‘Ѓ ,

                                               рќ‘ў в‰¤ рќ‘Ѓ рќ›ї в‰¤ рќђ»рќ‘ў1в€’рќ›ї/2 .
   For the displayed choices рќ›ї = 1/(20рќђЅ02 ) and рќњЊ16 = 1/(106 рќђЅ04 ), (4) holds for every рќђЅ0 в‰Ґ 1; any
constant loss is absorbed by the harmless initial shrinkage of рќњЊ16 .
   For fixed рќ‘ў, Cauchy-Schwarz gives

                           рќњЏрќђѕ1 (рќ‘ќ)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ўрќ‘ќ)1рќ‘Ѓ в€’рќ‘ўрќ‘ќ>0
                    в€‘пёЃ

                      #
                   рќ‘ќв€€рќђј16
                              вЋ›           вЋћ1/2 вЋ›                           вЋћ1/2

                                 рќњЏрќђѕ1 (рќ‘ќ)2 вЋ  вЋќ       рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ўрќ‘ќ)2 1рќ‘Ѓ в€’рќ‘ўрќ‘ќ>0 вЋ  .
                              в€‘пёЃ                 в€‘пёЃ
                           в‰¤вЋќ
                            вЋњ             вЋџ    вЋњ                           вЋџ
                                      #                     #
                                   рќ‘ќв€€рќђј16                 рќ‘ќв€€рќђј16

   The first factor is
                                                   1/2
                                              в‰Є рќ‘Њ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
   This is the ordinary рќ‘ћ = 1 divisor-function interval estimate; the X16 floor gives рќ‘Њ16 в‰Ґ
рќ‘‹рќ‘ѓ (log рќ‘Ѓ )в€’рќђµ16 , and in the present case рќ‘‹рќ‘ѓ > рќ‘Ѓ 1в€’рќ›ї , so the interval is far longer than any fixed
power needed for ShiuвЂ™s short-interval corollary.
   For the second factor, рќ‘Ѓ в€’ рќ‘ўрќ‘ќ lies in the residue class рќ‘Ѓ mod рќ‘ў in the interval рќђЅрќ‘ў of length рќђ»рќ‘ў ,
and the modulus condition has just been verified. X16-SH applied to рќ‘“ = рќњЏрќђѕ   2 gives
                                                                               3


                                   рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ўрќ‘ќ)2 1рќ‘Ѓ в€’рќ‘ўрќ‘ќ>0 в‰Є рќ‘Њ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в„°рќ‘ў,рќ‘Ѓ .
                           в€‘пёЃ

                              #
                           рќ‘ќв€€рќђј16

   Therefore
                                                                                         1/2
                              рќњЏрќђѕ1 (рќ‘ќ)рќњЏрќђѕ3 (рќ‘Ѓ в€’ рќ‘ўрќ‘ќ)1рќ‘Ѓ в€’рќ‘ўрќ‘ќ>0 в‰Є рќ‘Њ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) в„°рќ‘ў,рќ‘Ѓ .
                         в€‘пёЃ

                        #
                     рќ‘ќв€€рќђј16

   Summing over рќ‘ў в‰Ќ рќ‘€ and using the dyadic form of X16-LFA gives

                              рќ‘† в‰Є рќ‘Њ16 рќ‘€ (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) .
   This proves (2) in the large-carrier case.
   Combining the two cases proves the bilinear correlation estimate.
   вЂ”

X16C.6. Proof: Completion of X16-Core Substituting the bilinear estimate (2) into (1),
and absorbing all fixed coefficient and divisor exponents into рќђ¶16 = 100рќђЅ02 + 100, gives

                                                                 рќ‘Њ16
                         Massв„¬ (рќ‘ѓ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )рќђ¶16               + рќ‘Ѓ 1в€’рќњЊ16 (log рќ‘Ѓ )рќђ¶16 .
                                                                 рќ‘‹рќ‘ѓ
   This is X16-Core for one-side grouped product carriers.




                                                          63
   Complementary carriers рќ‘Ѓ в€’рќ‘ѓ , quotient carriers рќ‘  with рќђї = рќ‘‘рќ‘ , and controlled divisor quotients
рќђї/рќ‘‘0 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.
   вЂ”

X16C.7. Excluded Shortcut and Correct Routing              The following shortcut is not used:

                              рќ‘ѓ =рќ‘ќв€€рќђј       =в‡’    only average рќњЏ (рќ‘ќ).
    The actual remaining equation is

                                          рќ‘ќрќ‘ў + рќ‘„ = рќ‘Ѓ.
    The proof keeps the рќ‘„ = рќ‘Ѓ в€’рќ‘ќрќ‘ў correlation. For fixed рќ‘ќ, the рќ‘„-values form an AP modulo рќ‘ќ; for
fixed рќ‘ў, they form an AP modulo рќ‘ў. ShiuвЂ™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.
    вЂ”
Remark 5.20 (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: ShiuвЂ™s BrunвЂ“Titchmarsh theorem for
multiplicative functions, used in the divisor-function AP form stated in X16C.1.
   Children served: X16-BRS, BRS, TTH.


6     Package Dependency Ledger and Synchronization Notes
6.1    Compact Package Dependency Graph
CKP-X10-X16
+-- CKP branch
|   +-- G1a -> G2a -> G3a -> CKPD -> G4a/X10 -> G8a
|   +-- CKP zero frequency -> B1LD/H4 (imported local package output)
|   +-- X10ER excluded ranges -> C1P/C1A/C1; h=0 -> G8a/LPI -> H4
+-- X16/BRS branch
    +-- X16BRS -> X16C/X16
    +-- complementary carriers -> Edge, quotient-tag, local, or handled




                                                64
6.2     Local Ledger

      ID                            Uses                                 Exports
      X10                           DFI bilinear Kloosterman-fraction    central CKP nonzero-frequency sav-
                                    theorem                              ing
      X16                           Shiu/AP divisor-average theorem      X16C product-carrier estimate
      G1a                           CKP terminal structure               exact gcd splitting рќ‘ў = рќ‘”рќ‘Ћ, рќ‘ўвЂІ = рќ‘”рќ‘ћ
      G2a                           G1a, smooth Fourier/AP expansion     CKP frequency separation
      G3a                           G1a, G2a                             CKP-to-DFI phase рќ‘’(рќ‘џрќ‘Ћ/рќ‘ћ)
      CKPD                          G2a, G3a, X10                        actual two-variable weight derivative
                                                                         bounds
      G4a                           CKPD, X10, X10ER                     DFI application in the central CKP
                                                                         range
      G8a                           G1aвЂ“G4a,     CKPD,       LPI/B1LD,   CKP equals the LPI local projection
                                    C1P/C1A/C1                               plus рќ‘њ(рќ‘Ѓ ), later assembled by H4
      X16BRS                        BRS carrier definitions, X16C        reduction of BRS carriers to X16-
                                                                         Core or routed classes
      X16C                          X16, divisor second moment, AP       product-carrier Shiu/AP estimate
                                    CauchyвЂ“Schwarz




6.3     Synchronization Boundary
The package is synchronized with the one-file proof through the logical IDs listed above, not through
file names. File paths in this document are navigation aids identifying the current proof-source text.


7      Bibliography
This local bibliography lists only sources used by the CKP/X10/X16 analytic package.

    1. William Duke, John B. Friedlander, and Henryk Iwaniec, *Bilinear forms with Kloosterman
       fractions*, Inventiones Mathematicae 128 (1997), no. 1, 23вЂ“43. DOI: 10.1007/s002220050135.

    2. P. Shiu, A BrunвЂ“Titchmarsh theorem for multiplicative functions, Journal fuer die reine und
       angewandte Mathematik 313 (1980), 161вЂ“170. DOI: 10.1515/crll.1980.313.161.

    3. Gerald Tenenbaum, *Introduction to Analytic and Probabilistic Number Theory*, Gradu-
       ate Studies in Mathematics 163, American Mathematical Society, 3rd ed., 2015, Ch. II.5,
       Theorem 5.

    4. Jean-Marc Deshouillers and Henryk Iwaniec, *Kloosterman sums and Fourier coefficients of
       cusp forms*, Inventiones Mathematicae 70 (1982), no. 2, 219вЂ“288. This is background only;
       the active CKP/X10 estimate is the DukeвЂ“FriedlanderвЂ“Iwaniec theorem listed above.


8      References
The active references for this package are the four items listed in the local bibliography. The full
proof manuscript contains the global bibliography and source register for all five theorem packages.




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