﻿         TNG/TC1 No Rogue Short Interval Theorem Package

                                           Denis Saltykov

                                              May 2026


                                                 Abstract
        This note isolates the TC1 global-testing node from the full proof package. It proves, as
     a standalone structural theorem, that every actual B1-origin TC1 coarea test reaching the
     Liouville/Fourier input is near-global, with
                                           H в‰Ґ X(log X)в€’B ,
     and hence is covered by the Davenport/AP input X9L-GT. If a TC1 test is not near-global, it is
     routed away before X9L-GT is invoked. In particular, the proof never assumes pointwise shifted
     short-interval cancellation for О».


Contents
1 Scope                                                                                               1

2 Structural TC1 Coarea Tests                                                                         2

3 Structural Coarea Closure                                                                           2

4 Regular Branch                                                                                      3

5 Singular Branch                                                                                     3

6 The Near-Global-or-Routed Theorem                                                                   4

7 Decision Table                                                                                      4

8 TC1 Cancellation Output                                                                             5

9 Parameter Check                                                                                     5

10 Output for the Full Proof                                                                          6

11 Logical Dependencies                                                                               6


1   Scope
The package concerns the TC1 part of the GoodAWACK branch. Its logical task is to justify the
implication
        actual B1-origin TC1 test =в‡’ near-global X9L input or routed terminal output.

                                                   1
The package does not estimate Edge, LongAP/Local, CKP, or LocalDiag branches. Those branches
are merely the permitted outputs of the routed alternative.
    The external analytic input is only X9L-GT in its near-global Davenport/AP form: for polylog-
arithmic AP modulus and smoothness complexity, normalized Liouville/AP Fourier averages are
o(1) when
                                         H в‰Ґ X(log X)в€’B .
No arbitrary shifted short-interval theorem and no unused low-Оё polylog-modulus theorem is used.


2     Structural TC1 Coarea Tests
Fix a terminal TC1-GoodAWACK macro-template Оє. The data consist of:

    1. a B1 typed parent block;

    2. a B3 grouping record;

    3. an F3/F4 terminal routing history;

    4. a marked Liouville affine form Lm ;

    5. a TC1 tensor certificate;

    6. a C1-clean smooth box/coset cell в„¦в€— .

    TGT constructs coarea tests from the marked B1-origin form by

                                        n = Lm (z),         z в€€ в„¦в€— .

After only polylogarithmic scale, modulus, and smooth-weight subdivisions, a test has the form
                                              1 X
                                   Lp (О») =          О»(n)ПЃp (n)e(О±p n),                            (2.1)
                                              Hp nв€€I
                                                    p


where Ip is the structural image interval or AP image, Hp = |Ip |, the AP modulus is в‰¤ (log Xp )CОє ,
and ПЃp has polylogarithmic smoothness complexity.
   The word active means that the cell has not already been routed to Edge, LongAP/Local, CKP,
LocalDiag, or empty support.


3     Structural Coarea Closure
The interval Ip in (2.1) is not an arbitrary subinterval chosen after a larger structural image has been
found. It is the image piece produced by the TGT coarea construction from the terminal marked
B1 carrier, after only the normalizations needed for scale, modulus, and smoothness complexity.
    The formal closure principle is TTH-SC. It proves that every short subtest of a released coarea
test is either non-structural and reaggregated into the parent functional, or structural and routed
through TTD/ROC/BRS/X16BRS/X16C and C1P/C1A/C1 before the Liouville input is invoked.




                                                        2
4     Regular Branch
Assume first that the TC1 testing family is MRT-admissible. MRT supplies the start-pushforward
bound
                                                             dx
                                  (start)# ОЅОє в‰ЄОє (log N )CОє .                          (PACK)
                                                             X
For every active B1-origin coarea test in this same family, TTH supplies the near-global length
lower bound
                                      Hp в‰Ґ Xp (log Xp )в€’BОє .                               (4.1)
Together with the polylogarithmic AP modulus and smoothness bounds in Section 2, these are
exactly the hypotheses of X9L-GT in the near-global Davenport/AP form. X9L-GT gives
                                                               2
                              1 X
                      Z
                          sup         О»(gp u + bp )e(О±u)wp (u) dОЅОє (p) = o(1),                (4.2)
                           О± Up
                                1в‰¤uв‰¤U  p


where Up = Hp /gp .
    TGT says that a non-negligible TC1 macro-template would force a fixed positive lower bound
for the same averaged Fourier quantity. Hence the regular MRT-admissible branch contributes
o(N ).


5     Singular Branch
If MRT-admissibility fails, TTD identifies singular start concentration before any X9L input is
invoked. The model obstruction is a marked form moving through a short additive image while
transverse B1 variables appear to carry large volume.
   This singular geometry is structural, not analytic. It is handled as follows.

    1. ROC proves range-origin comparability for direct dyadic-coordinate origins and their con-
       trolled full-rank transports.

    2. ROC routes already tagged singular origins to Edge, LongAP/Local, CKP, LocalDiag, or
       empty support.

    3. BRS handles the complementary and quotient-origin cases. It uses the B1 range/slice estimate
       supplied by X16BRS and X16C.

    The BRS conclusion is:

                              short marked B1 image =в‡’ strict C1 Edge

unless the atom already carries a LongAP/Local, CKP, LocalDiag, Edge, empty, or nonterminal
routing tag.
   Consequently the singular branch is never sent to X9L-GT. It routes to

                          C1P/C1A/C1,          D1/H4,       G8a,   H4,    or 0.                   (5.1)




                                                    3
6     The Near-Global-or-Routed Theorem
Theorem 1 (TNG-A. TC1 tests are near-global or routed away). Fix a B1/B3/F3/F4 terminal
TC1-GoodAWACK macro-template Оє whose cell has not already been routed away, after C1 bound-
ary removal, fixed macro-template normalization, and polylogarithmic scale/modulus/smooth-weight
decomposition. Let Lp (О») be any active coarea test produced by TGT from the marked B1-origin
form.
   Then exactly one of the following alternatives holds.

    1. Near-global testing alternative. The test belongs to the regular MRT-admissible branch.
       PACK holds, the AP modulus and smoothness complexity are polylogarithmic, and

                                            Hp в‰Ґ Xp (log Xp )в€’BОє .

      Hence this test is an allowed input to X9L-GT in the near-global Davenport/AP form.

    2. Routed alternative. The test is not sent to X9L-GT. Before any Liouville/AP input is in-
       voked, TTD, ROC, BRS, and X16BRS/X16C route the corresponding cell to one of

                            C1P/C1A/C1,         D1/H4,        G8a,       H4,      or 0.

In particular, there is no third case consisting of an arbitrary shifted short interval or an unclassified
short AP fibre.

Proof. Start with the coarea tests constructed by TGT from the fixed macro-template Оє. MRT
separates regular testing families from singular start-concentration families.
    In the regular branch, MRT supplies PACK for the same testing family. The coarea tests retain
B1-origin because the allowed normalizations are only controlled CRT restrictions, fixed-divisor
quotients, full-rank transports, and polylogarithmic analytic subdivisions. These operations do
not replace the terminal marked B1 carrier. TTH-SC prevents replacement by a new arbitrary
short-interval family. TTH then gives the near-global length lower bound for every remaining
structural coarea image piece. The modulus and smoothness complexity bounds are polylogarithmic
by construction. Therefore the test satisfies the exact hypotheses of X9L-GT.
    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, quotient, and
carrier-slice cases are handled by BRS. In BRS, a genuinely short marked B1 image cannot hide
large 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 classifies refinements of an already released structural image. Non-structural
short pieces are aggregated back to the parent image, while genuine structural short-image children
are routed before X9L-GT. This excludes the pointwise shifted short-interval escape case. The
theorem follows.


7     Decision Table
For checking purposes, the theorem can be read as the following finite decision table.



                                                    4
 Test status after TGT         Structural source           Action before X9L        Result
 coarea
 Full-rank B1 marked car-      B1/B3/F3/F4 carrier         MRT + TTH-SC +           Near-global; X9L-GT
 rier, no short image                                      TTH                      may be invoked
 Direct B1 carrier with        same                        BRS + X16BRS/X16C        Strict C1 Edge
 short marked image
 Complementary carrier         F4/BRS      solved-affine   Reduce to product car-   Near-global or strict
 N в€’P                          origin                      rier P                   Edge
 Quotient carrier s in L =     F4 quotient tag             Transfer to controlled   Near-global or strict
 ds with tagged d                                          product carrier          Edge
 Untagged               quo-   unresolved F4 predicate     F4 blocks terminal TC1   Routed before X9L
 tient/divisor relation                                    release
 Singular start measure        TTD singular branch         ROC/BRS                  Routed before X9L
 from non-direct origin
 Artificial subdivision of     not structural      TGT     aggregate back           no X9L short-interval
 near-global image             coarea                                               input

    The only tests passed to X9L-GT are the first-row near-global tests.


8     TC1 Cancellation Output
Theorem 2 (TNG. TC1-GoodAWACK cancellation).
For every terminal TC1-GoodAWACK macro-template Оє, after C1 boundary removal and fixed
macro-template normalization,

                                      RTC1-GoodAWACK,Оє (N ) = o(N ).

Consequently
                                      RTC1-GoodAWACK (N ) = o(N ).

Proof. Aggregate terminal TC1 atoms by the fixed macro-template Оє as in TGT. Apply Theorem
TNG-A to every active coarea test.
   On the near-global alternative, MRT supplies PACK and TTH supplies the near-global length
lower bound. X9L-GT applies. This contradicts the fixed TGT testing lower bound unless the
contribution of Оє is o(N ).
   On the routed alternative, the cell is sent to Edge, LongAP/Local, CKP, LocalDiag, or empty
support by TTD/ROC/BRS. These outputs are handled by C1P/C1A/C1, D1/H4, G8a, H4, or zero,
and therefore do not contribute terminal TC1-GoodAWACK mass.
   There are only boundedly many structural TC1 macro-templates, depending on the fixed HeathвЂ“
Brown depth J0 . Summing over them gives the displayed o(N ) estimate.


9     Parameter Check
The parameter hierarchy required by the theorem is finite.

    1. The HeathвЂ“Brown depth J0 is fixed.

    2. The routing grammar and TC1 macro-template set are finite for this J0 .

    3. All CRT indices, divisor quotients, affine contents, AP moduli, and smoothness complexities
       reaching X9L are bounded by fixed powers of log N .

                                                      5
  4. The BRS short-image threshold BОє is chosen larger than the X16 slice-floor and C1 error-
     budget losses.

  5. X9L-GT then chooses the Davenport logarithmic saving exponent larger than the remaining
     polylogarithmic losses.

Thus the routed alternatives have o(N ) error budget, while the near-global alternative has o(1)
normalized TC1 testing average and hence o(N ) contribution.


10    Output for the Full Proof
This package proves the TC1 structural theorem

                      TNG-A + X9L-GT =в‡’ RTC1-GoodAWACK (N ) = o(N ).

It uses X9L-GT only in the near-global Davenport/AP form. It does not use:

  1. the old X8 inverse-Gowers input;

  2. pointwise shifted short-interval Liouville cancellation;

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


11    Logical Dependencies
External dependency: X9L-GT in the near-global Davenport/AP form.
   Internal dependencies: TGT, TTH-SC, MRT, TTD, ROC, BRS, TTH; X16BRS and X16C;
C1P/C1A/C1, D1/H4, G8a, E5, TGD; and the global parameter register.
   Children served: E10L and the TC1 part of the GoodAWACK branch.




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