﻿         TNG/TC1 No Rogue Short Interval Full Proof Package

                               Denis Saltykov (ds1678@gmail.com)

                                              May 2026


Contents
1 TNG/TC1 No Rogue Short Interval Full Proof Package                                                   2
  1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   2
  1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    2
  1.3 Included Proof-Source Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      2

2 Part 1. X9L-GT: Near-global Davenport/AP Liouville input                                             2
      2.0.1 X9L-GT. Davenport/AP Input for TC1 Testing . . . . . . . . . . . . . . . . .               2

3 Part 2. TGD: TC1 GoodAWACK dichotomy                                                                 9
      3.0.1 TGD. Terminal GoodAWACK True-Complexity Split . . . . . . . . . . . . .                    9

4 Part 3. TGT: TC1 global testing                                                   14
      4.0.1 TGT. Aggregated Testing Route for TC1-GoodAWACK . . . . . . . . . . . . 14

5 Part 4. TGT-MF: Measured Fourier transfer                                            19
      5.0.1 TGT-MF. Measured Fourier Transfer for TC1 Global Testing . . . . . . . . . 19

6 Part 5. TTH-SC: Structural coarea closure                                                        22
      6.0.1 TTH-SC. Structural Coarea Closure and No Artificial Short-Interval Refine-
            ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Part 6. TNG: TC1 near-global chain                                                   26
      7.0.1 TNG. B1-Origin TC1 Near-Global-or-Routed Theorem . . . . . . . . . . . . . 26

8 Part 7. TTD: TC1 testing dichotomy                                                       31
      8.0.1 TTD. TC1 Testing Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Part 8. MRT: MRT admissibility                                                         36
      9.0.1 MRT. PACK Interface for TC1 Global Testing . . . . . . . . . . . . . . . . . 36

10 Part 9. ROC: Singular-origin routing                                                  38
       10.0.1 ROC. Range-Origin Lemma for Singular TC1 Testing . . . . . . . . . . . . . 39

11 Part 10. BRS: B1 range/skeleton ROC slice                                               44
       11.0.1 BRS. B1 Range/Slice Closure for Singular TC1 Testing . . . . . . . . . . . . 44




                                                   1
12 Part 11. TTH: Near-global length theorem                                             48
       12.0.1 TTH. Internal Length Lower Bound for B1-Origin TC1 Coarea Tests . . . . . 49




                                            2
1     TNG/TC1 No Rogue Short Interval Full Proof Package
1.1     Abstract
This full-proof package contains the TC1 testing chain proving that active B1-origin TC1 tests are
near-global or routed away.

1.2     Scope
This package supplies the TC1/no-rogue-short-interval brick. X16 carrier-slice estimates used inside
BRS are proved in the CKP/X10/X16 analytic full-proof package.

1.3     Included Proof-Source Files
    1. External/x_9l_gt_avg_polylog_verification_ltx.md вЂ“ Near-global Davenport/AP Li-
       ouville input

    2. Lemmas/tc1_goodawack_dichotomy_ltx.md вЂ“ TC1 GoodAWACK dichotomy

    3. Lemmas/tc1_global_testing_ltx.md вЂ“ TC1 global testing

    4. Lemmas/tc1_measured_fourier_transfer_ltx.md вЂ“ Measured Fourier transfer

    5. Lemmas/tc1_structural_coarea_closure_ltx.md вЂ“ Structural coarea closure

    6. Lemmas/tc1_near_global_chain_ltx.md вЂ“ TC1 near-global chain

    7. Lemmas/tc1_testing_dichotomy_ltx.md вЂ“ TC1 testing dichotomy

    8. Lemmas/tc1_mrt_admissibility_ltx.md вЂ“ MRT admissibility

    9. Lemmas/tc1_singular_origin_roc_ltx.md вЂ“ Singular-origin routing

 10. Lemmas/b1_range_skeleton_roc_slice_ltx.md вЂ“ B1 range/skeleton ROC slice

 11. Lemmas/tc1_theta_1_3_ltx.md вЂ“ Near-global length theorem


2     Part 1. X9L-GT: Near-global Davenport/AP Liouville input
Source file: External/x_9l_gt_avg_polylog_verification_ltx.md.

2.0.1    X9L-GT. Davenport/AP Input for TC1 Testing
X9L-GT.0. Statement and Role           Lemma X9L-GT states and verifies the external input

                              X9L-GT     or   X9L-AVG-POLYLOG.
    It is the averaged Liouville/Fourier input used by TGT for MRT-admissible TC1 testing families.
    To avoid ambiguity, the statement has two logically separate layers.

    1. **General low-рќњѓ target.** This is the broad low-рќњѓ polylog-modulus AP-fibre estimate one
       might want for arbitrary рќђ» в‰Ґ рќ‘‹ рќњѓ , 0 < рќњѓ < 1/3. This proof does not claim a published
       citation for that general target.


                                                 3
  2. Near-global X9L-GT theorem. This is the theorem invoked by the proof tree after
     TTH. In that route, every surviving B1-origin TC1 coarea test has near-global length рќђ» в‰Ґ
     рќ‘‹(log рќ‘‹)в€’рќђµ , and Davenport/AP cancellation is sufficient.

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

  1. arithmetic progression fibres рќ‘› = рќ‘”рќ‘ў + рќ‘Џ;

  2. рќ‘” в‰¤ (log рќ‘‹)рќђ¶ ;

  3. linear phases depending on the fibre;

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

  5. fibre lengths рќ‘€ = рќђ»/рќ‘”, with the normalized sum divided by рќ‘€ .

   The unused general low-рќњѓ target is:


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

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


рќ‘‹9рќђї-рќђґрќ‘‰ рќђє-рќ‘ѓ рќ‘‚рќђїрќ‘Њ рќђїрќ‘‚рќђє is supplied for unrouted TC1 coarea tests by Davenport/AP whenever рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ

   Thus the broader unused target is:


X9L-POLYLOG-MOD<1/3 : prove the same averaged normalized AP-fibre Fourier estimate for every fixed 0 < рќњѓ

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

X9L-GT.1. External Source The external source is:
   H. Davenport, On some infinite series involving arithmetical functions (II), Quart. J. Math.
Oxford 8 (1937), 313вЂ“320, DOI 10.1093/qmath/os-8.1.313.
   We use the standard Davenport consequence: for every рќђґ > 0,
                                    вѓ’              вѓ’
                                    вѓ’ в€‘пёЃ           вѓ’
                                         рќњ‡(рќ‘›)рќ‘’(рќ›јрќ‘›)вѓ’вѓ’ в‰Єрќђґ рќ‘Њ (log рќ‘Њ )в€’рќђґ .
                                    вѓ’              вѓ’
                               sup вѓ’вѓ’                                                         (Dav)
                              рќ›јв€€R/Z вѓ’рќ‘›в‰¤рќ‘Њ           вѓ’

   The AP/interval form for рќњ† follows from рќњ† = рќњ‡ * 1в–Ў , a square-divisor split, additive-character
expansion of the AP condition, and summation by parts for smooth weights.




                                                 4
X9L-GT.2. Statement: Required Normalized AP-Fibre Form The form needed by TGT
can be abstracted as follows.
   Fix рќђ¶ > 0, 0 < рќњѓ < 1, and a testing measure рќњ€ whose start pushforward satisfies

                                                                 рќ‘‘рќ‘Ґ
                                       (start)# рќњ€ в‰Є (log рќ‘‹)рќђ¶        .                        (PACK)
                                                                 рќ‘‹
   For parameters рќ‘ќ in the test family, let

                   рќ‘”рќ‘ќ в‰¤ (log рќ‘‹)рќђ¶ ,           рќђ»рќ‘ќ в‰Ќ рќђ»,       рќ‘€рќ‘ќ = рќђ»рќ‘ќ /рќ‘”рќ‘ќ ,    рќђ»рќ‘ќ в‰Ґ рќ‘‹ рќњѓ .
   The needed Fourier test is of the shape
                                           вѓ’                              вѓ’
                                           вѓ’                              вѓ’
                                           вѓ’ 1    в€‘пёЃ                      вѓ’
                             в„’рќ‘ќ (рќњ†) = sup вѓ’вѓ’           рќњ†(рќ‘”рќ‘ќ рќ‘ў + рќ‘Џрќ‘ќ )рќ‘’(рќ›јрќ‘ў)вѓ’вѓ’ ,
                                     рќ›јв€€R/Z вѓ’ рќ‘€рќ‘ќ 1в‰¤рќ‘ўв‰¤рќ‘€рќ‘ќ                    вѓ’

   possibly with smooth weights of fixed/polylogarithmic complexity. The desired input is
                                        в€«пёЃ
                                             |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€(рќ‘ќ) = рќ‘њ(1).                       (X9L-GT)

    The normalization by рќ‘€рќ‘ќ = рќђ»рќ‘ќ /рќ‘”рќ‘ќ is essential. Bounds normalized by the ambient length рќђ»рќ‘ќ
are not enough unless they save a factor рќ‘”рќ‘ќ .
    вЂ”

X9L-GT.3. Scope Check: Limitation of the Unused Low-Theta Target                         The ordinary
Fourier theorem gives averaged cancellation for
                                         1   в€‘пёЃ
                                                   рќњ†(рќ‘›)рќ‘’(рќ›Ѕрќ‘›).
                                         рќђ» рќ‘Ґ<рќ‘›в‰¤рќ‘Ґ+рќђ»

   For an AP fibre,
                        в€‘пёЃ                             в€‘пёЃ
                             рќњ†(рќ‘”рќ‘ў + рќ‘Џ)рќ‘’(рќ›јрќ‘ў) =                   рќњ†(рќ‘›)рќ‘’(рќ›ј(рќ‘› в€’ рќ‘Џ)/рќ‘”).
                       рќ‘ўв‰¤рќ‘€                         рќ‘Џ<рќ‘›в‰¤рќ‘Џ+рќ‘”рќ‘€
                                                  рќ‘›в‰Ўрќ‘Џ (mod рќ‘”)

   Expanding the congruence by additive characters gives
                      вѓ’                    вѓ’         вѓ’                    вѓ’
                      вѓ’                    вѓ’         вѓ’                    вѓ’
                      вѓ’ 1 в€‘пёЃ               вѓ’         вѓ’1   в€‘пёЃ              вѓ’
                      вѓ’
                      вѓ’рќ‘€     рќњ†(рќ‘”рќ‘ў + рќ‘Џ)рќ‘’(рќ›јрќ‘ў)вѓ’ в‰¤ рќ‘” sup вѓ’
                                           вѓ’         вѓ’          рќњ†(рќ‘›)рќ‘’(рќ›Ѕрќ‘›)вѓ’вѓ’ .               (AP-loss)
                      вѓ’ рќ‘ўв‰¤рќ‘€                вѓ’      рќ›Ѕ вѓ’ рќђ» рќ‘Џ<рќ‘›в‰¤рќ‘Џ+рќђ»           вѓ’

    Thus a bare qualitative рќ‘њ(1) average for ordinary intervals does not imply the normalized AP-
fibre statement uniformly for рќ‘” в‰¤ (log рќ‘‹)рќђ¶ . One needs either:

  1. a logarithmic saving strong enough to absorb рќ‘”;

  2. a theorem stated directly relative to AP length рќ‘€ = рќђ»/рќ‘”;

  3. a proof that all TC1 moduli рќ‘” are bounded independently of рќ‘Ѓ .

   The TC1 route only gives рќ‘” в‰¤ (log рќ‘‹)рќђ¶ , not bounded рќ‘”.
   вЂ”


                                                       5
X9L-GT.4. Proof: Near-Global Davenport/AP Transfer                       The proof applies X9L-GT only
after TTH, where every surviving B1-origin coarea test has

                                          рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ .                                       (NG)
   For a fibre sum
                          в€‘пёЃ
                     рќ‘†=         рќњ†(рќ‘”рќ‘ў + рќ‘Џ)рќ‘’(рќ›јрќ‘ў),           рќђ» = рќ‘”рќ‘€,     рќ‘” в‰¤ (log рќ‘‹)рќђ¶ ,
                          рќ‘ўв‰¤рќ‘€

    expand the congruence рќ‘› в‰Ў рќ‘Џ (mod рќ‘”) by additive characters and transfer the phase рќ‘’(рќ›јрќ‘ў) to a
linear phase in рќ‘›. This gives a loss в‰¤ рќ‘”, and the normalization by рќ‘€ = рќђ»/рќ‘” gives a second factor
рќ‘”. DavenportвЂ™s bound, applied to global prefixes and then differenced over the interval of length
рќђ», gives
                              вѓ’                      вѓ’
                              вѓ’                      вѓ’
                              вѓ’ 1 в€‘пёЃ                         рќ‘‹
                                      рќњ‡(рќ‘”рќ‘ў + рќ‘Џ)рќ‘’(рќ›јрќ‘ў)вѓ’вѓ’ в‰Єрќђґ рќ‘” 2 (log рќ‘‹)в€’рќђґ .
                                                     вѓ’
                          sup вѓ’
                              вѓ’                                                             (Dav-AP)
                           рќ›ј вѓ’  рќ‘€ рќ‘ўв‰¤рќ‘€                вѓ’       рќђ»

    Under (NG), the factor рќ‘” 2 рќ‘‹/рќђ» is at most a fixed power of log рќ‘‹. By choosing the Davenport
saving exponent larger than this polylogarithmic loss, we obtain arbitrary logarithmic saving for
the normalized AP fibre.
    For рќњ†, use
                                                  в€‘пёЃ
                                         рќњ†(рќ‘›) =           рќњ‡(рќ‘›/рќ‘‘2 ).                               (Sq)
                                                  рќ‘‘2 |рќ‘›

    The terms рќ‘‘ в‰¤ (log рќ‘‹)рќђ· are handled by the same Davenport/AP argument after changing
variables рќ‘љ = рќ‘›/рќ‘‘2 ; the near-global condition is stable under this polylogarithmic square-divisor
division. The terms рќ‘‘ > (log рќ‘‹)рќђ· have PACK-averaged normalized contribution

                                         в‰Є (log рќ‘‹)рќ‘‚(рќђ¶) рќђ·в€’1 .
   Taking рќђ· large gives the required рќ‘њ(1) bound.
   Thus the theorem supplied here is:


X9L-GT-NG : normalized AP-fibre Fourier cancellation holds for all unrouted TC1 coarea tests satisfying рќђ» в‰Ґ рќ‘‹

   вЂ”

X9L-GT.5. Proof: Explicit AP/Congruence Transfer The Davenport step used above can
be isolated as follows.
    Let рќ‘ћ в‰¤ (log рќ‘‹)рќђ¶ , рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ , and рќђј = [рќ‘Ґ, рќ‘Ґ + рќђ»] вЉ‚ [рќ‘‹, 2рќ‘‹]. Then for every residue class
рќ‘Џ (mod рќ‘ћ) and every рќ›ј в€€ R/Z,
                               вѓ’                      вѓ’
                               вѓ’                      вѓ’
                               вѓ’                      вѓ’
                                                      вѓ’ в‰Єрќђґ рќ‘‹(log рќ‘‹)в€’рќђґ
                               вѓ’    в€‘пёЃ                вѓ’
                               вѓ’
                               вѓ’             рќњ‡(рќ‘›)рќ‘’(рќ›јрќ‘›)вѓ’
                               вѓ’рќ‘›в‰Ўрќ‘Џ рќ‘›в€€рќђј
                               вѓ’                      вѓ’
                                     (mod рќ‘ћ)
                                                      вѓ’

   with arbitrary рќђґ. Since the near-global range has рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ , this is рќђ»(log рќ‘‹)в€’рќђґ+рќђµ , and
the exponent рќђґ can be increased to absorb all fixed polylogarithmic losses.

                                                    6
   Indeed,
                                                 1 в€‘пёЃ
                              1рќ‘›в‰Ўрќ‘Џ   (mod рќ‘ћ) =               рќ‘’(рќ‘џ(рќ‘› в€’ рќ‘Џ)/рќ‘ћ),
                                                 рќ‘ћ рќ‘џ (mod рќ‘ћ)

   so the left side is bounded by
                                            вѓ’             вѓ’
                                            вѓ’ в€‘пёЃ          вѓ’
                                        sup вѓ’    рќњ‡(рќ‘›)рќ‘’(рќ›Ѕрќ‘›)вѓ’ .
                                            вѓ’             вѓ’
                                         рќ›Ѕ  вѓ’
                                              рќ‘›в€€рќђј
                                                          вѓ’

    The interval sum is the difference of two Davenport prefix sums, hence has arbitrary logarithmic
saving relative to the ambient scale рќ‘‹. Passing from the рќ‘›-sum to the normalized fibre sum with
рќ‘› = рќ‘ћрќ‘ў + рќ‘Џ divides by рќ‘€ = рќђ»/рќ‘ћ, so the normalization costs рќ‘ћрќ‘‹/рќђ», a polylogarithmic factor in
the near-global range. Smooth weights are removed by a fixed partition and summation by parts,
costing only another polylogarithmic factor. The more conservative bound (Dav-AP) above records
an allowable рќ‘ћ 2 рќ‘‹/рќђ» loss; this is still polylogarithmic after TTH.
                                           2                           рќђ· are handled by the same
                               рќ‘‘2 |рќ‘› рќњ‡(рќ‘›/рќ‘‘ ). The terms рќ‘‘ в‰¤ (log рќ‘‹)
                            в€‘пёЂ
    For рќњ†, insert рќњ†(рќ‘›) =
                                              2     2
congruence-transfer lemma at scale (рќ‘‹/рќ‘‘ , рќђ»/рќ‘‘ ); compatibility of рќ‘‘2 | рќ‘› with рќ‘› в‰Ў рќ‘Џ (mod рќ‘ћ)
only refines the residue class by a polylogarithmic modulus. The tail рќ‘‘ > (log рќ‘‹)рќђ· is bounded on
average by рќ‘‘>рќђ· рќ‘‘в€’2 , hence is рќ‘‚(рќђ·в€’1 ) after the PACK normalization.
            в€‘пёЂ

    This is the precise route:


        TTH near-global length =в‡’ Davenport AP/congruence transfer =в‡’ X9L-GT-NG.

   вЂ”

X9L-GT.6. Scope Check: Unused Low-Theta Extension For a general range рќђ» в‰Ґ рќ‘‹ рќњѓ ,
0 < рќњѓ < 1/3, the elementary Davenport/AP proof above loses the factor

                                                  рќ‘‹
                                                  рќ‘”2 .
                                                  рќђ»
    This is no longer polylogarithmic. A qualitative рќ‘њ(1) short-interval Fourier theorem for ordinary
intervals also does not imply the normalized AP-fibre statement uniformly for рќ‘” в‰¤ (log рќ‘‹)рќђ¶ , because
the ordinary-to-AP reduction (AP-loss) costs рќ‘”.
    Therefore the full low-рќњѓ theorem

                                     X9L-POLYLOG-MOD<1/3
   is not asserted here. This is harmless for this proof, since TTH routes every surviving B1-origin
coarea test into the near-global range before X9L-GT is invoked.
   вЂ”

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

                                    X9L-GT/X9L-AVG-POLYLOG
   interface as the following sharper pair:

  1. Near-global part:


                                                       7
                             рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ      =в‡’    рќ‘‹9рќђї-рќђєрќ‘‡ -рќ‘Ѓ рќђє
   by Davenport/AP, the square-divisor transfer to рќњ†, and polylog tail summation.

  1. Unused general low-theta part:


                                    X9L-POLYLOG-MOD<1/3

   namely the same normalized AP-fibre averaged Fourier estimate for рќђ» в‰Ґ рќ‘‹ рќњѓ , every fixed
0 < рќњѓ < 1/3, and рќ‘” в‰¤ (log рќ‘‹)рќђ¶ .
   There are two clean ways one could strengthen the unused general theorem:

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

  2. prove that the regular TC1 branch has bounded modulus рќ‘” = рќ‘‚рќњ… (1), so ordinary qualitative
     short-interval input loses only a fixed factor.

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

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


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

   The proof tree invokes only the following narrower theorem:


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

   Thus the conclusion is:

                      X9L-GT is proved in the near-global form used here.
   What is proved for this route:

           unrouted B1-origin coarea tests satisfy the cited averaged AP-fibre input.
   No low-рќњѓ external input is required, because TTH proves the stronger near-global range-origin
lower bound for every unrouted coarea test. The low-рќњѓ theorem

                                    X9L-POLYLOG-MOD<1/3
   remains an unused general target only.
   вЂ”

X9L-GT.9. External Theorem and Proof




                                                8
External sources         The external theorem package is:

  1. Davenport. H. Davenport, *On some infinite series involving arithmetical functions (II)*,
     Quart. J. Math. Oxford 8 (1937), 313вЂ“320, DOI 10.1093/qmath/os-8.1.313.

   We use the standard Davenport consequence: for every рќђґ > 0,
                                       вѓ’              вѓ’
                                       вѓ’ в€‘пёЃ           вѓ’
                                            рќњ‡(рќ‘›)рќ‘’(рќ›јрќ‘›)вѓ’вѓ’ в‰Єрќђґ рќ‘Њ (log рќ‘Њ )в€’рќђґ .
                                       вѓ’              вѓ’
                                  sup вѓ’вѓ’                                                     (Dav)
                                 рќ›јв€€R/Z вѓ’рќ‘›в‰¤рќ‘Њ           вѓ’

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

Exact input        For every fixed рќђ¶, рќђµ, рќђґ > 0, let a TC1 testing family satisfy:

  1. рќ‘”рќ‘ќ в‰¤ (log рќ‘‹рќ‘ќ )рќђ¶ ;

  2. рќ‘€рќ‘ќ = рќђ»рќ‘ќ /рќ‘”рќ‘ќ ;

  3. рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµ ;

  4. the start pushforward obeys

                                         (start)# рќњ€ в‰Є (log рќ‘‹)рќђ¶ рќ‘‘рќ‘Ґ/рќ‘‹;                      (PACK)

  5. all smooth weights have polylogarithmic рќђ¶ рќђЅ -complexity.

   Then
                       вѓ’                                   вѓ’2
                       вѓ’                                   вѓ’
                       вѓ’ 1
              в€«пёЃ             в€‘пёЃ                            вѓ’
                   sup вѓ’
                       вѓ’          рќњ†(рќ‘”рќ‘ќ рќ‘ў + рќ‘Џрќ‘ќ )рќ‘’(рќ›јрќ‘ў)рќ‘¤рќ‘ќ (рќ‘ў)вѓ’вѓ’ рќ‘‘рќњ€(рќ‘ќ) = рќ‘њ(1).          (X9L-GT-NG)
                    рќ›ј вѓ’ рќ‘€рќ‘ќ
                           1в‰¤рќ‘ўв‰¤рќ‘€рќ‘ќ                          вѓ’

Proof of the input. First remove the smooth weight by a fixed finite smooth partition and summa-
tion by parts. This only changes the logarithmic loss.
    For a fixed fibre, expand the congruence рќ‘› в‰Ў рќ‘Џрќ‘ќ (mod рќ‘”рќ‘ќ ) by additive characters. This costs at
most рќ‘”рќ‘ќ . Apply DavenportвЂ™s bound (Dav) to global prefixes and take differences. The AP fibre is
normalized by рќ‘€рќ‘ќ = рќђ»рќ‘ќ /рќ‘”рќ‘ќ , so the total polylogarithmic loss is at most

                                               рќ‘‹рќ‘ќ
                                         рќ‘”рќ‘ќ2      в‰¤ (log рќ‘‹рќ‘ќ )2рќђ¶+рќђµ .
                                               рќђ»рќ‘ќ
    Choosing the Davenport logarithmic saving exponent larger than 2рќђ¶ +рќђµ +рќђґ gives рќ‘‚((log рќ‘‹)в€’рќђґ )
for every near-global fibre, hence рќ‘њ(1) after PACK averaging.




                                                       9
Match to the proof tree       TTH proves in fact

                                       рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ…
  for every unrouted B1-origin coarea test not already routed to C1, CKP, LocalDiag, Lon-
gAP/Local or Impossible. Therefore the Davenport near-global part alone suffices for this proof.
  Thus X9L-GT is a proved external input for the proof tree.
  вЂ”

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


3    Part 2. TGD: TC1 GoodAWACK dichotomy
Source file: Lemmas/tc1_goodawack_dichotomy_ltx.md.

3.0.1   TGD. Terminal GoodAWACK True-Complexity Split
TGD.0. Statement and Role          Lemma TGD records a non-recursive refinement of the terminal
GoodAWACK class:

                  GoodAWACK = TC1-GoodAWACK вЉ” HighTC-GoodAWACK.
    The purpose is not to prove the HighTC contribution is small. The purpose is to make the split
finite and structural, so that the remaining HighTC class is a certified algebraic obstruction rather
than an indefinitely recurring tail.
    The guiding principle is:

                     TC1 is decided by a quadratic tensor independence test.
   If the test fails, the failure itself is the HighTC certificate.
   Logical dependencies are the F3/F4 terminal GoodAWACK interface, E5 content stability, BGS
normal-form data, and bounded tensor-linear algebra. TGD is used by TGT, TTD, TNG, HGO2R,
E10M, E10K, and E10L.
   вЂ”

TGD.1. Setup: Terminal GoodAWACK Data                   Let (рќ’њ, рќњЏ ) be a tagged terminal GoodAWACK
atom produced by

                                       рќђµ1 в†’ рќђµ3 в†’ рќђ№ 3/рќђ№ 4.
  By the F3/F4 terminal GoodAWACK interface and E5 content stabilization, the atom has a
model form

                                     в€‘пёЃ                      рќ‘Ў
                                                            в€ЏпёЃ
                                рќ’њ=         рќ‘Љ (рќ‘§)рќњ†(рќђї0 (рќ‘§))         рќ‘“рќ‘– (рќђїрќ‘– (рќ‘§)),
                                     рќ‘§в€€О©                    рќ‘–=1

    where:



                                                   10
  1. О© is a smooth box-like domain in a fixed-rank parameter lattice;

  2. рќ‘Љ is a smooth tagged weight of polylogarithmic complexity;

  3. рќђї0 , рќђї1 , . . . , рќђїрќ‘Ў are affine forms of bounded affine and Cauchy-Schwarz complexity;

  4. at least one active affine form carries a Liouville-type oscillatory factor;

  5. all active forms have controlled content;

  6. no terminal Edge, CKP, LongAP/Local, or LocalDiag predicate applies.

   For the true-complexity test, write

                                                   рќђїЛ™ рќ‘–
   for the homogeneous linear part of рќђїрќ‘– . Constants are irrelevant for the tensor test.
   Let

                                     рќ‘„рќ‘– := рќђїЛ™ рќ‘– вЉ™ рќђїЛ™ рќ‘– в€€ Sym2 (рќ‘‰Q* )
   be the quadratic tensor attached to рќђїрќ‘– on the active parameter space рќ‘‰ .
   Let

                                                 в„і(рќ’њ)
   be the finite set of marked Liouville-type affine forms in the atom. In E10 notation this set
contains the chosen marked form рќђї0 , but the refined split allows one to choose any marked form
that passes the TC1 test.
   вЂ”

TGD.2. Statement: Definition of TC1-GoodAWACK                          A terminal GoodAWACK atom
(рќ’њ, рќњЏ ) is called

                                         TC1-GoodAWACK
   if there exists a marked form рќђїрќ‘љ , рќ‘љ в€€ в„і(рќ’њ), such that

                                / spanQ {рќ‘„рќ‘– : рќ‘– Мё= рќ‘љ, рќђїрќ‘– active in рќ’њ}.
                             рќ‘„рќ‘љ в€€                                                             (TC1)
    Equivalently, the active affine system is true-complexity one relative to at least one marked
Liouville form.
    This is a deliberately relative condition. It is stronger than merely saying the forms are not
equal or proportional, and weaker than requiring all tensors рќ‘„рќ‘– to be linearly independent.
    The intended analytic consequence is the following replacement for the high-order E10 general-
ized von Neumann step:


  TC1-GoodAWACK non-small =в‡’ вЂ–рќњ†(рќђїрќ‘љ )вЂ–рќ‘€ 2 в‰« the corresponding normalized lower bound.

  In this proof this analytic consequence is supplied by the global testing chain recorded in Lemma
TNG:


                                                   11
                     TGT + MRT + TTD + ROC + BRS + TTH + X9L-GT.
   The present document only proves the structural split.
   вЂ”

TGD.3. Statement: Definition of HighTC-GoodAWACK                       A terminal GoodAWACK atom
(рќ’њ, рќњЏ ) is called

                                       HighTC-GoodAWACK
   if it is terminal GoodAWACK and no marked Liouville form satisfies (TC1).
   Equivalently, for every marked рќ‘љ в€€ в„і(рќ’њ),

                            рќ‘„рќ‘љ в€€ spanQ {рќ‘„рќ‘– : рќ‘– Мё= рќ‘љ, рќђїрќ‘– active in рќ’њ}.                    (HighTC)
    Thus each marked form has a quadratic dependence certificate. After clearing denominators,
for every marked рќ‘љ there are integers рќ‘ђрќ‘– , not all zero, with рќ‘ђрќ‘љ Мё= 0, such that
                                           в€‘пёЃ
                                                рќ‘ђрќ‘– рќ‘„рќ‘– = 0.                          (HighTC-cert)
                                            рќ‘–

   The relation (HighTC-cert) is the terminal HighTC obstruction. It is not a new unresolved
routing instruction.
   Examples of this kind include the four-term progression pattern

                                  рќ‘Ґ,   рќ‘Ґ + рќ‘џ,    рќ‘Ґ + 2рќ‘џ,     рќ‘Ґ + 3рќ‘џ,
   for which

                                    рќђї20 в€’ 3рќђї21 + 3рќђї22 в€’ рќђї23 = 0.
    This pattern is not a mere equality/proportionality collision. It is a higher true-complexity
affine configuration.
    вЂ”

TGD.4. Proof: Finite TC1/HighTC Dichotomy

Lemma 3.1 (Lemma TGD.1). Every tagged terminal GoodAWACK atom belongs to exactly one
of

                        TC1-GoodAWACK,             HighTC-GoodAWACK.
    Moreover, if it belongs to HighTC-GoodAWACK, then it carries the explicit finite algebraic
certificate (HighTC-cert) for every marked Liouville-type form.

Proof. Fix a tagged terminal GoodAWACK atom (рќ’њ, рќњЏ ).
   By the GoodAWACK terminal predicate, the set of active forms is finite and has bounded car-
dinality depending only on рќђЅ0 . The set of marked Liouville-type forms is also finite and nonempty.
   For each marked рќ‘љ в€€ в„і(рќ’њ), form the quadratic tensor

                                         рќ‘„рќ‘љ = рќђїЛ™ рќ‘љ вЉ™ рќђїЛ™ рќ‘љ


                                                 12
   in the finite-dimensional rational vector space

                                                Sym2 (рќ‘‰Q* ).
   There are two possibilities.
   First, for at least one marked рќ‘љ,

                                        / spanQ {рќ‘„рќ‘– : рќ‘– Мё= рќ‘љ}.
                                     рќ‘„рќ‘љ в€€
   Then (рќ’њ, рќњЏ ) is TC1-GoodAWACK by definition.
   Second, for every marked рќ‘љ,

                                     рќ‘„рќ‘љ в€€ spanQ {рќ‘„рќ‘– : рќ‘– Мё= рќ‘љ}.
    Then (рќ’њ, рќњЏ ) is HighTC-GoodAWACK by definition. Since the vector space and the active set
are finite-dimensional and rational, each span membership gives a rational linear relation among
the рќ‘„рќ‘– . Clearing denominators gives an integer relation
                                       в€‘пёЃ
                                            рќ‘ђрќ‘– рќ‘„рќ‘– = 0,     рќ‘ђрќ‘љ Мё= 0,
                                       рќ‘–

     which is exactly (HighTC-cert).
     The two alternatives are mutually exclusive by the law of excluded middle applied to the finite
list of marked tensors. They are exhaustive because every marked tensor either is or is not in the
rational span of the remaining active tensors.
     Therefore the dichotomy is finite, disjoint and non-recursive. Lemma proved.
     вЂ”


Parameter check 3.2 (TGD.5. Parameter Check: No Infinite Tail). The class HighTC-GoodAWACK
is not defined by saying "whatever remains after another analytic decomposition." It is defined by
the explicit algebraic condition (HighTC).
    Thus a HighTC atom is terminal at the level of this split. Future work has only three legitimate
options:

  1. prove an analytic estimate for all atoms satisfying (HighTC-cert);

  2. prove that some certified HighTC patterns are actually CKP, Edge, or genuine LocalDiag
     under additional already-terminal criteria;

  3. refine the terminal predicate by a new finite invariant that strictly decreases.

   What is not allowed is an unmeasured iteration

                      HighTC в†’ smaller HighTC в†’ smaller HighTC в†’ В· В· В· .
   Such an iteration would need a separate well-founded complexity measure. The present split
avoids that problem by making HighTC a certified finite obstruction class.
   вЂ”




                                                    13
TGD.6. Compatibility with LocalDiag The HighTC certificate must not automatically be
routed to LocalDiag.
   Lemma F3 defines LocalDiag as forced equality, proportionality, gcd-local dependence, or un-
avoidable collision that makes the contribution a canonical local term. A quadratic tensor relation
such as

                                    рќђї20 в€’ 3рќђї21 + 3рќђї22 в€’ рќђї23 = 0
   does not by itself produce a canonical local main term.
   Therefore:

                              HighTC-GoodAWACK Мёв‡’ LocalDiag.
   Only those HighTC atoms whose certificate also forces a genuine local/main degeneracy may
be passed to H4. Otherwise they remain in the HighTC-GoodAWACK branch.
   This resolves the ambiguity between the broad B3 phrase "affine dependence among active
forms" and the narrower F3/H4 terminal meaning of LocalDiag.
   вЂ”

TGD.7. Output for E10         After this split, E10 should be treated as two sub-branches:

               рќ‘…GoodAWACK (рќ‘Ѓ ) = рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) + рќ‘…HighTC-GoodAWACK (рќ‘Ѓ ).

TC1 branch      The TC1 branch is handled by the global-testing route:

                   TC1 =в‡’ рќ‘€ 2 -generalized von Neumann =в‡’ TNG =в‡’ рќ‘њ(рќ‘Ѓ ).
   This replaces X8 on the TC1 sub-branch. The orthogonality input is X9L-GT.

HighTC branch       The HighTC branch is the explicit algebraic obstruction:

                                     HighTC-GoodAWACK
    It is closed structurally: origin-degenerate HighTC is rerouted by HGO2R, and the free-affine
residual is excluded by E10M plus E10K.
    The important gain is conceptual: HighTC is an explicit finite algebraic obstruction, not an
open-ended residual tail, and it is discharged by HGO2R/E10M/E10K/E10L.
    вЂ”
Remark 3.3 (TGD.8. Output).

                   TC1/HighTC dichotomy proved as a finite structural split.

   TGD does not by itself close E10. It supplies the stable interface:

                 GoodAWACK = TC1-GoodAWACK вЉ” HighTC-GoodAWACK.
   The TC1 branch is handled by TNG. The HighTC branch is handled by HGO2R/E10Y/E10X/E10K
and then by E10L.




                                                14
TGD.9. Logical Dependencies Internal dependencies: the F3/F4 terminal GoodAWACK
interface, E5, BGS, and bounded tensor-linear algebra.
    Children served: TGT, TTD, TNG, HGO2R, E10M, E10K, and E10L.


4     Part 3. TGT: TC1 global testing
Source file: Lemmas/tc1_global_testing_ltx.md.

4.0.1    TGT. Aggregated Testing Route for TC1-GoodAWACK
TGT.0. Statement and Role Lemma TGT records the global-testing replacement for the
pointwise short-interval TC1 route.
   The statement is:


after aggregation over a fixed TC1 macro-template, every regular testing family is closed by the averaged near-gl

   Equivalently, one first aggregates all TC1 atoms with the same structural macro-template and
only then tests Liouville against the induced measured family of intervals or arithmetic progressions.
This avoids selecting a single bad fibre before the averaging structure has been exposed.


TC1-TESTING-DICHOTOMY : every B1-origin TC1 testing family is either MRT-admissible, or its singular pa

    Logical dependencies are TGD, TGT-MF, MRT, TTD, TTH-SC, ROC, BRS, TTH, E5, X9L-
GT, and the parameter register. TGT is used by TNG and E10L; E10L is a downstream consumer
of the TC1 testing route, not an input to it.
    вЂ”

TGT.1. Setup: Macro-Template Aggregation                Fix a structural TC1 macro-template рќњ…. The
template fixes:

    1. the B1 typed parent pattern;

    2. the B3 grouping skeleton;

    3. the F3/F4 routing grammar;

    4. the marked Liouville origin;

    5. the affine coefficient transport type;

    6. the TC1 tensor certificate.

   It does not select a single dyadic atom. Instead, it contains all dyadic, CRT, divisor, and
smoothing cells compatible with the same structural template.
   Write the corresponding terminal TC1 atoms as

                                          рќ’њрќ‘— ,   рќ‘— в€€ рќђЅрќњ… (рќ‘Ѓ ),
    with effective volumes рќ‘‰рќ‘— , domains О©рќ‘— , marked forms рќђїрќ‘љ,рќ‘— , and normalized contributions

                                                 15
                                  1 в€‘пёЃ                      в€ЏпёЃ
                          рќ‘Ћрќ‘— :=          рќ‘Љрќ‘— (рќ‘§)рќњ†(рќђїрќ‘љ,рќ‘— (рќ‘§))      рќ‘“рќ‘–,рќ‘— (рќђїрќ‘–,рќ‘— (рќ‘§)).
                                  рќ‘‰рќ‘— рќ‘§в€€О©                   рќ‘–Мё=рќ‘љ
                                            рќ‘—

   The aggregated contribution is
                                                          в€‘пёЃ
                                            рќ‘…рќњ… (рќ‘Ѓ ) =                 рќ‘‰рќ‘— рќ‘Ћрќ‘— .
                                                        рќ‘—в€€рќђЅрќњ… (рќ‘Ѓ )

   Since the number of structural macro-templates is bounded in terms of рќђЅ0 , if the total TC1
contribution is not рќ‘њ(рќ‘Ѓ ), then along an infinite sequence there is a fixed рќњ… and рќњЂ > 0 such that

                                                |рќ‘…рќњ… (рќ‘Ѓ )| в‰Ґ рќњЂрќ‘Ѓ.                                           (1)
   No dyadic polylogarithmic pigeonhole is used at this stage.
   вЂ”
Remark 4.1 (TGT.2. Proof: Global TC1 Generalized von Neumann Output). For each atom рќ‘—, the
TC1 weighted generalized von Neumann step gives

                                   |рќ‘Ћрќ‘— | в‰¤ рќђ¶рќњ… вЂ–рќњ†(рќђїрќ‘љ,рќ‘— )вЂ–рќ‘ђрќ‘€рќњ…2 (О©вЂІ ) + рќ‘њрќњ… (1),                              (2)
                                                                      рќ‘—

   after the usual C1 boundary removals and content normalizations.
   Multiply by рќ‘‰рќ‘— , sum over рќ‘—, and use (1). Since
                                                 в€‘пёЃ
                                                      рќ‘‰рќ‘— в‰Єрќњ… рќ‘Ѓ
                                                  рќ‘—

   for a fixed macro-template, (1)вЂ“(2) imply
                                  1 в€‘пёЃ
                                             рќ‘‰рќ‘— вЂ–рќњ†(рќђїрќ‘љ,рќ‘— )вЂ–рќ‘ђрќ‘€рќњ…2 (О©вЂІ ) в‰«рќњ…,рќњЂ 1.                              (3)
                                  рќ‘Ѓ рќ‘—в€€рќђЅ (рќ‘Ѓ )                     рќ‘—
                                        рќњ…

    After replacing рќ‘ђрќњ… by a harmless bounded power and using 0 в‰¤ вЂ– В· вЂ–рќ‘€ 2 в‰¤ 1, this gives the
fixed-threshold global obstruction

                                     Eрќ‘—в€јрќ‘‰рќ‘— вЂ–рќњ†(рќђїрќ‘љ,рќ‘— )вЂ–4рќ‘€ 2 (О©вЂІ ) в‰«рќњ…,рќњЂ 1.                               (GT-U2)
                                                                  рќ‘—


   This proves the internal aggregation step.
   вЂ”

TGT.3. Proof: Measured Fourier Transfer Apply Lemma TGT-MF to the normalized
box/coset models and the obstruction (GT-U2). The lemma uses the Fourier normalization
                                                          в€‘пёЃ
                                            вЂ–рќђ№рќ‘— вЂ–4рќ‘€ 2 =        |рќђ№М‚пёЂрќ‘— (рќњ‰)|4
                                                          рќњ‰

    and the finite coarea normal form of the marked affine form рќђїрќ‘љ,рќ‘— . It constructs a finite proba-
bility measure рќњ€рќњ… on tests
                             1 в€‘пёЃ
                  в„’рќ‘ќ (рќњ†) =          рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),                    рќ‘ќ = (рќ‘—, рќњ‰, coarea piece),       (4)
                             рќђ»рќ‘ќ рќ‘›в€€рќђј
                                    рќ‘ќ




                                                        16
   where рќђјрќ‘ќ is a shifted interval or AP image, рќђ»рќ‘ќ = |рќђјрќ‘ќ |, the AP modulus/content and the weight
complexity are polylogarithmically controlled, and C1 boundary pieces have already been discarded.
TGT-MF gives the fixed testing lower bound
                                  в€«пёЃ
                                                 |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰«рќњ…,рќњЂ 1,              (GT-Test)
                                       рќ’«рќњ… (рќ‘Ѓ )

   for the induced probability measure рќњ€рќњ… .
   This is the global replacement for a pointwise shifted short-interval input:

              not one bad interval, but a whole measured family of Liouville tests.
   вЂ”
Parameter check 4.2 (TGT.4. Parameter Check: MRT-Admissible Testing Families). The av-
eraged Liouville input can only apply if the testing measure genuinely averages over starts/scales.
Define a testing family (рќ’«рќњ… , рќњ€рќњ… ) to be MRT-admissible if, after partitioning into рќ‘‚рќњ… ((log рќ‘Ѓ )рќђ¶ )
scale/modulus/weight-complexity classes, the pushforward of рќњ€рќњ… to interval starts is dominated by
a polylogarithmic multiple of normalized counting/Lebesgue measure:

                                                                      рќ‘‘рќ‘Ґ
                                       (start)# рќњ€рќњ… в‰Єрќњ… (log рќ‘Ѓ )рќђ¶                           (PACK)
                                                                      рќ‘‹
   on each dyadic рќ‘Ґ в‰Ќ рќ‘‹, with

                                                    рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќрќњѓрќњ…
   outside C1-negligible boundary pieces.
   This condition is the common form of:
  1. E7 pushforward regularity;
  2. coarea image regularity when the marked image sweeps many starts;
  3. absence of rank-one/point-mass short-image concentration.
    PACK is not supplied by TTH alone. The verification of PACK and the routing of PACK failures
are recorded in MRT. TGT invokes X9L-GT only on the branch selected there as MRT-admissible.
    Assume the external averaged Liouville theorem in the qualitative form:
                                  в€«пёЃ
                                                 |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) = рќ‘њрќњ… (1)             (X9L-GT)
                                   рќ’«рќњ… (рќ‘Ѓ )
   for every MRT-admissible TC1 testing family.
   Then (GT-Test) contradicts (X9L-GT).
Lemma 4.3 (Lemma TGT.1. Admissible global testing closure). For a fixed structural macro-
template рќњ…, if the induced TC1 testing family is MRT-admissible in the sense of MRT and X9L-GT
holds, then

                                                 рќ‘…рќњ… (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).
Proof. Assume not. Then (1) holds for some рќњЂ > 0. TGT.2 and TGT-MF give the fixed lower
bound (GT-Test). MRT-admissibility allows the averaged Liouville input X9L-GT, giving рќ‘њ(1) for
the same left side. This is a contradiction. Lemma proved.
   вЂ”


                                                        17
TGT.5. Singular testing measures The route does not close arbitrary TC1 testing families.
If the test measure is concentrated on one or very few interval starts, then averaged Liouville
theorems say nothing.
    The model obstruction is exactly the earlier SAI/rank-one model:

                    О© = [рќ‘‹, рќ‘‹ + рќ‘Њ ] Г— [1, рќ‘Ђ ],    рќђїрќ‘љ (рќ‘ў, рќ‘Ј) = рќ‘ў,       рќ‘Њ рќ‘Ђ в‰Ќ рќ‘Ѓ.                   (5)
   The coarea image is the single interval [рќ‘‹, рќ‘‹ + рќ‘Њ ]. Averaging over рќ‘Ј increases the weight of the
same interval; it does not create an average over starts. The pushforward in (PACK) is a point
mass, so the family is not MRT-admissible.
   Thus global aggregation does not require pointwise X9L-SI. Instead, it isolates the exact struc-
tural branch:

          singular testing measure   в‡ђв‡’      rank-one / short affine-image concentration.
   This is the structural obstruction handled by the singular branch of TTD.
   вЂ”

TGT.6. Output Form: Structural Closure                The structural replacement for individual local
tails is:

                                 TC1-TESTING-DICHOTOMY.
   For every actual B1/B3/F3/F4 terminal TC1 macro-template рќњ…, after C1 boundary removal,
exactly one of the following holds:

  1. the induced global testing family is MRT-admissible, so Lemma TGT.1 closes it using aver-
     aged Liouville cancellation;

  2. the non-admissible/singular part has an origin tag forcing strict C1 Edge;

  3. it is a genuine H4-admissible LongAP/Local main term;

  4. it exposes a CKP grouping handled by G8a;

  5. it exposes LocalDiag;

  6. it is empty/impossible by parent B1 scale or congruence constraints.

   This theorem is supplied in the consolidated form TNG-A. Internally TNG-A uses TTD, TTH-
SC, ROC, BRS, X16BRS/X16C, and TTH. It replaces:

  1. pointwise X9L-SI;

  2. atomwise E7-REG-CARRIER;

  3. TC1-SAI-ROUTE;

  4. ad hoc coarea short-image routing.

    It asks for regularity or origin-routing of the global testing measure, not of each presentation
of the same local tail.
    вЂ”

                                                 18
TGT.7. Compatibility with Auxiliary Reductions The E7 averaged-fibre argument proves
the averaged slicing part for one coordinate presentation. In the present language, it constructs
part of рќ’«рќњ… .
    The E7 regular-pushforward check concerns condition (PACK) for E7 fibres and finds that
rank-one carriers are exactly the non-admissible case.
    The TC1 coarea Fourier step constructs the coarea tests (4). Theorem TNG-A says that
near-global images are closed by X9L-GT, while genuinely short or singular images are routed by
TTD/ROC/BRS using X16BRS/X16C before X9L-GT is invoked.
    The TC1-SAI route shows that short image alone is not enough to route an atom by the terminal
predicates. In the present language, it says that non-admissible testing measure is not automatically
C1/D1/G8a/LocalDiag.
    X9L-GT is the external averaged input. The global testing formulation explains why a qualita-
tive рќ‘њ(1) theorem may suffice: after macro-template aggregation, the lower bound in (GT-Test) is
a fixed в‰«рќњЂ 1, not a polylogarithmic threshold.
    вЂ”
Remark 4.4 (TGT.8. Output).

                  The global testing route is a genuine conceptual improvement.

   Together with Theorem TNG-A and X9L-GT, it gives the TC1 closure:

                              TC1 macro-templates contribute рќ‘њ(рќ‘Ѓ ).
   After TNG-A, the singular structural branch is not a residual. Lemma TTH supplies the near-
global length information in the near-global alternative,

                                         рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ

for B1-origin coarea tests. Therefore the only analytic рќ‘‹9рќђї input required by the TC1 branch is
the near-global Davenport/AP form X9L-GT.
    The single-source statement of this chain is Lemma TNG. TGT supplies the aggregation and
testing lower bound; Lemma TNG verifies that the unrouted tests seen by X9L-GT are exactly the
MRT-admissible, near-global B1-origin coarea tests.
    вЂ”

TGT.9. External Input Check            X9L-GT records the external input: Davenport closes the
near-global AP-fibre range

                                         рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ ,
   and this is the only X9L input used by this proof.
   The proof does not invoke a normalized AP-fibre estimate for arbitrary shifted intervals through-
out the range рќђ» в‰Ґ рќ‘‹ рќњѓ , 0 < рќњѓ < 1/3. The only AP-fibre estimate required after the TNG reduction
to B1-origin coarea tests is the near-global Davenport/AP estimate stated above.

TGT.10. Logical Dependencies External dependency: X9L-GT in the near-global Daven-
port/AP range.
   Internal dependencies: TGD, TGT-MF, MRT, TTD, TTH-SC, ROC, BRS, TTH, E5, and the
parameter register.

                                                 19
    Children served: TNG, E10L, and the TC1-GoodAWACK closure.
    Direction note: TGT.2 and TGT-MF construct the measured testing family, while TGT.4 closes
only the MRT-admissible regular branch. The full TC1 closure uses the later TNG-A interface to
dispose of singular or short-image tests. Thus references from TTD/TTH/TNG back to the TGT
construction do not mean that those lemmas assume the full TGT closure theorem.


5    Part 4. TGT-MF: Measured Fourier transfer
Source file: Lemmas/tc1_measured_fourier_transfer_ltx.md.

5.0.1   TGT-MF. Measured Fourier Transfer for TC1 Global Testing
TGT-MF.0. Statement and Role Lemma TGT-MF is the measure-theoretic and Fourier
normalization step used inside TGT. It turns the global рќ‘€ 2 -obstruction produced by TC1 aggre-
gation into a finite measured family of Liouville tests.
    The statement is:

                                       Eрќ‘—в€јрќ‘‰рќ‘— вЂ–рќђ№рќ‘— вЂ–4рќ‘€ 2 (О©вЂІ ) в‰Ґ рќ‘ђ
                                                                рќ‘—

                                                     =в‡’                                                      (TGT-MF)
                                        в€«пёЃ
                                                                2
         в€ѓ (рќ’«рќњ… (рќ‘Ѓ ), рќњ€рќњ… ) such that                 |в„’рќ‘ќ (рќњ†)| рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰Ґ рќ‘ђ/рќђ¶MF (рќњ…) в€’ рќ‘њрќњ… (1).
                                          рќ’«рќњ… (рќ‘Ѓ )

   Here рќђ№рќ‘— (рќ‘§) = рќњ†(рќђїрќ‘љ,рќ‘— (рќ‘§)) on the normalized box/coset model О©вЂІрќ‘— , after the C1 boundary and
content-normalization removals used in TGT.2. The constant рќђ¶MF (рќњ…) depends only on the fixed
TC1 macro-template рќњ…, the bounded dimension of its boxes, and the fixed coarea complexity of the
template. It is independent of рќ‘Ѓ .
   Logical dependencies are the TGT.1вЂ“TGT.2 setup, the C1 boundary removal interface, E5
content/affine transport control, and the finite F3/F4 coarea normal form. The lemma is used by
TGT, TTD, TNG, and TTH.
   вЂ”

TGT-MF.1. Setup: Normalized Fourier Models For every atom рќ‘— в€€ рќђЅрќњ… (рќ‘Ѓ ), let О©вЂІрќ‘— be the
finite box/coset model remaining after C1-negligible boundary pieces and controlled content factors
have been removed. It is endowed with normalized counting measure
                                                                1 в€‘пёЃ
                                               EО©вЂІрќ‘— рќ‘“ :=                рќ‘“ (рќ‘§).
                                                               |О©вЂІрќ‘— | вЂІ
                                                                     рќ‘§в€€О©рќ‘—

   Let рќђєрќ‘— be the finite abelian group obtained by completing the box/coset model with the same
periods, and let рќђє
                 М‚пёЂ рќ‘— be its character group. Fourier coefficients are normalized by


                                             рќђ№М‚пёЂрќ‘— (рќњ‰) := Eрќ‘§в€€рќђєрќ‘— рќђ№рќ‘— (рќ‘§)рќњ‰(рќ‘§).                                        (1)
    With this normalization,
                                       в€‘пёЃ                            в€‘пёЃ
                   вЂ–рќђ№рќ‘— вЂ–4рќ‘€ 2 (рќђєрќ‘— ) =           |рќђ№М‚пёЂрќ‘— (рќњ‰)|4 ,                 |рќђ№М‚пёЂрќ‘— (рќњ‰)|2 = Eрќђєрќ‘— |рќђ№рќ‘— |2 в‰¤ 1.        (2)
                                       рќњ‰в€€рќђє
                                         М‚пёЂрќ‘—                         рќњ‰в€€рќђє
                                                                       М‚пёЂрќ‘—



                                                                20
    Replacing the box by its completed coset model changes the рќ‘€ 2 -quantity only by the рќ‘њрќњ… (1)
boundary term already assigned to C1. Thus the TGT lower bound may be read with рќђєрќ‘— in place
of О©вЂІрќ‘— .
    Set

                                                                  рќ‘‰рќ‘—
                                                 рќ‘¤рќ‘— = в€‘пёЂ                    .                       (3)
                                                             рќ‘–в€€рќђЅрќњ… (рќ‘Ѓ ) рќ‘‰рќ‘–

   The global obstruction entering this lemma is
                                              в€‘пёЃ           в€‘пёЃ
                                                     рќ‘¤рќ‘—           |рќђ№М‚пёЂрќ‘— (рќњ‰)|4 в‰Ґ рќ‘ђ.                  (4)
                                         рќ‘—в€€рќђЅрќњ… (рќ‘Ѓ )        рќњ‰в€€рќђє
                                                            М‚пёЂрќ‘—
   вЂ”

TGT-MF.2. Setup: Coarea Normal Form for One Fourier Coefficient For each pair
(рќ‘—, рќњ‰), the marked form рќђїрќ‘љ,рќ‘— and the finite F3/F4 coarea normal form decompose the Fourier
coefficient as

                                                         рќ›Ѕрќ‘—,рќњ‰,рќ‘џ в„’рќ‘—,рќњ‰,рќ‘џ (рќњ†)рќ‘‚рќњ… (рќ‘Ѓ в€’100 ),
                                                в€‘пёЃ
                                 рќђ№М‚пёЂрќ‘— (рќњ‰) =                                                         (5)
                                              рќ‘џв€€в„›(рќ‘—,рќњ‰)

   where:

  1. #в„›(рќ‘—, рќњ‰) в‰¤ рќђµco (рќњ…);

        рќ‘џ |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ | в‰¤ рќђµco (рќњ…);
       в€‘пёЂ
  2.

  3. every в„’рќ‘—,рќњ‰,рќ‘џ has the normalized form

                                                     1       в€‘пёЃ
                                 в„’рќ‘—,рќњ‰,рќ‘џ (рќњ†) =                          рќњ†(рќ‘›)рќњЊрќ‘—,рќњ‰,рќ‘џ (рќ‘›)рќ‘’(рќ›јрќ‘—,рќњ‰,рќ‘џ рќ‘›);   (6)
                                                 рќђ»рќ‘—,рќњ‰,рќ‘џ рќ‘›в€€рќђј
                                                              рќ‘—,рќњ‰,рќ‘џ



  4. рќђјрќ‘—,рќњ‰,рќ‘џ is a shifted interval or arithmetic-progression image of the marked form;

  5. the AP modulus/content and the derivative complexity of рќњЊрќ‘—,рќњ‰,рќ‘џ are bounded by fixed powers
     of log рќ‘Ѓ determined by рќњ…;

  6. the discarded coarea boundary pieces have total contribution рќ‘њрќњ… (1) after the рќ‘—-average and
     are already C1-admitted.

   Equation (5) is a finite identity on the normalized box/coset model. It is not a pigeonhole over
dyadic atoms. The constants in (5) depend on the fixed dimension and routing grammar of рќњ…, not
on the number of dyadic cells inside the macro-template.
   вЂ”




                                                             21
TGT-MF.3. Construction of the Testing Measure                                          Let
                                               в€‘пёЃ           в€‘пёЃ                     в€‘пёЃ
                                     рќ‘†рќњ… :=            рќ‘¤рќ‘—           |рќђ№М‚пёЂрќ‘— (рќњ‰)|2              |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ |.                            (7)
                                                 рќ‘—         рќњ‰в€€рќђє
                                                             М‚пёЂрќ‘—                 рќ‘џв€€в„›(рќ‘—,рќњ‰)

   By (2) and the coarea bound in TGT-MF.2,

                                                          0 < рќ‘†рќњ… в‰¤ рќђµco (рќњ…).                                                       (8)
   The strict positivity follows from (4). Define the finite parameter set

                                                             М‚пёЂ рќ‘— , рќ‘џ в€€ в„›(рќ‘—, рќњ‰), |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ | > 0}.
                    рќ’«рќњ… (рќ‘Ѓ ) := {(рќ‘—, рќњ‰, рќ‘џ) : рќ‘— в€€ рќђЅрќњ… (рќ‘Ѓ ), рќњ‰ в€€ рќђє                                                                    (9)
   Define the probability measure рќњ€рќњ… by

                                                                   рќ‘¤рќ‘— |рќђ№М‚пёЂрќ‘— (рќњ‰)|2 |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ |
                                              рќњ€рќњ… (рќ‘—, рќњ‰, рќ‘џ) =                                .                                    (10)
                                                                             рќ‘†рќњ…
   This is an ordinary finite probability measure. Hence all measurability assertions are literal:
every subset of рќ’«рќњ… (рќ‘Ѓ ) is measurable.
   For рќ‘ќ = (рќ‘—, рќњ‰, рќ‘џ), set

                              в„’рќ‘ќ (рќњ†) := в„’рќ‘—,рќњ‰,рќ‘џ (рќњ†),                рќђјрќ‘ќ = рќђјрќ‘—,рќњ‰,рќ‘џ ,            рќђ»рќ‘ќ = рќђ»рќ‘—,рќњ‰,рќ‘џ .                        (11)
   вЂ”

TGT-MF.4. Proof of the Lower Bound                                 From (5) and CauchyвЂ™s inequality,

                                                                   |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ | |в„’рќ‘—,рќњ‰,рќ‘џ (рќњ†)|2 + рќ‘‚рќњ… (рќ‘Ѓ в€’100 ).
                                                          в€‘пёЃ
                          |рќђ№М‚пёЂрќ‘— (рќњ‰)|2 в‰¤ 2рќђµco (рќњ…)                                                                                 (12)
                                                      рќ‘џв€€в„›(рќ‘—,рќњ‰)

   Multiplying (12) by рќ‘¤рќ‘— |рќђ№М‚пёЂрќ‘— (рќњ‰)|2 and summing over рќ‘—, рќњ‰ gives
          в€‘пёЃ        в€‘пёЃ                               в€‘пёЃ        в€‘пёЃ                   в€‘пёЃ
               рќ‘¤рќ‘—        |рќђ№М‚пёЂрќ‘— (рќњ‰)|4 в‰¤ 2рќђµco (рќњ…)           рќ‘¤рќ‘—        |рќђ№М‚пёЂрќ‘— (рќњ‰)|2              |рќ›Ѕрќ‘—,рќњ‰,рќ‘џ | |в„’рќ‘—,рќњ‰,рќ‘џ (рќњ†)|2 + рќ‘њрќњ… (1).   (13)
          рќ‘—         рќњ‰                                 рќ‘—        рќњ‰                  рќ‘џв€€в„›(рќ‘—,рќњ‰)

   Using (4), (7), and (10), (13) implies

                                                                                   рќ‘ђ в€’ рќ‘њрќњ… (1)
                                         в€«пёЃ
                                                     |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰Ґ                      .                                  (14)
                                          рќ’«рќњ… (рќ‘Ѓ )                                 2рќђµco (рќњ…)рќ‘†рќњ…
   By (8),

                                                                                   рќ‘ђ
                                    в€«пёЃ
                                               |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰Ґ                          в€’ рќ‘њрќњ… (1)                             (15)
                                     рќ’«рќњ… (рќ‘Ѓ )                                 рќђ¶MF (рќњ…)
   with

                                                      рќђ¶MF (рќњ…) = 2рќђµco (рќњ…)2 .                                                      (16)
   This proves the measured Fourier transfer.
   вЂ”



                                                                    22
Parameter check 5.1 (TGT-MF.5. Parameter Check: Complexity and Normalizations). The
construction preserves the exact normalization needed later by MRT and X9L-GT:

    1. рќњ€рќњ… is a probability measure by (10).

    2. Each test is normalized by рќђ»рќ‘ќв€’1 .

    3. The AP modulus/content is polylogarithmic because the F3/F4/E5 transport operations have
       controlled content and the macro-template рќњ… is fixed.

    4. The weight рќњЊрќ‘ќ has polylogarithmic derivative complexity inherited from the original smooth
       dyadic and CRT cutoffs.

    5. Boundary components are not part of рќ’«рќњ… ; they are routed to C1 before this lemma is invoked.

    6. No single dyadic fibre, interval start, or Fourier frequency is selected as the obstruction. The
       obstruction is carried by the finite probability measure рќњ€рќњ… .

   The start-pushforward regularity of рќњ€рќњ… is not asserted here. It is the separate PACK/MRT
question handled by MRT, TTD, ROC, BRS, and TTH.
   вЂ”

TGT-MF.6. Output Form             For use in TGT, TTD, TNG, and TTH, the output is:

                                           GT-U2 =в‡’ GT-Test
    where
                                              в€«пёЃ
                             GT-Test :                   |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰«рќњ…,рќ‘ђ 1.
                                               рќ’«рќњ… (рќ‘Ѓ )

    The implicit constant is рќ‘ђ/рќђ¶MF (рќњ…) up to the C1-negligible рќ‘њрќњ… (1) boundary term. This is the
closed measure-theoretic/Fourier bridge required by the TC1 global testing route.


6     Part 5. TTH-SC: Structural coarea closure
Source file: Lemmas/tc1_structural_coarea_closure_ltx.md.

6.0.1    TTH-SC. Structural Coarea Closure and No Artificial Short-Interval Refine-
         ment
TTH-SC.0. Statement and Role Lemma TTH-SC is the formal closure principle used in
the TC1 near-global route. It proves that a released near-global structural coarea image cannot be
replaced by arbitrary short shifted intervals inside the active TC1 testing family.
    Fix a TC1 macro-template рќњ… after the B1/B3/F3/F4 routing interface, C1 boundary removal,
and the TGT.2/TGT-MF coarea construction. Let рќ’«рќњ… (рќ‘Ѓ ) be the finite family of structural coarea
tests released by TGT-MF, with probability measure рќњ€рќњ… . For рќ‘ќ в€€ рќ’«рќњ… (рќ‘Ѓ ), write
                                             1 в€‘пёЃ
                                  в„’рќ‘ќ (рќњ†) =          рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),
                                             рќђ»рќ‘ќ рќ‘›в€€рќђј
                                                     рќ‘ќ

    where рќђјрќ‘ќ is the marked B1-origin coarea image piece and рќђ»рќ‘ќ = |рќђјрќ‘ќ |.

                                                     23
   Then every refinement of a released test that can occur inside the proof is classified by exactly
one of the following alternatives.

  1. Controlled structural refinement. The refinement is generated by the finite TGT-MF
     coarea algebra: dyadic scale subdivision, AP/modulus normalization, smooth-weight par-
     tition, controlled CRT restriction, fixed divisor quotienting, full-rank affine transport, or
     primitive slicing. It produces at most (log рќ‘Ѓ )рќђ¶рќњ… child tests, and each child has length

                                          рќђ»рќ‘ќвЂІ в‰Ґ рќђ»рќ‘ќ (log рќ‘Ѓ )в€’рќђ¶рќњ… .                              (SC1)

     Therefore a near-global parent remains near-global after enlarging the logarithmic exponent.

  1. Non-structural analytic subdivision. The subdivision is a partition of рќђјрќ‘ќ chosen after the
     structural coarea test has already been released and is not one of the generators in the TGT-
     MF coarea algebra. Such pieces are not elements of рќ’«рќњ… (рќ‘Ѓ ), carry no independent testing
     mass, and are reassembled into the parent functional before the X9L-GT input is invoked.

  1. Genuine structural short-image alternative. The refinement is structural and produces
     a child image shorter than the controlled lower bound (SC1). Then this is not an artificial
     subdivision of an already released near-global test. It is a genuine short-image B1-origin cell,
     and TTD/ROC/BRS, with X16BRS/X16C and C1P/C1A/C1, routes the cell to one of

                           рќђ¶1рќђґ/рќђ¶1,        рќђ·1/рќђ»4,         рќђє8рќ‘Ћ,      рќђ»4,    or 0

     before X9L-GT is applied.

    Consequently no arbitrary shifted short interval can survive as an active unrouted TC1 input
to X9L-GT.
    Logical dependencies are TGT-MF, TGD, F3/F4, C1P/C1A/C1, TTD, ROC, BRS, X16BRS, X16C,
E5, and the parameter register. The lemma does not use TTH and does not use the full TNG
closure theorem.
    вЂ”

TTH-SC.1. Setup: The Structural Coarea Algebra For a fixed macro-template рќњ…, let
рќ‘љрќ‘Ћрќ‘Ўв„Ћрќ‘ђрќ‘Ћрќ‘™рќђґрќњ… be the finite coarea algebra generated by the operations that are already present in the
TGT-MF construction and the preceding F3/F4 routing interface:

  1. fixing/projection of bounded coordinates;

  2. CRT restriction by polylogarithmic moduli;

  3. fixed divisor quotienting by controlled divisors;

  4. full-rank affine transport with controlled content;

  5. dyadic scale and AP/modulus normalization;

  6. smooth-weight partition of bounded differentiability complexity;

  7. primitive slicing;

  8. post-terminal Fourier/cube subdivisions that preserve the marked Liouville origin.

                                                 24
    The number of atoms produced by this algebra inside a fixed рќњ…-cell is bounded by (log рќ‘Ѓ )рќђ¶рќњ… .
This follows from the fixed macro-template complexity, the polylogarithmic modulus bounds, and
the parameter register.
    A released TC1 coarea test is an atom of
рќ‘љрќ‘Ћрќ‘Ўв„Ћрќ‘ђрќ‘Ћрќ‘™рќђґрќњ… which has not been routed to Edge, LongAP/Local, CKP, LocalDiag, empty support,
or an impossible support class before the TGT-MF testing measure is formed.
    Thus рќ’«рќњ… (рќ‘Ѓ ) is supported only on released atoms of
рќ‘љрќ‘Ћрќ‘Ўв„Ћрќ‘ђрќ‘Ћрќ‘™рќђґрќњ… .
    вЂ”

TTH-SC.2. Proof: Controlled Structural Refinements Let рќ‘ќ в€€ рќ’«рќњ… (рќ‘Ѓ ) be released and
suppose that a child рќ‘ќвЂІ is obtained by applying further generators of
рќ‘љрќ‘Ћрќ‘Ўв„Ћрќ‘ђрќ‘Ћрќ‘™рќђґрќњ… which are allowed after release only for scale, modulus, smoothness, or bounded prim-
itive normalization.
    Each such generator has one of two effects.
    First, it may restrict to one of finitely many residue or smooth-weight classes. The number of
classes is at most (log рќ‘Ѓ )рќђ¶рќњ… , and empty or boundary classes are routed to C1P/C1A/C1.
    Second, it may change the AP modulus or the smooth weight while preserving the marked
image рќђїрќ‘љ (О©* ) up to a polylogarithmic partition. Again the number of nonempty pieces is at most
(log рќ‘Ѓ )рќђ¶рќњ… .
    Therefore every non-routed child satisfies

                                         рќђ»рќ‘ќвЂІ в‰Ґ рќђ»рќ‘ќ (log рќ‘Ѓ )в€’рќђ¶рќњ… .
   If the parent satisfies

                                        рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… ,
   then, after absorbing the fixed polylogarithmic losses and the height distortion рќ‘‹рќ‘ќвЂІ = рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )рќ‘‚рќњ… (1) ,
the child satisfies
                                                                  вЂІ
                                        рќђ»рќ‘ќвЂІ в‰Ґ рќ‘‹рќ‘ќвЂІ (log рќ‘‹рќ‘ќвЂІ )в€’рќђµрќњ…
   for a larger exponent рќђµрќњ…вЂІ . Thus controlled structural refinement does not create a low-theta
short-interval input.
   вЂ”

TTH-SC.3. Proof: Non-Structural Analytic Subdivisions                   Suppose that рќђјрќ‘ќ is partitioned
into subintervals or AP subpieces
                                                     вЁ†пёЃ
                                             рќђјрќ‘ќ =          рќђјрќ‘ќ,рќњ”
                                                    рќњ”в€€О©рќ‘ќ

   after рќ‘ќ has already been released, and assume that this partition is not generated by
рќ‘љрќ‘Ћрќ‘Ўв„Ћрќ‘ђрќ‘Ћрќ‘™рќђґрќњ… .
   Then the subpieces рќђјрќ‘ќ,рќњ” are not elements of рќ’«рќњ… (рќ‘Ѓ ). In particular, TGT-MF assigns no inde-
pendent testing mass to them, and the global lower bound supplied by TGT-MF is not a statement
about these subpieces. The only functional exported by TGT-MF at this location is the parent
functional в„’рќ‘ќ .
   Algebraically, after splitting the sum one has

                                                    25
                                            в€‘пёЃ рќђ»рќ‘ќ,рќњ”
                                 в„’рќ‘ќ (рќњ†) =               в„’рќ‘ќ,рќњ” (рќњ†; рќњЊрќ‘ќ , рќ›јрќ‘ќ )
                                            рќњ”в€€О©рќ‘ќ
                                                   рќђ»рќ‘ќ

    up to the harmless smoothing errors already included in the C1 boundary accounting. This
identity is used only for internal estimates if needed; it does not create a new released testing
family. Before invoking X9L-GT the pieces are reassembled into в„’рќ‘ќ .
    Thus an arbitrary shifted short interval obtained in this way is not an active TC1 test.
    вЂ”

TTH-SC.4. Proof: Genuine Structural Short Images Are Routed It remains to consider
a structural child рќ‘ќвЂІ whose image is genuinely shorter than the controlled bound (SC1). Since рќ‘ќвЂІ is
structural, the shortness is not an analytic refinement chosen after release. It is a property of the
marked B1-origin image on a routed subcell.
    The TTD/ROC/BRS chain applies to exactly this situation.

  1. TTD separates the regular start-distribution branch from singular short-image concentration.

  2. ROC handles direct dyadic-coordinate and tagged full-rank origins, routing failures to an
     already admitted terminal class.

  3. BRS reduces the remaining complementary, quotient, and carrier-slice cases to the B1 carrier-
     slice estimate.

  4. X16BRS and X16C supply the carrier-slice bound.

  5. C1P/C1A/C1 admits the strict Edge alternative created by a genuinely short marked image.

   Therefore a genuine structural short-image child is routed to

                        рќђ¶1рќђґ/рќђ¶1,        рќђ·1/рќђ»4,           рќђє8рќ‘Ћ,        рќђ»4,      or 0,
   and is not passed to X9L-GT.
   вЂ”

Parameter check 6.1 (TTH-SC.5. Parameter Check). The only loss exported by TTH-SC is
polylogarithmic. If рќђµрќњ… is the near-global exponent before structural refinement, choose рќђµрќњ…вЂІ so that

                                 рќђµрќњ…вЂІ в‰Ґ рќђµрќњ… + рќђ¶рќњ… + рќђ¶height (рќњ…) + 10.
    The parameter register chooses the TTH exponent after the TGT-MF coarea complexity, the
BRS/X16 constants, and the smooth-weight decomposition constants. Hence this enlargement is
already absorbed in the final exponent used by TTH.
    No power-saving estimate is weakened by TTH-SC: alternatives 2 and 3 are not estimated
by X9L-GT, while alternative 1 remains inside the same near-global Davenport/AP input after
enlarging рќђµрќњ… .
    вЂ”




                                                   26
TTH-SC.6. Output Form             For use in TTH and TNG, the lemma exports the following closed
barrier:


Every short subtest of a released TC1 coarea test is either non-structural and reaggregated, or structural and ro

    Equivalently, every test that is actually passed to X9L-GT is a structural TGT-MF coarea
test, up to controlled polylogarithmic subdivision, and satisfies the near-global length lower bound
supplied by TTH.


7     Part 6. TNG: TC1 near-global chain
Source file: Lemmas/tc1_near_global_chain_ltx.md.

7.0.1    TNG. B1-Origin TC1 Near-Global-or-Routed Theorem
TNG.0. Statement and Role Lemma TNG is the bridge lemma for the TC1 branch of
GoodAWACK. It packages the route


        рќђµ1-origin coarea в†’ рќ‘‡ рќ‘‡ рќђ»-рќ‘†рќђ¶ в†’ рќ‘Ђ рќ‘…рќ‘‡ /рќ‘‡ рќ‘‡ рќђ· в†’ рќ‘…рќ‘‚рќђ¶ + рќђµрќ‘…рќ‘† в†’ рќ‘‡ рќ‘‡ рќђ» в†’ рќ‘‹9рќђї-рќђєрќ‘‡

   into a single checkable source statement.
   It introduces no new analytic estimate. Its role is to make explicit that the TC1 branch never
invokes a pointwise shifted short-interval theorem for рќњ†. The only X9L input used in the proof is
the near-global Davenport/AP form

                                          рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ
   after the structural B1-origin reductions have been applied.
   Here an active or unrouted coarea test means a structural TGT.2/TGT-MF test whose cell
has not already been sent to Edge, LongAP/Local, CKP, LocalDiag, or empty support. Logical
dependencies are the TGT.2/TGT-MF global-testing construction, TTH-SC, MRT, TTD, ROC,
BRS, TTH, C1P/C1A/C1, D1/H4, G8a, X16BRS, X16C, E5, TGD, X9L-GT, and the parameter register.
TNG is used by E10L.
   вЂ”

TNG.1. Setup: Active B1-Origin Coarea Tests              Fix a terminal TC1-GoodAWACK macro-
template рќњ…. It consists of:

    1. a B1 typed parent block;

    2. a B3 grouping record;

    3. the F3/F4 routing history;

    4. a marked Liouville affine form рќђїрќ‘љ ;

    5. the TC1 tensor certificate;

    6. the C1-clean smooth box/coset cell О©* on which the TC1 Fourier/coarea argument is per-
       formed.

                                                 27
   An active B1-origin coarea test is a test produced from this data by the coarea decomposition

                                       рќ‘› = рќђїрќ‘љ (рќ‘§),      рќ‘§ в€€ О©* ,
   after only the following normalizations:

  1. polylogarithmically many scale, modulus, and smooth-weight subdivisions;

  2. controlled CRT restrictions;

  3. fixed divisor quotienting with controlled divisor;

  4. full-rank affine transports with controlled content;

  5. removal of C1 boundary pieces.

   Thus a test has the form
                                            1 в€‘пёЃ
                                 в„’рќ‘ќ (рќњ†) =          рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),                  (TNG-test)
                                            рќђ»рќ‘ќ рќ‘›в€€рќђј
                                                  рќ‘ќ

   where:


       рќ‘”рќ‘ќ в‰¤ (log рќ‘‹рќ‘ќ )рќђ¶рќњ… ,     рќђ»рќ‘ќ = |рќђјрќ‘ќ |,     рќњЊрќ‘ќ has polylogarithmic smoothness complexity.

  The word active excludes cells already routed to Edge, LongAP/Local, CKP, LocalDiag, or
empty support. Those cells are handled by C1P/C1A/C1, D1/H4, G8a, H4, or contribute zero.
  вЂ”

TNG.2. Structural Coarea Closure The coarea interval рќђјрќ‘ќ is a structural image piece of
the terminal marked B1-origin carrier рќђїрќ‘љ (О©* ). The formal barrier against rogue short-interval
refinements is Lemma TTH-SC.
    More precisely, TTH-SC classifies every refinement of a released coarea test. Controlled scale,
AP/modulus, and smooth-weight subdivisions remain structural and lose only a fixed power of
log рќ‘‹. A subdivision chosen after release which is not generated by the structural coarea alge-
bra is not an element of the TGT-MF testing family and is reaggregated into its parent func-
tional before X9L-GT is invoked. A genuinely structural short-image child is routed through
TTD/ROC/BRS/X16BRS/X16C and C1P/C1A/C1 before any Liouville/AP input is applied.
    Thus arbitrary shifted short intervals are not active TC1 tests, and this is a closure lemma
rather than a convention of exposition.
    вЂ”

TNG.3. Proof: Regular Branch Assume that the TC1 testing family for рќњ… is MRT-admissible.
Then MRT supplies the start-pushforward bound

                                                            рќ‘‘рќ‘Ґ
                                    (start)# рќњ€рќњ… в‰Єрќњ… (log рќ‘Ѓ )рќђ¶рќњ…  .                          (PACK)
                                                            рќ‘‹
   For every active B1-origin coarea test in this family, TTH supplies

                                        рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… .                                (TTH)

                                                  28
   Together with the polylogarithmic modulus and smoothness bounds in TNG.1, this is exactly
the hypothesis set of the near-global X9L-GT theorem:

             PACK + {рќ‘”рќ‘ќ в‰¤ (log рќ‘‹рќ‘ќ )рќђ¶рќњ… } + {рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… } =в‡’ X9L-GT-NG.
   Indeed X9L-GT uses DavenportвЂ™s estimate in AP form. The loss in passing from global prefixes
to AP fibres is bounded by

                                         (log рќ‘‹рќ‘ќ )2рќђ¶рќњ… +рќђµрќњ… +рќ‘‚рќњ… (1) .
    Choosing the Davenport logarithmic saving exponent larger than this loss and the required final
saving gives
                                    в€«пёЃ
                                         |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) = рќ‘њрќњ… (1).                     (X9L-NG)

    By the TGT.2/TGT-MF global-testing construction, a non-small TC1 macro-template would
force a fixed lower bound for the same left side. Therefore the MRT-admissible branch contributes
рќ‘њ(рќ‘Ѓ ).
    вЂ”

TNG.4. Proof: Singular Branch Routes Before X9L If MRT-admissibility fails, the testing
measure has singular start concentration. TTD identifies the only possible unrouted singular geom-
etry: the marked form moves through a short additive image while transverse B1-origin variables
carry the volume.
    The route is then structural, not analytic.
    First, ROC proves range comparability for direct dyadic-coordinate origins and controlled full-
rank transports. It also routes tagged failures to the already existing terminal classes.
    Second, the complementary solved-affine or quotient-origin case is handled by BRS. BRS applies
the B1 carrier-slice estimate, supplied by X16BRS and X16C, and proves the dichotomy

                             short marked image =в‡’ strict C1 Edge
    unless the failure already carries a LongAP/Local, CKP, LocalDiag, Edge, or empty routing
tag.
    Thus a singular TC1 testing family is never sent to X9L-GT. It is routed to:

                        рќђ¶1рќђґ/рќђ¶1,          рќђ·1/рќђ»4,          рќђє8рќ‘Ћ,         рќђ»4,   or 0.
   вЂ”

TNG.5. Output Theorem: TC1 Near-Global-or-Routed The TTH/BRS/X16 part of the
TC1 proof is used through the following single theorem-interface. It is intentionally stronger as an
interface than the individual component lemmas: it classifies the actual tests that reach the TC1
global-testing stage.
Theorem 7.1 (Theorem TNG-A. TC1 tests are near-global or routed away). Fix a B1/B3/F3/F4
terminal TC1-GoodAWACK macro-template рќњ… whose cell has not already been routed away, after C1
boundary removal, fixed macro-template normalization, and polylogarithmic scale/modulus/smooth-
weight decomposition. Let в„’рќ‘ќ (рќњ†) be any unrouted coarea test produced by TGT from the marked
B1-origin form.
   Then exactly one of the following alternatives holds.

                                                    29
  1. Near-global testing alternative. The test belongs to the regular MRT-admissible branch.
     The start-pushforward satisfies PACK, the AP modulus and smoothness complexity are poly-
     logarithmic, and TTH gives
                                        рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… .
      Hence this test is an allowed input to the near-global Davenport/AP theorem X9L-GT.

  2. Routed alternative. The test is not sent to X9L-GT. Before any Liouville/AP input is
     invoked, TTD, ROC, BRS, and the X16BRS/X16C carrier-slice estimate route the corre-
     sponding cell to one of

                                  рќђ¶1рќђґ/рќђ¶1,             рќђ·1/рќђ»4,         рќђє8рќ‘Ћ,         рќђ»4,         or 0.

    In particular, there is no third case consisting of an arbitrary shifted short interval or an un-
classified short AP fibre. The exclusion of that third case is supplied by TTH-SC.

Proof. Start with the coarea tests constructed in TGT from the fixed macro-template рќњ…. MRT first
separates the regular branch from the singular start-concentration branch.
    In the regular branch, MRT supplies PACK for the same testing family. The coarea test still has
B1-origin in the sense of TTH.2, because the only normalizations are controlled CRT restrictions,
fixed-divisor quotients, full-rank transports, and post-terminal analytic subdivisions that do not
replace the terminal marked carrier. TTH-SC prevents the released test from being replaced by a
new arbitrary short-interval family. TTH then gives the near-global length lower bound for every
remaining coarea image piece. The modulus and smoothness complexity bounds are those recorded
in TNG.1. Thus the test is exactly an X9L-GT input.
    In the singular branch, TTD identifies a singular-origin mechanism. Direct dyadic-coordinate
and tagged full-rank transport cases are handled by ROC. The complementary solved-affine, quo-
tient, and carrier-slice cases are handled by BRS. In BRS, a genuinely short marked B1 image
cannot carry uncontrolled transverse mass: X16BRS reduces all BRS carrier types to X16-Core,
and X16C proves X16-Core. Therefore a short B1 image is a strict C1 Edge contribution unless it
already carries a LongAP/Local, CKP, LocalDiag, Edge, empty, or nonterminal routing tag. These
are precisely the routed alternatives listed above.
    Finally, TTH-SC gives the closure barrier for refinements of an already released near-global
structural image. Non-structural short pieces are aggregated back to the parent piece, while genuine
structural short-image children are routed before X9L-GT. Hence no pointwise shifted short-interval
escape case remains. The theorem follows.
    For publication checking, the component bridge behind the theorem is

                                            BRS/X16 =в‡’ TTH =в‡’ X9L
    is the following finite decision table on an unrouted TC1 coarea test.

  Test status after TGT           Structural source              BRS/X16 action                Result before X9L
  coarea
  Direct B1 product carrier,      B1/B3/F3/F4 marked car-        BRS range comparability       рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµрќњ… ; X9L-
  full-rank transport, no short   rier                           holds                         GT may be invoked.
  image
  Direct B1 product carrier       same                           X16BRS/X16C carrier-slice     strict C1 Edge via C1A E6;
  with short marked image                                        estimate bounds the short-    no X9L invocation.
                                                                 image mass




                                                            30
  Complementary carrier рќ‘Ѓ в€’      F4/BRS solved-affine origin         Replace      by      product    near-global or strict Edge.
  рќ‘ѓ                                                                  carrier   рќ‘ѓ     and    apply
                                                                     X16BRS/X16C
  Quotient carrier рќ‘  in рќђї = рќ‘‘рќ‘    F4 quotient tag                     Transfer рќ‘  в€€ рќђј to рќђї в€€ рќ‘‘рќђј;       near-global or strict Edge.
  with tagged рќ‘‘                                                      controlled divisor sum is ab-
                                                                     sorbed
  Untagged   quotient/divisor    unresolved F4 ordinary divi-        F4 does not release the cell    routed to Edge, LocalDiag,
  relation                       sor predicate                       to TC1 testing                  CKP, GoodAWACK with
                                                                                                     tag, or nonterminal de-
                                                                                                     crease.
  Singular start measure from    TTD singular branch                 ROC handles direct/tagged       routed before X9L.
  non-direct origin                                                  origins;     BRS    handles
                                                                     solved-affine complement
  Artificial subdivision of an   TTH-SC     non-structural           Aggregated back to the      no pointwise shifted short-
  already near-global image      case                                structural image piece      interval input is created.
  Genuine structural short-      TTH-SC structural short-            TTD/ROC/BRS/X16BRS/X16C     no unclassified short AP fi-
  image refinement               image case                          and C1P/C1A/C1 route it         bre remains.




    Thus the only tests actually passed to X9L-GT are the first row: unrouted structural coarea
image pieces whose length is near-global after BRS/TTH. The second row is the critical use of
X16. It says that a genuinely short marked B1 image cannot hide a large transverse mass: the B1
carrier-slice estimate converts it into a strict C1 Edge contribution.
    This formulation also fixes the quantifiers. TTH is not a theorem about arbitrary E7 directional
fibres or arbitrary shifted subintervals. It is a theorem about the unrouted B1-origin coarea tests
selected by TGT after F3/F4 routing, C1 boundary removal, and TTD/MRT normalization.
    вЂ”


TNG.6. Output: TC1 Cancellation Theorem

Theorem 7.2 (Theorem TNG). For every unrouted B1/B3/F3/F4 terminal TC1-GoodAWACK
macro-template рќњ…, after C1 boundary removal and fixed macro-template normalization, Theorem
TNG-A applies to every TC1 coarea test. Consequently:

  1. every test sent to X9L-GT is near-global and MRT-admissible;

  2. every non-near-global or singular test is routed to an already handled terminal class before
     X9L-GT is invoked.

    Consequently

                                           рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).

Proof. Aggregate terminal TC1 atoms by the fixed macro-template рќњ… as in TGT. Apply Theorem
TNG-A to the unrouted coarea tests.
   On the near-global alternative, MRT supplies PACK and TTH supplies the near-global length
lower bound for the same B1-origin coarea tests. Hence the near-global X9L-GT theorem applies
with the parameters listed in TNG.3. This contradicts the fixed TGT testing lower bound unless
the рќњ…-contribution is рќ‘њ(рќ‘Ѓ ).
   On the routed alternative, the cell is sent to Edge, LongAP/Local, CKP, LocalDiag, or empty
support by TTD/ROC/BRS using X16BRS/X16C where the carrier-slice estimate is needed. These


                                                                31
outputs are outside terminal TC1-GoodAWACK and are handled by C1P/C1A/C1, D1/H4, G8a, H4,
or zero. Therefore the routed branch contributes no terminal TC1-GoodAWACK mass.
    There are only boundedly many structural TC1 macro-templates, depending on the fixed pa-
rameter рќђЅ0 . Summing over them gives the displayed рќ‘њ(рќ‘Ѓ ) estimate. Theorem proved.
    вЂ”


Remark 7.3 (TNG.7. Output).

                        TNG-A + X9L-GT =в‡’ рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).

   Here TNG-A is the single TC1 structural theorem packaging TGT/TTH-SC/MRT/TTD/ROC/BRS/X16BRS/
The chain uses X9L-GT only in the near-global Davenport/AP form. It does not use:

    1. X8 inverse-Gowers input;

    2. pointwise shifted short-interval Liouville cancellation;

    3. a low-рќњѓ polylog-modulus theorem for arbitrary short AP fibres.

TNG.8. Logical Dependencies External dependency: X9L-GT in the near-global Daven-
port/AP form.
   Internal dependencies: the TGT.2/TGT-MF global-testing construction, TTH-SC, MRT, TTD,
ROC, BRS, TTH, C1A, C1, D1, H4, G8a, X16BRS, X16C, E5, TGD, and the parameter register.
   Children served: E10L and the GoodAWACK TC1 branch.


8     Part 7. TTD: TC1 testing dichotomy
Source file: Lemmas/tc1_testing_dichotomy_ltx.md.

8.0.1    TTD. TC1 Testing Dichotomy
TTD.0. Statement and Role Lemma TTD is the testing-dichotomy reduction. Regular TC1
testing families close by TGT and the near-global X9L input, while singular B1-origin cases are
closed by ROC, BRS, and TTH.
    The target isolated in TGT is:

                                   TC1-TESTING-DICHOTOMY.
   The desired statement is:
   For every actual B1/B3/F3/F4 terminal TC1 macro-template рќњ…, after C1 boundary removal,
the induced global Liouville testing family is either:

    1. averaged-admissible in the sense of TGT; or

    2. its singular part has a B1-origin route to strict C1 Edge, D1/H4 LongAP/Local, G8a CKP,
       LocalDiag, or empty/impossible.




                                                   32
   The theorem supplied by TTD is:


       regular families close by X9L-GT, and singular families route by ROC/BRS/TTH.

   The singular-origin component is:


 TC1-SINGULAR-ORIGIN : every singular TC1 testing measure has an existing routing origin.

    This is narrower than the pointwise X9L-SI obstruction. It is a structural B1-origin problem,
not an analytic short-interval estimate.
    Logical dependencies are the TGT.2/TGT-MF global-testing construction, MRT, ROC, BRS,
TTH, C1, D1/H4, G8a, X16BRS, X16C, and X9L-GT. TTD is used by the full TGT closure
statement, TTH-SC, TTH, TNG, and E10L.
    вЂ”

TTD.1. Setup and Regular Branch                Let рќњ… be a fixed TC1 macro-template. By TGT, if

                                              |рќ‘…рќњ… (рќ‘Ѓ )| в‰Ґ рќњЂрќ‘Ѓ                                     (1)
  along an infinite sequence, then the global TC1 generalized von Neumann step and Lemma
TGT-MF produce a measured testing family

                                                (рќ’«рќњ… (рќ‘Ѓ ), рќњ€рќњ… )
   whose tests have the form
                                              1 в€‘пёЃ
                                в„’рќ‘ќ (рќњ†) =             рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),                          (2)
                                              рќђ»рќ‘ќ рќ‘›в€€рќђј
                                                     рќ‘ќ

   and satisfy the fixed lower bound
                                   в€«пёЃ
                                              |в„’рќ‘ќ (рќњ†)|2 рќ‘‘рќњ€рќњ… (рќ‘ќ) в‰«рќњ…,рќњЂ 1.                          (3)
                                    рќ’«рќњ… (рќ‘Ѓ )

   If (рќ’«рќњ… , рќњ€рќњ… ) is MRT-admissible, then the averaged Liouville input gives the opposite bound рќ‘њ(1).
Hence:

Lemma 8.1 (Lemma TTD.1. Regular testing branch). Assume the averaged Liouville input X9L-
GT for MRT-admissible testing families. If the TC1 testing family induced by рќњ… is MRT-admissible,
then

                                              рќ‘…рќњ… (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).

Proof. This is the regular-branch part of the TGT/TGT-MF construction, with MRT-admissibility
checked through MRT. The point is that the macro-template aggregation gives a fixed lower bound
(3), so qualitative averaged cancellation suffices. Lemma proved.
    Thus the only remaining case is non-admissible, or singular, testing measure.
    вЂ”




                                                     33
TTD.2. Setup: Geometry of Singular Testing Measure After the standard polyloga-
rithmic decomposition into dyadic scale, AP modulus, and smooth weight-complexity classes, the
failure of MRT-admissibility means that the start pushforward

                                             (start)# рќњ€рќњ…
    is not dominated by a polylogarithmic multiple of normalized measure on its dyadic start range.
    At the affine-geometric level this can happen only if a positive fraction of the testing mass is
carried by pieces where the marked affine image has too few independent start directions. The
model is

                   О© = [рќ‘‹0 , рќ‘‹0 + рќђ»] Г— [1, рќ‘Ђ ],        рќђїрќ‘љ (рќ‘ў, рќ‘Ј) = рќ‘ў,    рќђ»рќ‘Ђ в‰Ќ рќ‘Ѓ,                  (4)
   with

                                         рќђ» < рќ‘‹0 (log рќ‘‹0 )в€’рќђµ .                                     (5)
    The transverse coordinate рќ‘Ј supplies volume, but it does not move the Liouville interval. The
testing measure is concentrated on essentially one short interval near рќ‘‹0 .
    This is the common geometric content of:

  1. the rank-one/nonregular E7 carrier;

  2. the short affine-image residual in the TC1 coarea/Fourier decomposition;

  3. the singular affine-image model;

  4. the singular testing measure of TGT.

   вЂ”

TTD.3. Proof: Tagged Singular Origins Route Away                    The singular geometry is harmless
if it comes from an already tagged origin.

Lemma 8.2 (Lemma TTD.2. Tagged singular origin routes away). Suppose a singular TC1 test-
ing subfamily arises because the marked affine image has lost start directions through one of the
following tagged operations:

  1. fixing/projection with short residual volume;

  2. fixed divisor quotient with a short quotient range;

  3. variable quotient residual whose quotient range is short;

  4. local/diagonal forcing;

  5. CKP balanced grouping;

  6. strict C1 Edge origin;

  7. impossible or inconsistent fibre;

  8. post-terminal primitive slicing that does not create a new terminal GoodAWACK skeleton.


                                                  34
   Then the subfamily contributes only to C1, D1/H4, G8a, LocalDiag, or zero, and does not
remain in terminal TC1-GoodAWACK.

Proof. This follows directly from the routing interface fixed by B3, F3, F4, E5, and the terminal-
operation rule stated in TGD.
    Items 1, 2, 3, and 6 are strict C1 or F4 short-volume/Type-I cases. F4.6 routes short divisor
or short quotient cases to Edge, and C1 counts only those Edge cases with an explicit summable
budget.
    Item 4 is F4.7/F3 LocalDiag detection.
    Item 5 is the CKP route handled by G8a.
    Item 7 is empty.
    Item 8 is terminal-interface clean: post-terminal primitive slicing, Cauchy/cube operations, and
Fourier expansion do not generate new terminal GoodAWACK skeletons. The terminal TC1/HighTC
test uses the pre-slicing affine vectors.
    Thus every tagged singular origin is routed away from terminal TC1-GoodAWACK. Lemma
proved.
    вЂ”


TTD.4. Statement: Singular-Origin Criterion The only possible obstruction to the regular
branch is a singular testing subfamily whose marked Liouville form moves through a short additive
image while transverse B1-origin variables carry the volume. In model form this geometry is

                  О© = [рќ‘‹0 , рќ‘‹0 + рќђ»] Г— [1, рќ‘Ђ ],        рќђїрќ‘љ (рќ‘ў, рќ‘Ј) = рќ‘ў,   рќђ»рќ‘Ђ в‰Ќ рќ‘Ѓ,                  (6)
   where:

  1. рќђ» в‰Ґ рќ‘Ѓ рќњѓ , so no short-direction C1 predicate is automatic;

  2. рќђ» < рќ‘‹0 (log рќ‘‹0 )в€’рќђµ , so the Liouville image is a shifted short interval;

  3. the full effective volume is рќђ»рќ‘Ђ в‰Ќ рќ‘Ѓ , so C1 short-volume Edge is not automatic;

  4. рќђїрќ‘љ has controlled content;

  5. no LocalDiag or CKP relation is forced at the terminal interface;

  6. the marked рќњ†(рќђїрќ‘љ ) factor remains a nonlocal oscillatory coefficient, so LongAP/Local does
     not apply.

   The singular-origin assertion is:


TC1-SINGULAR-ORIGIN : model (6), and every equivalent singular testing measure, cannot arise from an actu

   вЂ”




                                                 35
TTD.5. Proof: Range-Origin Comparability and BRS Closure

Lemma 8.3 (Lemma ROC). For every actual terminal GoodAWACK marked Liouville form рќђїрќ‘љ ,
after C1 boundary removal and after passing to a fixed TC1 macro-template, either:

  1. the affine image satisfies near-global range comparability


          |рќђїрќ‘љ (О©)| в‰Ґ рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђ¶ ,     рќ‘‹рќ‘љ в‰Ќ max(2, dist(рќђїрќ‘љ (О©), 0) + |рќђїрќ‘љ (О©)|);            (ROC)

  1. or the failure of (ROC) is caused by a tagged origin from Lemma TTD.2.

  If ROC holds, the marked image is closed by the near-global coarea argument and the near-global
X9L-GT input. If ROC fails, Lemma TTD.2 routes it away.
  Therefore ROC proves the direct-origin part of the TC1 testing dichotomy.

Proof of Lemma ROC. B1 begins with dyadically localized product variables, whose value and
additive range are comparable. Controlled CRT restrictions and fixed divisor quotients only lose
polylogarithmic factors. Variable quotient residuals are routed by F4 if the quotient is short, local,
or CKP; otherwise the quotient is central-long. E10M and E10K forbid untagged rank-dropping
affine changes in a terminal GoodAWACK skeleton.


BRS closure of the complementary singular case Lemma ROC proves range-origin compa-
rability for direct dyadic-coordinate origins and their controlled CRT/divisor/full-rank transports.
It also confirms that tagged failures route by Lemma TTD.2. The part not covered by direct
comparability is the complementary affine-origin case, where a short marked image carries hidden
transverse B1 multiplicity.
    Lemma BRS closes exactly that case using X16BRS/X16C. It proves:

                              B1-RANGE-SKELETON/ROC-SLICE.
   Thus TC1-SINGULAR-ORIGIN is supplied by ROC plus BRS.
   вЂ”

TTD.6. Output Theorem           Use:

  1. X9L-GT: averaged Liouville cancellation for MRT-admissible testing families;

  2. ROC and BRS, which prove TC1-SINGULAR-ORIGIN;

  3. TTH, which puts unrouted tests in the cited X9L-GT range.

   Then

                                   рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).




                                                 36
Proof. Aggregate by TC1 macro-template. Lemma MRT first selects the regular MRT-admissible
branch or a singular-origin branch. If a template is MRT-admissible, Lemma TTD.1 closes it. If
it is singular, ROC plus BRS gives either ROC, Edge by slice mass, or one of the tagged origins
in Lemma TTD.2. Hence the singular part is closed by the near-global coarea argument or routes
to C1, D1/H4, G8a, LocalDiag, or zero. Summing over the bounded number of macro-templates
gives the claim. Theorem proved using the BRS/X16-Core input supplied by Lemmas X16BRS
and X16C.
     вЂ”


Remark 8.4 (TTD.7. Output).

TC1-TESTING-DICHOTOMY is proved, with MRT selection explicit and BRS/X16-Core supplied by X16BRS
    What is proved here:


MRT-admissible testing families close by X9L-GT, and singular origins route away by ROC/BRS/TTH.
    The low-theta alternative is not used. Lemma TTH supplies the near-global bound рќђ» в‰Ґ
рќ‘‹(log рќ‘‹)в€’рќђµ , and X9L-GT supplies the averaged Liouville input for that range. The singular
structural branch is not a residual.

TTD.8. Logical Dependencies External dependency: X9L-GT after TTH supplies the near-
global range.
   Internal dependencies: the TGT.2/TGT-MF global-testing construction, MRT, ROC, BRS,
TTH, C1, D1/H4, G8a, X16BRS, and X16C.
   Children served: TGT, TTH-SC, TTH, TNG, and E10L.


9     Part 8. MRT: MRT admissibility
Source file: Lemmas/tc1_mrt_admissibility_ltx.md.

9.0.1    MRT. PACK Interface for TC1 Global Testing
MRT.0. Statement and Role Lemma MRT verifies the PACK interface required before TGT
applies the averaged Liouville input.
   The important distinction is:
    1. TTH supplies a near-global length lower bound рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµрќњ… .
    2. MRT-admissibility also needs a start-pushforward bound:
                                                                рќ‘‘рќ‘Ґ
                                      (start)# рќњ€рќњ… в‰Є (log рќ‘Ѓ )рќђ¶      .                   (PACK)
                                                                рќ‘‹
   Length alone does not imply PACK. The exact interface is: regular TC1 macro-templates satisfy
PACK by finite B1/B3/F3/F4 multiplicity; failure of PACK is a singular-origin event and must
route through TTD/ROC/BRS before X9L-GT is invoked.
   Logical dependencies are E5, TGD, TGT-MF, TTD, ROC, BRS, TTH, X16BRS, and X16C.
MRT is used by TGT, TNG, and E10L.
   вЂ”

                                               37
MRT.1. Setup: Testing Family            Fix a TC1 macro-template рќњ…. It consists of:

  1. a parent B1 block;

  2. a B3 grouping candidate;

  3. the F3/F4 routing history;

  4. a marked affine Liouville form рќђїрќ‘љ ;

  5. the measured coarea/Fourier decomposition supplied by TGT-MF.

   Each test in the family has the form
                          1 в€‘пёЃ
               в„’рќ‘ќ (рќњ†) =          рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),      рќђјрќ‘ќ = [рќ‘Ґрќ‘ќ , рќ‘Ґрќ‘ќ + рќђ»рќ‘ќ ] в€© рќђїрќ‘љ (О©рќ‘ќ ).
                          рќђ»рќ‘ќ рќ‘›в€€рќђј
                                рќ‘ќ

    The measure рќњ€рќњ… is the normalized volume weight inherited from the tagged B1/B3/F3/F4 cell.
    For Lemma MRT, a regular TC1 macro-template means a B1-origin TC1 macro-template
for which the marked coarea map рќђїрќ‘љ has rank one on an active full-rank direction of the current
parameter lattice and the induced start map is not collapsed onto a lower-dimensional/rank-deficient
image. Rank-deficient, point-mass, or short-image failures are by definition sent to the singular
branch in MRT.3.
    вЂ”

MRT.2. Proof: Regular Multiplicity Condition               For a fixed macro-template рќњ…, say that the
start map is regular if for every interval рќђЅ вЉ‚ [рќ‘‹, 2рќ‘‹]

                                                                   |рќђЅ|
                                    рќњ€рќњ… {рќ‘ќ : рќ‘Ґрќ‘ќ в€€ рќђЅ} в‰Є (log рќ‘Ѓ )рќђ¶рќњ…       .                    (REG-start)
                                                                    рќ‘‹
    This is exactly PACK, with рќђ¶ = рќђ¶рќњ… .
    In the B1-origin setting, REG-start follows from finite routing multiplicity when the start
coordinate is the image of a full-rank dyadic coordinate map. The source of the full-rank condition
is the regular branch just defined: rank-deficient coarea maps are not regular MRT inputs but
singular-origin inputs handled by TTD/ROC/BRS.
    Full-rank affine transports distort length and lattice index only by polylogarithmic factors by
E5. Specifically, E5.2 controls CRT restrictions, E5.4 controls primitive slicing by writing the image
as рќ‘”рќ‘ў + рќ‘Џ with controlled рќ‘”, and E5.5 states that full-rank affine changes preserve content up to
controlled factors. The parent B1 variables are dyadically localized, and the routing grammar
has at most (log рќ‘Ѓ )рќђ¶0 cells. Hence a subinterval of relative length |рќђЅ|/рќ‘‹ captures at most a
polylogarithmic multiple of that relative volume.
    This proves MRT-admissibility for the regular full-rank B1-origin TC1 testing families.
    вЂ”

MRT.3. Proof: Failure of PACK is Singular If REG-start fails, the TC1 tests concentrate
too much mass on too few starts. For a B1-origin macro-template, this can only happen through
one of the structural singular mechanisms already named in TTD/ROC:

  1. rank-one/nonregular E7 carrier;

  2. short image of the marked B1 carrier;

                                                  38
  3. fixed or variable quotient range collapse;

  4. forced local dependence or diagonal collision;

  5. impossible/empty support.

   These are not sent to X9L-GT. They are routed by TTD to ROC/BRS, and then to C1 Edge,
LongAP/Local, CKP, LocalDiag, or empty support.
   Thus X9L-GT is invoked only on tests satisfying PACK.
   вЂ”

MRT.4. Statement: Interface Lemma Lemma MRT. Let рќњ… be a B1-origin TC1 macro-
template after F3/F4 routing and C1 boundary removal.
   Then exactly one of the following holds:

  1. the induced testing family (рќ’«рќњ… , рќњ€рќњ… ) is MRT-admissible in the sense of TGT.4;

  2. рќњ… has a singular-origin certificate and is routed by the TNG-A structural branch before X9L-
     GT is invoked.

Proof. If the start map satisfies REG-start, PACK follows by MRT.2. This regular case uses only
the E5/TGD full-rank transport control and the finite B1/B3/F3/F4 routing multiplicity; it does
not use X16-Core.
    If REG-start fails, the failure is a concentration/rank-collapse event in the B1-origin coarea map.
The finite B1/B3/F3/F4 grammar leaves only the five mechanisms listed in MRT.3, and each is one
of the singular-origin cases handled by Theorem TNG-A through TTD/TTH-SC/ROC/BRS/TTH.
In this singular branch, BRS invokes the X16-Core input proved by Lemma X16C. Lemma proved.
    вЂ”


Remark 9.1 (MRT.5. Output).

               X9L-GT is applied only after Lemma MRT selects the MRT branch.

   The singular branch is handled by BRS/X16-Core, explicitly supplied by Lemmas X16BRS and
X16C, not by a pointwise short-interval Liouville theorem.

MRT.6. Logical Dependencies Internal dependencies: E5, TGD, TTD, ROC, BRS, TTH,
X16BRS, and X16C.
   Children served: TGT, TNG, and E10L.


10     Part 9. ROC: Singular-origin routing
Source file: Lemmas/tc1_singular_origin_roc_ltx.md.




                                                  39
10.0.1    ROC. Range-Origin Lemma for Singular TC1 Testing
ROC.0. Statement and Role Lemma ROC is the singular-origin reduction feeding the
BRS/TTH route. Direct B1-origin short-image cases are closed by BRS and X16BRS, while re-
maining tagged origins route to Edge, CKP, LocalDiag, LongAP/Local, or Impossible.
   The lemma addresses the singular-origin block from TTD:

                                TC1-SINGULAR-ORIGIN/ROC.
   The desired range-origin comparability statement is:
   For every actual B1-origin terminal TC1-GoodAWACK marked form рќђїрќ‘љ , after C1 boundary
removal and fixed macro-template normalization, either

          |рќђїрќ‘љ (О©)| в‰Ґ рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђ¶ ,    рќ‘‹рќ‘љ в‰Ќ max(2, dist(рќђїрќ‘љ (О©), 0) + |рќђїрќ‘љ (О©)|),        (ROC)
   or the failure of (ROC) has an existing C1/D1/G8a/LocalDiag/empty origin tag.
   The direct-origin part proved in ROC is:


direct dyadic-coordinate origins and their controlled CRT/divisor/full-rank transports satisfy ROC.

   The complementary/solved affine origins are supplied by the subsequent BRS carrier-slice the-
orem:


B1-RANGE-SKELETON/ROC-SLICE : enrich terminal GoodAWACK skeletons with additive image length and

   Logical dependencies are B1, BGS, TTD, BRS.1, X16BRS, X16C, and the E10Y/E10M/E10K
terminal-affine grammar interface. The dependency on BRS is noncircular: BRS uses only ROC.1
and ROC.2, while the full ROC closure is obtained after BRS is invoked. ROC is used by BRS,
TTH-SC, TTH, TNG, and E10L.
   вЂ”

ROC.1. Proof: Clean Dyadic-Coordinate Origins Satisfy ROC Suppose рќђїрќ‘љ is a surviving
parent/grouped coordinate origin in the sense of Lemma BGS, Type A, and that no Goldbach
complement or quotient-solving step is used to define it.
   At the B1/B3 level, the corresponding grouped variable рќ‘ў is dyadically localized:

                                             рќ‘ў в‰Ќ рќ‘€.
   If рќ‘ў is terminal GoodAWACK and not C1-routed, it is long:

                                            рќ‘€ в‰Ґ рќ‘Ѓ рќњѓ.
   Its additive image on the dyadic cell has length

                                           |рќ‘ў(О©)| в‰Ќ рќ‘€.
   Also

                                            рќ‘‹рќ‘љ в‰Ќ рќ‘€.


                                               40
   Therefore

                                          |рќђїрќ‘љ (О©)| в‰Ќ рќ‘‹рќ‘љ ,
   which is stronger than (ROC).

Stability under controlled transports         The same conclusion survives the following operations:

  1. controlled CRT restriction, losing at most a polylogarithmic index;

  2. fixed divisor quotient рќђї в†¦в†’ рќђї/рќ‘‘ with рќ‘‘ в‰¤ (log рќ‘Ѓ )рќђ¶ ;

  3. full-rank affine coordinate changes with polylogarithmically bounded minors and inverse mi-
     nors, as normalized by the E10Y/E10M/E10K terminal-affine grammar interface;

  4. removal of C1 boundary pieces.

    Indeed, each operation changes additive image length and height by at most a polylogarithmic
factor unless it is rank-dropping. Rank-dropping operations are tagged by the terminal-affine
grammar interface and are handled by TTD, Lemma TTD.2.
    Hence:

Lemma 10.1 (Lemma ROC.1. Direct-origin range comparability). Direct dyadic-coordinate marked
forms and their controlled full-rank CRT/divisor transports satisfy (ROC), after C1 boundary re-
moval.

Proof. Dyadic localization gives value and additive range comparable to the same scale. Controlled
CRT/divisor/full-rank transports distort both by only a polylogarithmic factor. If the transport
loses the direction responsible for the image length, it is a tagged rank drop, not a direct-origin
case. Lemma proved.
    вЂ”


ROC.2. Proof: Tagged Singular Origins Route Away                  If (ROC) fails because one of the
following tagged origins is present:

  1. short residual volume;

  2. Type I error budget;

  3. short fixed divisor or short quotient;

  4. forced local dependence;

  5. CKP balanced multiplicative origin;

  6. impossible/inconsistent support;

  7. post-terminal primitive slicing that does not create a new terminal GoodAWACK skeleton;

   then the singular testing family is already routed by Lemma TTD.2.
   Thus the only possible obstruction to ROC is an **untagged range-defective origin**.
   вЂ”

                                                 41
ROC.3. Setup: Complementary Affine-Origin Problem The dangerous case is not a direct
dyadic coordinate. It is a marked affine form obtained from the Goldbach relation or from solving
a grouped equation.
    The schematic source is:

                                         рќ‘ѓрќђґ (рќ‘Ћ) + рќ‘ѓрќђµ (рќ‘Џ) = рќ‘Ѓ.
   After grouping and partial solving, a surviving marked form may look like

                                          рќђїрќ‘љ = рќ‘Ѓ в€’ рќђїother ,                                     (1)
   or, in a two-group presentation,

                                                     рќ‘Ѓ в€’ рќ‘ўрќ‘Ј
                              рќ‘ўрќ‘Ј + рќ‘ўвЂІ рќ‘Ј вЂІ = рќ‘Ѓ,        рќђїрќ‘љ = рќ‘ўвЂІ =.                          (2)
                                                          рќ‘ЈвЂІ
   Here the absolute height of рќђїрќ‘љ can be рќ‘‹рќ‘љ в‰Ќ рќ‘Ѓ or рќ‘Ѓ/рќ‘Ј вЂІ , while its additive image length is
controlled by the variation of the other side:

                                        |рќђїрќ‘љ (О©)| в‰Ќ |рќђїother (О©)|.
   Thus it is possible at the interface level to have

                                      |рќђїрќ‘љ (О©)| в‰Є рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђµ
   without any immediate contradiction.
   The model is:

                  О© = [рќ‘‡, рќ‘‡ + рќђ»] Г— [1, рќ‘Ђ ],          рќђїрќ‘љ (рќ‘Ў, рќ‘џ) = рќ‘Ѓ в€’ рќ‘Ў,   рќђ»рќ‘Ђ в‰Ќ рќ‘Ѓ,               (3)
   with

                                       рќ‘Ѓ рќњѓ в‰¤ рќђ» < рќ‘Ѓ (log рќ‘Ѓ )в€’рќђµ .                                 (4)
   The рќ‘Ў-range is long, so C1 short-direction Edge is not automatic. The full abstract box volume
can be в‰Ќ рќ‘Ѓ because of the transverse рќ‘џ-range. The marked image is a shifted short interval near
рќ‘Ѓ.
   This is exactly the singular testing model from TTD.
   вЂ”

ROC.4. Setup: Why Actual B1 Saves This Case Although model (3) is allowed by the
abstract terminal interface, it may be impossible as an actual B1 descendant.
   The reason is that the transverse variable рќ‘џ cannot be an arbitrary free volume direction if it
only records factorizations of a fixed integer рќђїрќ‘љ = рќ‘›. In the true B1 finite-convolution expansion,
once the marked integer рќ‘› and the complementary product are fixed, the remaining factorization
multiplicity is divisor-type, not a free interval of length рќ‘Ђ в‰Ќ рќ‘Ѓ/рќђ».
   If one could prove a uniform coarea slice bound of the form
                                        в€‘пёЃ
                                                 |рќ‘Љ (рќ‘§)| в‰Є (log рќ‘Ѓ )рќђ¶                        (Slice)
                                        рќ‘§в€€О©
                                      рќђїрќ‘љ (рќ‘§)=рќ‘›




                                                     42
   or an averaged version strong enough after summing over рќ‘› в€€ рќђїрќ‘љ (О©), then every short-image
complementary case would be Edge:
                            в€‘пёЃ      в€‘пёЃ
                                             |рќ‘Љ (рќ‘§)| в‰Є |рќђїрќ‘љ (О©)|(log рќ‘Ѓ )рќђ¶ = рќ‘њ(рќ‘Ѓ )
                         рќ‘›в€€рќђїрќ‘љ (О©) рќђїрќ‘љ (рќ‘§)=рќ‘›

   whenever

                                         |рќђїрќ‘љ (О©)| в‰¤ рќ‘Ѓ (log рќ‘Ѓ )в€’рќђµ
    with рќђµ chosen large.
    This is the bridge supplied below by the BRS component between the actual B1 factorization
origin and the terminal affine/coarea interface.
    вЂ”

ROC.5. Interface Passed to BRS Lemma B1 records that elementary coefficients are polylog-
arithmically bounded per tuple. It does not state a coarea slice-multiplicity theorem for terminal
affine forms рќђїрќ‘љ .
    Lemma BGS records:

  1. parent block;

  2. grouping;

  3. routing history;

  4. current lattice/domain;

  5. current affine forms and their origin types.

   It does not record, for each marked form:

  1. additive height рќ‘‹рќ‘љ ;

  2. additive image length рќ‘Њрќ‘љ = |рќђїрќ‘љ (О©)|;

  3. coarea slice mass                                    в€‘пёЃ
                                             рќ‘Ђрќ‘љ (рќ‘›) =              |рќ‘Љ (рќ‘§)|;
                                                        рќђїрќ‘љ (рќ‘§)=рќ‘›

  4. whether рќ‘Ђрќ‘љ (рќ‘›) is divisor-bounded by the parent product origin or can genuinely have free
     transverse volume.

   The E10Y/E10M/E10K terminal-affine grammar interface forbids untagged rank-dropping affine
maps, but model (3) need not be a rank-drop of the terminal affine span. It is a range/slice-
multiplicity defect.
   Therefore full ROC is supplied in the TC1 route through the BRS augmentation of the terminal
range/slice data.
   вЂ”




                                                     43
ROC.6. Statement: B1 Range Skeleton Lemma               The needed strengthening is:

                                   B1-RANGE-SKELETON.
   Every terminal GoodAWACK skeleton is augmented in BRS with, for every marked form рќђїрќ‘љ :

  1. a height scale рќ‘‹рќ‘љ ;

  2. an image-length scale рќ‘Њрќ‘љ ;

  3. a coarea slice-mass majorant рќ‘†рќ‘љ ;

  4. an origin tag explaining whether рќ‘†рќ‘љ is divisor-bounded or a genuine free-volume direction.

   The BRS theorem asserts:
Lemma 10.2 (Lemma BRS). For every actual B1/F3/F4 terminal TC1-GoodAWACK macro-
template and marked form рќђїрќ‘љ , after C1 boundary removal, one of the following holds:

  1. рќ‘Њрќ‘љ в‰Ґ рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђ¶ , so ROC holds;

  2. рќ‘Њрќ‘љ рќ‘†рќ‘љ = рќ‘њ(рќ‘Ѓ ), so the short image is strict C1 Edge;

  3. the free-volume slice origin exposes LongAP/Local, CKP, LocalDiag, or impossible support;

  4. the singular slice is a tagged quotient/divisor/rank-drop already handled by Lemma TTD.2.

    This lemma proves the BRS component of TC1-SINGULAR-ORIGIN/ROC. It uses the direct-
origin comparability sublemma ROC.1 but not the full ROC closure statement.
    вЂ”

ROC.7. Proof: Closure from BRS           Assume Theorem BRS.1. Then:

  1. direct dyadic-coordinate origins satisfy ROC by Lemma ROC.1;

  2. tagged singular origins route away by Section ROC.2;

  3. complementary/solved affine origins are controlled by BRS: either near-global, Edge by slice
     mass, or routed to D1/G8a/LocalDiag/empty.

   Therefore every singular TC1 testing measure is routed or impossible. Combining with TGT
and TTD gives

                                  рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).
   вЂ”
Remark 10.3 (ROC.8. Output).

ROC reduces the remaining singular-origin case to B1-RANGE-SKELETON/ROC-SLICE, which is supplied by

   What is proved:


        direct dyadic-coordinate origins and controlled full-rank transports satisfy ROC.

                                               44
     Block supplied by BRS:

                               B1-RANGE-SKELETON/ROC-SLICE.
    This is a smaller and more concrete target than pointwise X9L-SI: prove that a terminal marked
affine form with short image cannot carry large hidden transverse B1 multiplicity unless the origin
is already Edge, LongAP/Local, CKP, LocalDiag, or impossible.
    вЂ”

ROC.9. Output for the Proof Tree Lemma BRS proves the BRS block stated above, using
X16BRS/X16C. Therefore the combined status is:

                    TC1-SINGULAR-ORIGIN/ROC is supplied by ROC + BRS.
     The proof tree uses ROC and BRS together.

ROC.10. Logical Dependencies Internal dependencies: B1, BGS, TTD, Theorem BRS.1,
X16BRS, X16C, and the E10Y/E10M/E10K terminal-affine grammar interface. The dependency
is noncircular: BRS uses only the direct-origin sublemma ROC.1 and the tagged-origin routing in
ROC.2, while the full ROC closure is obtained in Sections ROC.7вЂ“ROC.9 after Theorem BRS.1
has been invoked.
    Children served: sublemma ROC.1 serves BRS; the full ROC+BRS closure serves TTH-SC,
TTH, MRT, TGT, TNG, and E10L.


11       Part 10. BRS: B1 range/skeleton ROC slice
Source file: Lemmas/b1_range_skeleton_roc_slice_ltx.md.

11.0.1     BRS. B1 Range/Slice Closure for Singular TC1 Testing
BRS.0. Statement and Role             Lemma BRS proves the structural block isolated in Lemma
ROC:

                               B1-RANGE-SKELETON/ROC-SLICE.
     The point is to rule out the artificial model

                     О© = [рќ‘‹, рќ‘‹ + рќ‘Њ ] Г— [1, рќ‘Ђ ],       рќђїрќ‘љ (рќ‘ў, рќ‘Ј) = рќ‘ў,   рќ‘Њ рќ‘Ђ в‰Ќ рќ‘Ѓ,
   when рќђїрќ‘љ is an actual B1-origin terminal marked form. In the genuine B1 descendant, the
transverse variable is not arbitrary free mass. It is tied to boundedly many finite-convolution
product variables. Restricting the marked carrier to a short additive image therefore cuts the B1
tuple mass by the same relative factor, up to the standard divisor-sum losses already recorded as
X16 in the ledger.
   The result is:


the singular short-image B1-origin residual is Edge unless it already has a LongAP/Local, CKP, LocalDiag, imp




                                                     45
    Thus BRS closes the structural singular-origin branch using the BRS-specific divisor-sum es-
timate X16BRS. Lemma X16BRS reduces the four BRS carrier types to the fixed-depth divisor-
correlation input X16-Core, and Lemma X16C proves X16-Core.
    Equivalently, BRS supplies the routed alternative in Theorem TNG-A: a TC1 coarea test with
a genuinely short B1-origin marked image is routed to strict Edge or to an already handled tagged
class before X9L-GT is invoked.
    Logical dependencies are B1, C1, C1A, F3, F4, TTD, ROC.1, X16BRS, and X16C. BRS is used
by ROC, TTH-SC, TTH, TNG, and E10L.
    вЂ”

BRS.1. Statement: X16-B1 Dyadic Carrier Estimate                                       Let в„¬ be a fixed B1 typed dyadic
block. Its parent variables are

                               рќ‘Ґ1 , . . . , рќ‘Ґрќ‘џ , рќ‘¦1 , . . . , рќ‘¦рќ‘  ,          рќ‘џ, рќ‘  в‰¤ 2рќђЅ0 ,
   with dyadic supports and parent equation

                                            рќ‘ѓрќђґ (рќ‘Ґ) + рќ‘ѓрќђµ (рќ‘¦) = рќ‘Ѓ.
    Let рќђ¶(рќ‘Ґ, рќ‘¦) be a B1 carrier attached to a terminal marked form. It is one of the following,
after a bounded number of controlled CRT restrictions, fixed-divisor quotients, and full-rank affine
coordinate changes:
  1. a grouped product carrier;
  2. a Goldbach complementary carrier рќ‘Ѓ в€’ рќ‘ѓ ;
  3. a quotient carrier рќ‘  occurring in a recorded equation рќђї = рќ‘‘рќ‘ ;
  4. a controlled divisor quotient of one of the previous carriers.
   For quotient carriers, the divisor in рќђї = рќ‘‘рќ‘  is always tagged before BRS is invoked. This is
the quotient-tag completeness statement of F4.9/F4.11: an untagged variable divisor would still
be an unresolved ordinary divisor predicate and could not pass the F3.13 terminal GoodAWACK
labelling step.
   Let рќ‘‹рќђ¶ be its dyadic height and let рќђј be an additive interval. Put

                               рќ‘Њ16 := max{|рќђј в€© Z|, рќ‘‹рќђ¶ (log рќ‘Ѓ )в€’рќђµ16 },
   where рќђµ16 is the X16 slice-floor exponent fixed in the parameter register. Then the total absolute
B1 tuple mass on the subcell

                                                        рќђ¶в€€рќђј
   satisfies

                                                                      рќ‘Њ16
                                                                 (пё‚         )пё‚
                                                            рќђ¶1
                      Massв„¬ (рќђ¶ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )                                 + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶1 ,
                                                                      рќ‘‹рќђ¶
   more precisely,

                                                    рќ‘Њ16
                        Massв„¬ (рќђ¶ в€€ рќђј) в‰Є рќ‘Ѓ (log рќ‘Ѓ )рќђ¶1    + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶1 ,                                (BRS-slice)
                                                    рќ‘‹рќђ¶
   for constants рќђ¶1 , рќњЊ > 0 depending only on рќђЅ0 and the fixed routing architecture.

                                                           46
Proof. This is the B1 form of the divisor-sum input X16. The exact statement is X16BRS; its
carrier reductions are recorded in Lemma X16BRS, while the analytic core is proved in Lemma
X16C.
    For a grouped product carrier, fixing рќђ¶ = рќ‘ђ does not leave a one-variable divisor average. The
remaining variables still contain the same-side complementary product рќ‘ў, and the opposite B1 side
is forced to have product рќ‘Ѓ в€’ рќ‘ђрќ‘ў. Thus the required bound is the fixed-depth correlation
                             в€‘пёЃ                  в€‘пёЃ
                                   рќњЏрќ‘‚(рќђЅ0 ) (рќ‘ђ)         рќњЏрќ‘‚(рќђЅ0 ) (рќ‘ў)рќњЏрќ‘‚(рќђЅ0 ) (рќ‘Ѓ в€’ рќ‘ђрќ‘ў),
                             рќ‘ђв€€рќђј                 рќ‘ўв‰Ќрќ‘€

    with positive support рќ‘Ѓ в€’ рќ‘ђрќ‘ў > 0, not merely рќ‘ђв€€рќђј рќњЏрќ‘‚(рќђЅ0 ) (рќ‘ђ). This is exactly X16-Core, proved
                                                            в€‘пёЂ

in Lemma X16C by Shiu AP divisor averages. It gives the main term proportional to рќ‘Њ16 /рќ‘‹рќђ¶ , plus
a power-saving boundary error. The smooth dyadic weights are handled by partial summation, and
the elementary B1 coefficient types рќњ‡, 1, log cost only (log рќ‘Ѓ )рќђ¶1 .
    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, so
the same estimate applies to the product carrier рќ‘ѓ .
    For a quotient carrier рќђї = рќ‘‘рќ‘ , put рќђ¶ = рќ‘ , рќ‘  в‰Ќ рќ‘‹рќђ¶ , and рќ‘‘ в‰Ќ рќђ·. The restriction рќ‘  в€€ рќђј restricts
the product рќ‘‘рќ‘  to a set whose X16 length has the same ratio рќ‘Њ16 /рќ‘‹рќђ¶ inside its dyadic carrier scale
рќђ·рќ‘‹рќђ¶ . Applying the grouped-product estimate to рќ‘‘рќ‘  gives

                                                   рќ‘Њ16
                                   рќ‘Ѓ (log рќ‘Ѓ )рќђ¶1        + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶1 ,
                                                   рќ‘‹рќђ¶
    which is (BRS-slice). If рќ‘‘ is controlled/fixed, this is just fixed-divisor absorption. If рќ‘‘ varies
over a tagged dyadic divisor family, the divisor boundedness of the B1/F4 coefficient contributes
only an additional (log рќ‘Ѓ )рќ‘‚рќђЅ0 (1) factor, as recorded in Lemma X16BRS. If the variable quotient
equation instead forces local dependence, balanced multiplicative structure, short residual volume,
or an impossible support, F4 routes the atom to LocalDiag, CKP, Edge, or empty before it reaches
terminal GoodAWACK.
    Controlled CRT restrictions and full-rank affine coordinate changes alter the lattice index,
carrier height, and interval length by at most polylogarithmic factors. These losses are absorbed
in (log рќ‘Ѓ )рќђ¶1 . C1 boundary pieces are discarded before the estimate is applied. Lemma proved.
    вЂ”


BRS.2. Proof: Short Image Implies Strict Edge                        Let рќђїрќ‘љ be a terminal TC1-GoodAWACK
marked form with B1 carrier рќђ¶рќ‘љ . Let

                                    рќ‘‹рќ‘љ в‰Ќ рќ‘‹рќђ¶рќ‘љ ,                 рќ‘Њрќ‘љ = |рќђїрќ‘љ (О©)|.
   Assume the marked image is singular:

                                          рќ‘Њрќ‘љ < рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђµ .                                   (SAI)
   Choose

                              рќђµрќњ… > рќђµ16 + рќђ¶0 + рќђ¶1 + рќђ¶16 + рќњЊв€’1
                                                          16 + 20,

    where рќђ¶0 is the C1 saving budget, рќђ¶1 is the internal C1/B1 coefficient loss, рќђµ16 is the X16 slice-
floor exponent, and рќђ¶16 , рќњЊ16 come from X16-BRS as registered in the parameter register. Then



                                                          47
                   Mass(рќђїрќ‘љ (О©)) в‰Є рќ‘Ѓ (log рќ‘Ѓ )в€’рќђ¶0 в€’10 + рќ‘Ѓ 1в€’рќњЊ (log рќ‘Ѓ )рќђ¶1 = рќ‘њ(рќ‘Ѓ ).
   Therefore the singular short-image subcell satisfies the strict C1 short residual volume predicate
E6 and is registered in the C1A admission ledger.

Proof. Apply (BRS-slice) with рќђ¶ = рќђ¶рќ‘љ and рќђј = рќђїрќ‘љ (О©). The singular image condition gives
|рќђј в€© Z|/рќ‘‹рќ‘љ в‰¤ (log рќ‘Ѓ )в€’рќђµрќњ… , after harmless polylogarithmic renormalization of рќ‘‹рќ‘љ . The X16 floor
contributes only (log рќ‘Ѓ )в€’рќђµ16 . The displayed choice of рќђµрќњ… and рќђµ16 makes the first term logarith-
mically saved, while the second term has power saving. This is exactly the strict C1 E6 budget.
Lemma proved.
    вЂ”


Parameter check 11.1 (BRS.3. Parameter Check: Compatibility with Routing Tags). The
previous lemma applies only to terminal GoodAWACK descendants that actually reach the B1
carrier estimate. If the short image is caused by any of the following, the atom does not need BRS:

  1. short residual volume or Type I error budget;

  2. short fixed divisor or short quotient;

  3. forced local dependence or proportionality;

  4. CKP-balanced multiplicative structure;

  5. impossible congruence or support;

  6. tagged rank drop or quotient/divisor origin already present in the routing record.

   These cases are exactly the tagged alternatives of Lemma TTD.2 and the F4 decision tree.
   Thus BRS only handles the previously untagged complementary/solved affine case. In that case
the carrier remains a genuine B1 product or quotient carrier, so BRS.1 applies.
   вЂ”

BRS.4. Output Theorem

Theorem 11.2 (Theorem BRS.1. B1 range/slice dichotomy). For every actual B1/B3/F3/F4
terminal TC1-GoodAWACK macro-template and every marked form рќђїрќ‘љ , after C1 boundary removal
and fixed macro-template normalization, one of the following holds:

  1. рќђїрќ‘љ satisfies range-origin comparability

                                       |рќђїрќ‘љ (О©)| в‰Ґ рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђµ ;

  2. the short-image subcell is strict C1 Edge by BRS.2;

  3. the origin is tagged and routes to LongAP/Local, CKP, LocalDiag, Edge, or empty by F3/F4
     and Lemma TTD.2.




                                                 48
Proof. If рќђїрќ‘љ is a direct dyadic-coordinate origin or a controlled full-rank transport of one, the
direct-origin comparability sublemma ROC.1 gives case 1 unless the image is restricted to a smaller
subcell. In that subcell, BRS.2 gives case 2.
    If рќђїрќ‘љ is a fixed divisor quotient, the carrier scale is changed by a polylogarithmic factor only,
so BRS.1 and BRS.2 apply.
    If рќђїрќ‘љ is a variable quotient residual or complementary solved affine origin, F4 first removes all
short quotient, forced local, balanced CKP, and impossible cases. If any such tag is present, we are
in case 3. Otherwise the quotient/complement carrier remains an actual B1 product carrier with
controlled content. BRS.1 applies, and a failure of range comparability gives case 2.
    Finally, the E10Y/E10M/E10K terminal-affine grammar interface excludes arbitrary untagged
rank-dropping affine regrouping. Full-rank affine transports preserve BRS up to polylogarithmic
loss; rank drops carry one of the tags already covered by case 3. These cases exhaust the B1-to-
GoodAWACK skeleton. Theorem proved.
    вЂ”


BRS.5. Output for Singular TC1 Testing             Combining Theorem BRS.1 with the direct-origin
and tagged-origin parts of ROC gives

                                 TC1-SINGULAR-ORIGIN/ROC.
    Indeed, a singular testing measure is precisely a concentration on marked forms whose image
fails near-global range comparability. By BRS.1, such a failure is either strict C1 Edge or an existing
routing tag. Hence it cannot remain as an untagged terminal TC1-GoodAWACK contribution.
    Together with Lemma TTD, the singular branch of the TC1 global-testing route is closed. The
MRT-admissible branch is still the branch handled by TGT using the averaged Liouville input
X9L-GT.
    вЂ”
Remark 11.3 (BRS.6. Output).

           B1-RANGE-SKELETON/ROC-SLICE is proved using X16-BRS/X16-Core.

   This does not prove a pointwise shifted short-interval theorem for рќњ†. It shows that the only
TC1 situations where such a pointwise theorem appeared to be needed are not genuine terminal
B1-origin GoodAWACK mass: short image mass is Edge after the B1 carrier slice estimate, and all
non-Edge failures carry existing routing tags.
   The structural reduction in BRS is separate from the analytic carrier-slice input. The analytic
input is discharged by Lemma X16C.

BRS.7. Logical Dependencies Internal dependencies: B1, C1, C1A, F3, F4, TTD, the direct-
origin sublemma ROC.1, X16BRS, and X16C. BRS does not depend on the full ROC closure
theorem; full ROC is obtained only after BRS is combined with ROC.1 and the tagged-origin
routing part of ROC.
    Children served: ROC, TTH-SC, TTH, MRT, TGT, and TNG.


12     Part 11. TTH: Near-global length theorem
Source file: Lemmas/tc1_theta_1_3_ltx.md.

                                                  49
12.0.1   TTH. Internal Length Lower Bound for B1-Origin TC1 Coarea Tests
TTH.0. Statement and Role          Lemma TTH proves the internal bypass

                                        TC1-THETA-1/3
   in the form needed by the TGT route.
   The conclusion is:


every unrouted B1-origin coarea test in the TC1 testing family has рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµрќњ… and hence рќђ» в‰Ґ рќ‘‹ 1/3+рќњЂрќњ… .

   Consequently the low-theta external input

                                   X9L-POLYLOG-MOD<1/3
    is not needed for the coarea-normalized TC1 route. The near-global Davenport/AP input X9L-
GT applies.
    TTH is not an independent analytic estimate. It is a structural consequence of BRS. The X16-
BRS/X16-Core input is supplied by Lemmas X16BRS and X16C; the parameter consequences are
recorded in the parameter register.
    In the TC1 proof TTH is used through Theorem TNG-A, the near-global-or-routed theorem:
TTH supplies the near-global alternative, while BRS/X16BRS/X16C route the complementary
short-image alternatives away before X9L-GT is invoked.
    Here "unrouted" means that the cell has not already been sent to Edge, LongAP/Local, CKP,
LocalDiag, or empty support by the preceding routing lemmas. Logical dependencies are the
TGT.2/TGT-MF coarea-test construction, TTH-SC, TTD, ROC, BRS, MRT, E5, X16BRS, X16C,
and the parameter register. TTH is used by the full TGT closure statement, TNG, and E10L; the
external input X9L-GT is invoked only downstream, after TTH has supplied the near-global length
lower bound.
    вЂ”

TTH.1. Scope Restriction         The following stronger statement is outside the scope of Lemma
TTH:

                       every possible E7 directional fibre has рќђ» в‰Ґ рќ‘‹ 1/3+рќњЂ .
     That statement is too strong at the level of the abstract E7 interface. A box may have a short
but long-enough direction рќ‘€ = рќ‘Ѓ рќњѓ and a transverse base sweeping many starts. Such a directional
slicing can be MRT-admissible while still having рќ‘€ < рќ‘‹ 1/3 .
     The proof after TGT and the BRS/ROC reductions does not need that stronger E7-fibre state-
ment. It uses the coarea tests produced from the marked affine image

                                           рќ‘› = рќђїрќ‘љ (рќ‘§),
  not an arbitrary directional fibre selected before coarea.
  Thus TC1-THETA-1/3 is proved below for the coarea testing family that is actually fed into
X9L-GT.
  вЂ”



                                                50
TTH.2. Setup: Why B1-Origin Coarea Tests Are the Relevant Tests                       The quantifier
"B1-origin coarea test" is sufficient for the proof route for the following reason.
   In TGT, a fixed TC1 macro-template рќњ… fixes:

  1. the B1 typed parent pattern;

  2. the B3 grouping skeleton;

  3. the F3/F4 routing grammar;

  4. the marked Liouville origin;

  5. the affine coefficient transport type;

  6. the TC1 tensor certificate.

   The tests used in the averaged Liouville input are then produced in TGT.3 by Fourier/coarea
decomposition along the same marked form:

                                              рќ‘› = рќђїрќ‘љ,рќ‘— (рќ‘§).
    Thus the Liouville argument in every unrouted test is still the terminal marked B1-origin car-
rier, possibly after controlled CRT restriction, fixed divisor quotienting, full-rank transport, and
Cauchy/cube/Fourier post-terminal operations. These post-terminal operations do not create a
new non-B1 Liouville carrier:

  1. Lemma E5 preserves controlled content under Cauchy/cube operations;

  2. the TGD/TGT terminal-interface lemma treats Cauchy/cube operations, primitive slicing,
     and Fourier expansion as post-terminal analytic operations, not new terminal origins;

  3. if a post-terminal operation creates a collision, rank drop, local dependence, CKP structure,
     Edge piece, or impossible support, that piece is routed away before entering the TC1 testing
     family to which X9L-GT is applied.

    Therefore every unrouted test to which X9L-GT is applied has a B1 carrier in the sense required
by Lemma BRS.
    The exclusion of arbitrary post hoc short-interval refinements is not a convention. It is the
closure principle TTH-SC:


    a short subtest is either non-structural and reaggregated, or structural and routed away.

   Thus the only tests released to X9L-GT are structural TGT-MF coarea image pieces, after
the controlled polylogarithmic scale/modulus/smoothness decomposition needed to normalize the
weights.
   вЂ”




                                                   51
TTH.3. Setup Fix a TC1 macro-template рќњ… and an actual terminal B1/B3/F3/F4 GoodAWACK
atom in the Branch B route, after:

  1. C1 boundary and strict Edge pieces have been removed;

  2. LongAP/Local pieces have been passed to D1/H4;

  3. CKP pieces have been passed to G8a;

  4. LocalDiag pieces have been passed to H4;

  5. impossible or empty routing tags have been discarded.

   Let

                                                 рќђїрќ‘љ (рќ‘§)
     be the marked Liouville affine form. Let О©* denote the C1-clean smooth box/coset cell on which
the coarea test is taken. This may be the original terminal cell or a post-WGVN/Fourier subcell
О©вЂІрќ‘— , but it is still a subcell of the same B1-origin carrier and has the same marked Liouville origin.
Let

                                              рќђјрќ‘љ = рќђїрќ‘љ (О©* )
   be its marked affine image on this cell.
   Write

                          рќ‘Њрќ‘љ := |рќђјрќ‘љ |,      рќ‘‹рќ‘љ в‰Ќ max(2, dist(рќђјрќ‘љ , 0) + рќ‘Њрќ‘љ )
   for the image length and height.
   The coarea testing step of TGT produces tests
                                            1 в€‘пёЃ
                                 в„’рќ‘ќ (рќњ†) =          рќњ†(рќ‘›)рќњЊрќ‘ќ (рќ‘›)рќ‘’(рќ›јрќ‘ќ рќ‘›),
                                            рќђ»рќ‘ќ рќ‘›в€€рќђј
                                                   рќ‘ќ

   where рќђјрќ‘ќ is a coarea image interval or AP image piece of рќђїрќ‘љ on О©* , with polylogarithmic
content/modulus and polylogarithmic smooth partition losses.
   After fixing one scale/modulus/weight-complexity class, TTH-SC gives

                         рќђ»рќ‘ќ в‰«рќњ… рќ‘Њрќ‘љ (log рќ‘Ѓ )в€’рќђ¶рќњ… ,         рќ‘‹рќ‘ќ в‰Ќрќњ… рќ‘‹рќ‘љ (log рќ‘Ѓ )рќ‘‚рќњ… (1) .                  (1)
    This is the controlled-structural-refinement output of TTH-SC. Pieces shorter than this are not
released X9L inputs: if they are non-structural, TTH-SC reaggregates them into the parent coarea
piece; if they are structural, TTH-SC routes them through TTD/ROC/BRS/X16BRS/X16C and
C1P/C1A/C1 before X9L-GT is invoked.
    вЂ”




                                                   52
TTH.4. Proof: BRS Gives Near-Global Image for Every Unrouted Test Theorem
BRS.1 says that for every actual B1/B3/F3/F4 terminal TC1-GoodAWACK macro-template and
every marked form рќђїрќ‘љ , after C1 boundary removal and fixed macro-template normalization, the
B1 carrier slice estimate applies to any surviving carrier subcell. Therefore, applied to the interval
рќђјрќ‘љ = рќђїрќ‘љ (О©* ), one of the following holds:

  1. range-origin comparability:
                                                   рќ‘Њрќ‘љ в‰Ґ рќ‘‹рќ‘љ (log рќ‘‹рќ‘љ )в€’рќђµрќњ… ;                                    (ROC)

  2. the short-image subcell is strict C1 Edge;

  3. the origin is tagged and routes to LongAP/Local, CKP, LocalDiag, Edge, or empty.

     In the TC1 coarea testing family to which X9L-GT is applied, cases 2 and 3 have already
been removed by the routing assumptions in TTH.2. Therefore every remaining test comes from a
marked image satisfying (ROC).
     Combining (ROC) with (1), and absorbing all polylogarithmic distortions into a larger exponent
рќђµрќњ…вЂІ , gives
                                                                     вЂІ
                                                 рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… .                                             (2)
   This is the near-global lower bound needed by the Davenport/AP X9L input. We record it as
the TTH conclusion:
                                                                     вЂІ
                                                 рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ…                                        (TTH)

    The exponent рќђµрќњ…вЂІ is chosen after the BRS/X16 constants. In the notation of the parameter
register, it must dominate
                               рќђµ16 + рќђ¶0 + рќђ¶1 + рќђ¶16 + рќњЊв€’1
                                                      16 + 20.

Thus the near-global conclusion uses the X16-Core constants fixed in Lemma X16C.
   It is also stronger than a 1/3-power lower bound.
   вЂ”

Parameter check 12.1 (TTH.5. Parameter Check: No Hidden Short-Fibre Quantifier). The
proof above uses the following exact quantifier structure.


   Object                                  Allowed to enter X9L?            Reason
   Structural TGT coarea image of the      Yes, after TTH                   BRS proves near-global length unless
   marked B1 carrier                                                        the mass is Edge/tagged.
   Polylogarithmic                         Yes                              It loses only a fixed power of log рќ‘‹,
   AP/modulus/smoothness          subdi-                                    absorbed into рќђµрќњ… .
   vision of that image
   Artificial subdivision into arbitrary   No                               TTH-SC classifies it as non-
   shifted short intervals                                                  structural and reaggregates it.
   Genuine structural short-image child    No                               TTH-SC       routes    it    through
                                                                            TTD/ROC/BRS/X16BRS/X16C
                                                                            and C1P/C1A/C1 before X9L.
   Singular start concentration            No                               TTD/ROC/BRS routes it before
                                                                            X9L.
   Unresolved quotient/divisor origin      No                               F4 must tag or route it before TTH
                                                                            is invoked.




                                                          53
   Therefore the proof never needs the statement
                                          вѓ’             вѓ’
                                          вѓ’ в€‘пёЃ          вѓ’
                              sup              рќњ†(рќ‘›)рќ‘’(рќ›јрќ‘›)вѓ’ = рќ‘њ(|рќђј|)   (рќњѓ < 1/3),
                                          вѓ’             вѓ’
                                          вѓ’
                        рќђјвЉ‚[рќ‘‹,2рќ‘‹], |рќђј|=рќ‘‹ рќњѓ вѓ’             вѓ’
                                        рќ‘›в€€рќђј

    nor any polylog-modulus analogue for arbitrary short shifted intervals. The only Liouville input
is the near-global averaged AP form after the B1-origin coarea normalization.
    вЂ”

TTH.6. Output: The 1/3 Lower Bound                 Choose any fixed
                                                  2
                                          0 < рќњЂрќњ… < .
                                                  3
   For definiteness one may take рќњЂрќњ… = 1/6. Since
                                                     вЂІ
                                     рќ‘‹рќ‘ќ (log рќ‘‹рќ‘ќ )в€’рќђµрќњ… в‰Ґ рќ‘‹рќ‘ќ1/3+рќњЂрќњ…
   for all sufficiently large рќ‘‹рќ‘ќ , (TTH) implies

                                            рќђ»рќ‘ќ в‰Ґ рќ‘‹рќ‘ќ1/3+рќњЂрќњ…

    for every unrouted coarea test рќ‘ќ in the TC1 testing family, outside the already routed C1/tagged
pieces.
    Small bounded рќ‘‹рќ‘ќ values are harmless and can be absorbed into the finite initial range of the
final sufficiently-large-рќ‘Ѓ theorem.
    вЂ”

TTH.7. Output for X9L-GT X9L-GT supplies the normalized polylog-modulus averaged AP-
fibre/Fourier input for the TC1 range. Its cited form uses the near-global Davenport/AP input
whenever

                                         рќђ» в‰Ґ рќ‘‹(log рќ‘‹)в€’рќђµ .
    By (TTH), every unrouted B1-origin coarea test produced by the TC1 global-testing route lies
in the near-global range. Therefore the low-theta residual

                                    X9L-POLYLOG-MOD<1/3
   is bypassed for the coarea-normalized TC1 branch.
   Combining:

  1. TGT;

  2. TTD;

  3. TTH-SC;

  4. ROC;

  5. BRS;

  6. X9L-GT in the near-global Davenport/AP range;

                                                   54
  7. the present length lemma;

   gives

                                  рќ‘…TC1-GoodAWACK (рќ‘Ѓ ) = рќ‘њ(рќ‘Ѓ ).
    No pointwise shifted short-interval theorem and no low-theta polylog-modulus external theorem
are used in this route.
    вЂ”
Remark 12.2 (TTH.8. Output).

           TC1-THETA-1/3 is proved for the unrouted B1-origin coarea testing family.

    The proof is not an independent analytic estimate. It is a structural length consequence of the
BRS/ROC slice theorem: an actual terminal TC1 marked image is either near-global relative to
its B1 carrier height, or it has already left the GoodAWACK branch.

TTH.9. Logical Dependencies Internal dependencies: the TGT.2/TGT-MF coarea-test con-
struction, TTH-SC, TTD, ROC, BRS, MRT, E5, X16BRS, X16C, and the parameter register.
    External dependency: none for the structural length lower bound. X9L-GT is used only down-
stream after the near-global range has been established.
    Children served: TGT, TNG, and E10L.




                                                55

