> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# SRBRD-21
## Fixed-$a$ Spectral Dependency / Goldbach-Shift Uniformization Audit
### Bezout Small-Phase Breakdown, Spectral-Frequency Inflation, Naive Residue Reduction Barrier and Minimal Replacement Lemma

**Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program  
**Branch:** SRBRD — Symmetric Root-Bounded Relation Domain  
**Round:** SRBRD-21  
**Date:** 2026-09-30  
**Version:** v0.1  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Type:** Proof-Dependency Audit / Spectral-Uniformity Barrier / Low-Row Replacement Target  
**Prerequisites:** SRBRD-00 through SRBRD-20  
**Principal external sources:** Maynard I / II / III large-moduli trilogy  
**Status:** Merely tracking the fixed-$a$ implied constant has been proved insufficient to uniformize Maynard-II low-row Type-II; three explicit fixed-$a$ proof breakpoints have been found; Goldbach remains unproved

---

# 0. The Question of This Round

SRBRD-20 compressed the external analytic gap into the low row:

$$
\boxed{
X^{1/3+\eta}
\lesssim
N
\lesssim
X^{2/5-\eta}.
}
$$

Maynard-II Proposition 5.1 / Proposition 8.2：

- The variable range matches;
- Prime coefficients can be placed in its Siegel–Walfisz / divisor-bounded framework;
- The well-factorable modulus technology is very strong;

but its residue:

$$
a
$$

is fixed.

SRBRD requires:

$$
\boxed{
a_X=2m=X+3\asymp X.
}
$$

The previous round left two possibilities:

### Optimistic route

Perhaps the published proof merely did not track the constant for:

$$
a
$$

as displayed above,

while actually:

$$
C(a_X)\le X^{o(1)}.
$$

### Hard route

The proof genuinely uses:

$$
|a|\ll X^\theta
$$

or even:

$$
|a|=O(1),
$$

and:

$$
a_X\asymp X
$$

destroys a key estimate.

After auditing each dependency, this round obtains:

$$
\boxed{
\text{Hard route is real.}
}
$$

---

# 1. External Confirmation from Maynard-I

Maynard-I's introduction explicitly states:

Deshouillers–Iwaniec / Kuznetsov-type Kloosterman estimates restrict it to:

$$
\boxed{
a\text{ fixed}
}
$$

or:

$$
\boxed{
a\text{ a small power of }X.
}
$$

This is not merely bookkeeping in the theorem statement.

The subsequent proof indeed absorbs the absolute size of:

$$
a
$$

into a small error or spectral variable range at multiple points.

The Goldbach shift:

$$
a_X=X+3
$$

has exponent:

$$
1,
$$

and is outside the small-power regime.

---

# 2. Maynard-II Fixed-$a$ Declaration

Maynard-II's notation in §4 states explicitly:

$$
\boxed{
a
\text{ throughout the paper is a fixed positive integer}
}
$$

and all asymptotic implied constants are allowed to depend on:

$$
a.
$$

Thus the form of Proposition 8.2:

$$
\ll_{a,A,\varepsilon}
\frac{X}{(\log X)^A}
$$

cannot itself be applied directly to:

$$
a_X=X+3
$$

as displayed above.

But this round focuses not on that declaration,

but on finding the actual dependence inside the proof.

---

# 3. Dependency Graph

The Maynard-II low-row route can be simplified as:

$$
\boxed{
\text{Prop 5.1}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Prop 8.2}
}
$$

Prop 8.2 is split via:

$$
\boxed{
\text{Prop 6.8}
}
$$

into:

### E1 channel

$$
\boxed{
\text{Lemma 6.10}
}
$$

Its source:

$$
\boxed{
\text{Maynard-I Lemma 18.3}.
}
$$

### E2 channel

$$
\boxed{
\text{Lemma 6.9}
}
$$

Its source:

$$
\boxed{
\text{Maynard-I Lemma 14.5},
}
$$

then reduced to:

$$
\boxed{
\text{Maynard-II Lemma 8.1}
}
$$

and finally uses:

$$
\boxed{
\text{Deshouillers–Iwaniec spectral estimate}.
}
$$

This round finds explicit fixed-$a$ breakpoints at three locations.

---

# 4. Breakpoint A — Bezout Small-Phase Approximation in Lemma 6.10

Maynard-II Lemma 6.10 directly cites Maynard-I Lemma 18.3.

Its core phase:

$$
e
\left(
\frac{
a h\overline{n_1}
}{q}
\right).
$$

Bezout reciprocity gives:

$$
\boxed{
\frac{
a h\overline{n_1}
}{q}
\equiv
-
\frac{
a h\overline q
}{n_1}
+
\frac{
a h
}{
n_1q
}
\pmod1.
}
$$

---

# 5. Size of the Dropped Fraction

In Maynard-I:

$$
|h|
\ll
\frac{
QN(\log X)^5
}{X}.
$$

while:

$$
n_1\asymp N,
\qquad
q\asymp Q.
$$

Thus:

## Theorem SRBRD-21-T01

$$
\boxed{
\left|
\frac{
a h
}{
n_1q
}
\right|
\ll
\frac{
|a|(\log X)^5
}{X}.
}
$$

Maynard fixes:

$$
a
$$

as displayed above,

and therefore writes:

$$
O
\left(
\frac{
(\log X)^5
}{X}
\right).
$$

---

# 6. Goldbach Shift Breaks T01 Approximation

For:

$$
a_X=X+3,
$$

T01 becomes:

$$
\boxed{
\left|
\frac{
a_X h
}{
n_1q
}
\right|
\ll
(\log X)^5.
}
$$

This does not tend to:

$$
0.
$$

Thus one can no longer use:

$$
e(\theta+\delta)
=
e(\theta)
+
O(|\delta|)
$$

to treat the final fraction as a negligible perturbation.

Thus:

## Verdict A

$$
\boxed{
\text{Maynard-I Lemma 18.3 / Maynard-II Lemma 6.10
does not uniformize to }a_X\asymp X
\text{ by tracking constants alone.}
}
$$

---

# 7. Breakpoint B — Bezout Simplification in Lemma 6.9

Maynard-II Lemma 6.9 cites Maynard-I Lemma 14.5.

Its phase simplification likewise uses Bezout reciprocity.

schematically：

$$
\boxed{
e
\left(
\frac{
a h\overline{n_1}r_2
}{
q r_1
}
+
\cdots
\right)
}
$$

is replaced by a simplified reciprocal phase,

and a residual fraction is discarded.

---

# 8. Lemma 6.9 Dropped Fraction Scale

The Maynard-I proof explicitly states:

Since:

$$
|h|
\le
H
=
\frac{
QR(\log X)^5
}{M},
$$

and:

$$
NM\asymp X,
$$

the final fraction, for fixed-$a$, is:

$$
O
\left(
\frac{
(\log X)^5
}{X}
\right)
$$

or, in the next step:

$$
O
\left(
\frac{
(\log X)^6
}{X}
\right).
$$

Restoring the hidden $a$:

## Theorem SRBRD-21-T02

the small-phase error has scale:

$$
\boxed{
O
\left(
\frac{
|a|(\log X)^{O(1)}
}{X}
\right).
}
$$

Thus:

$$
a_X\asymp X
$$

likewise loses smallness.

---

# 9. Verdict B

$$
\boxed{
\text{Maynard-II Lemma 6.9
also uses fixed }a
\text{ at the phase-simplification level.}
}
$$

Thus:

> Even if Lemma 8.1 is uniformized separately,
> Prop 8.2's upstream simplification still requires replacement.

---

# 10. Breakpoint C — Spectral Frequency Inflation in Lemma 8.1

After the off-diagonal square expansion in Maynard-II Lemma 8.1,

Define:

$$
\boxed{
\ell
=
h_1s_2-h_2s_1.
}
$$

Nonzero diagonal:

$$
\ell\neq0.
$$

a new spectral numerator variable is then defined:

$$
\boxed{
y
=
a f\ell.
}
$$

and:

$$
\boxed{
z=s_1s_2.
}
$$

---

# 11. Fixed-$a$ Frequency Range

For fixed-$a$, Maynard writes:

$$
\boxed{
Y
\ll
X^{o(1)}
\frac{
NR^2S^3
}{M}.
}
$$

But this actually absorbs:

$$
|a|.
$$

Restoring the dependency:

## Theorem SRBRD-21-T03

$$
\boxed{
Y_a
\ll
|a|
X^{o(1)}
\frac{
NR^2S^3
}{M}.
}
$$

---

# 12. Goldbach Frequency Inflation

For:

$$
a_X=X+3,
$$

$$
\boxed{
Y_{a_X}
\asymp
X
\times
Y_{\mathrm{fixed}}
}
$$

adds a full factor on the exponent scale:

$$
X^1.
$$

This is not a:

- logarithmic loss；
- divisor-function loss；
- $X^{o(1)}$ loss。

---

# 13. Deshouillers–Iwaniec Bound Depends on $Y$

Lemma 8.1 ultimately uses Maynard-II Lemma 6.12,

whose bound's:

$$
J
$$

explicitly contains:

$$
Y.
$$

The proof writes:

$$
\boxed{
J^2
\ll
C(Z+Y)(C+BZ)
+
C^2B\sqrt{(Z+Y)Z}
+
B^2YZ.
}
$$

Therefore:

$$
\boxed{
Y
}
$$

is an analytic conductor / spectral-range parameter,

not something appearing only in a unit-modulus coefficient.

---

# 14. A Second Use of Fixed-$a$: $Z>Y$

Maynard then uses the original parameter inequalities to obtain:

$$
\boxed{
Z>Y.
}
$$

and greatly simplifies:

$$
J.
$$

But, after restoring:

$$
a
$$

we have:

$$
\boxed{
\frac{Y_a}{Z}
\asymp
|a|
\frac{
NR^2S
}{M}
X^{o(1)}.
}
$$

The original:

$$
M>R^2SN
$$

is sufficient to control only fixed:

$$
a.
$$

For:

$$
a_X\asymp X,
$$

cannot imply:

$$
Z>Y_{a_X}.
$$

---

# 15. Verdict C

$$
\boxed{
\text{Lemma 8.1 spectral reduction has a genuine
absolute-size }a\text{ barrier.}
}
$$

Thus:

$$
\boxed{
\text{Goldbach-shift uniformization is not a matter
of replacing }\ll_a\text{ by }\ll.
}
$$

The proof geometry itself must change.

---

# 16. Three Breakpoints Summary

| Stage | fixed-$a$ use | restored dependence | $a_X\asymp X$ |
|---|---|---|---|
| Lemma 6.10 / Maynard-I 18.3 | dropped Bezout fraction | $|a|\log^5X/X$ | not small |
| Lemma 6.9 / Maynard-I 14.5 | dropped reciprocal phase | $|a|\log^{O(1)}X/X$ | not small |
| Lemma 8.1 | spectral numerator $y=af\ell$ | $Y\propto|a|$ | range inflates by $X$ |

Therefore:

$$
\boxed{
\text{DIRECT CONSTANT TRACKING FAIL}.
}
$$

---

# 17. Consistency with Maynard-I Introduction

Maynard-I explicitly states:

The Deshouillers–Iwaniec/Kuznetsov route requires:

$$
\boxed{
a\text{ fixed}
}
$$

or:

$$
\boxed{
a\text{ a small power of }X.
}
$$

T01–T03 now explain where this statement actually arises in the proof.

---

# 18. Why a Small Power Can Still Work

Suppose:

$$
|a|
\le
X^\kappa
$$

with：

$$
0<\kappa<1
$$

small.

Bezout error：

$$
\frac{
|a|(\log X)^C
}{X}
\le
X^{-1+\kappa}
(\log X)^C
\to0.
$$

Although the spectral frequency is enlarged by:

$$
X^\kappa,
$$

if:

$$
\kappa
$$

is sufficiently small,

it may be absorbed by the original proof's power margins.

This corresponds precisely to Maynard-I's:

> fixed or a small power of $X$.

Goldbach：

$$
\kappa=1
$$

lies in an entirely different regime.

---

# 19. Naive Residue-Representative Rescue

One might try:

Since:

$$
nm\equiv a\pmod q
$$

depends only on:

$$
a\bmod q,
$$

for each:

$$
q
$$

choose a centered representative:

$$
\boxed{
a_q
\equiv a_X\pmod q,
\qquad
|a_q|\le q/2.
}
$$

If:

$$
q\le X^{3/5},
$$

then:

$$
|a_q|
\le
X^{3/5},
$$

which seems to re-enter the small-power region.

---

# 20. Local Bezout Error Would Indeed Become Small

For a single fixed:

$$
q
$$

as displayed above,

$$
\boxed{
\frac{
|a_q|(\log X)^C
}{X}
\ll
X^{-2/5}
(\log X)^C.
}
$$

Thus:

> Naive residue reduction indeed repairs the small-phase size of **a single q term**.

This is a genuine observation.

---

# 21. Why This Is Not a Black-Box Fix

The Maynard-II proof subsequently:

1. Applies Cauchy–Schwarz to:
   $$
   q
   $$
   as displayed above;
2. Fourier expansion；
3. Uses:
   $$
   n_1-n_2=qdf;
   $$
4. Changes the:
   $$
   q
   $$
   summation to an:
   $$
   f
   $$
   summation;
5. Finally assembles the numerator:
   $$
   y=af\ell.
   $$

If this is changed to:

$$
a_q,
$$

then the numerator becomes:

$$
\boxed{
a_q f\ell
}
$$

while:

$$
a_q
$$

itself depends on:

$$
q=\frac{n_1-n_2}{df}.
$$

Thus:

$$
\boxed{
\text{the numerator and }n_1,n_2,d,f\text{ become strongly coupled again}.
}
$$

---

# 22. Residue-Reduction Separability Barrier

## Research Verdict SRBRD-21-R01

$$
\boxed{
\text{Choosing a small representative }a_q
\text{ repairs local phase size,
but destroys the fixed-numerator separability
used by the q-to-f switch and spectral grouping.}
}
$$

Therefore:

$$
\boxed{
\text{no black-box residue-representative repair was found.}
}
$$

This explains precisely why:

# Uniform-residue dispersion

requires a genuinely different argument.

---

# 23. Maynard-III as a Replacement Architecture

Maynard-III's uniform Type-II propositions directly contain:

$$
\boxed{
\sup_{(a,q)=1}
}
$$

or compatible residue suprema in the balanced case.

Thus its argument cannot rely on a core step requiring:

$$
\boxed{
|a|\ll X^\kappa
}
$$

such absolute smallness of a representative.

Therefore:

> Maynard-III does not merely track constants for Maynard-II;
> It provides a different analytic architecture capable of retaining variable residues.

---

# 24. Why Low Row Still Matters

Maynard-III's substantial uniform Type-II coverage mainly extends from:

$$
N\approx X^{2/5}
$$

toward:

$$
X^{1/2}
$$

as displayed above.

But the SRBRD low row:

$$
\boxed{
X^{1/3}
\lesssim
N
<
X^{2/5}
}
$$

remains the most natural strong-provider region of the fixed-residue Maynard-II range.

Thus this problem cannot be completely bypassed.

---

# 25. Direct GSHIFT-TII v0.1 Is Too Optimistic

SRBRD-20 target：

$$
\boxed{
\text{take Maynard-II Prop 5.1 and make }a=X+3
\text{ uniform}
}
$$

If this means:

> Retaining the original proof and proving only:
> $$
> C(a_X)\le X^{o(1)},
> $$

this round has already proved:

$$
\boxed{
\text{FAIL}.
}
$$

Since:

- Small-phase errors are no longer small;
- The spectral frequency range is inflated;
- The parameter inequalities are altered.

---

# 26. Revised Low-Row Target

The new target should be:

# GSHIFT-LOW-UNIFORM

not:

> Maynard-II-with-better-constants,

but:

> **Within the same low-row variable / modulus range,
> prove anew a Goldbach-shift uniform convolution estimate,
> without using fixed-$a$ reciprocal simplifications.**

---

# 27. Target Statement

For:

$$
A_X=X+3,
$$

$$
X^{1/3+\eta}
\le
N
\le
X^{2/5-\eta},
$$

$$
NM\asymp X,
$$

and target factorable modulus weights:

$$
\lambda_q,
$$

the desired estimate is:

$$
\boxed{
\sum_{\substack{
q\le Q\\
(q,A_X)=1
}}
\lambda_q
\sum_{n\sim N}
\alpha_n
\sum_{v\sim M}
\beta_v
\left[
\mathbf1_{nv\equiv A_X\pmod q}
-
\frac{
\mathbf1_{(nv,q)=1}
}{
\varphi(q)
}
\right]
\ll
\frac{
X
}{
(\log X)^A
}
}
$$

with a uniform implied constant.

---

# 28. Minimal Internal Replacement Gates

If one still wants to retain the shell of Maynard-II Prop 8.2,

at least three replacement gates are needed.

## GS-E1

uniform replacement for Lemma 6.10：

> Bezout truncation not using:
> $$
> |a|/X\to0
> $$
> as displayed above.

## GS-E2

uniform replacement for Lemma 6.9：

> No use of a small absolute residue representative.

## GS-SPEC

uniform replacement for Lemma 8.1：

> The spectral bound must not take the numerator range as:
> $$
> Y\propto |a|.
> $$

---

# 29. One Combined Replacement Is More Realistic

Rather than repairing separately:

$$
GS\mbox{-}E1,
\quad
GS\mbox{-}E2,
\quad
GS\mbox{-}SPEC,
$$

it is more natural to use:

$$
\boxed{
\text{Maynard-III-style uniform dispersion}
}
$$

to replace the entire:

$$
\boxed{
\text{Prop 6.8 → Lemma 6.9/6.10 → Lemma 8.1}
}
$$

chain.

That is:

# Uniform Low-Row Dispersion Lemma

---

# 30. Goldbach Special Structure

SRBRD is not an arbitrary growing residue:

$$
a(X).
$$

but:

$$
\boxed{
A_X=X+3=2m.
}
$$

and the row product:

$$
nv
$$

satisfies, in this application:

$$
\boxed{
nv\le X.
}
$$

Thus:

$$
\boxed{
A_X-nv
\ge3.
}
$$

---

# 31. Complement Promotion

congruence：

$$
nv\equiv A_X\pmod q
$$

is exactly equivalent to:

$$
\boxed{
q
\mid
(A_X-nv).
}
$$

Define:

$$
\boxed{
w=A_X-nv.
}
$$

then:

$$
\boxed{
nv+w=A_X,
\qquad
w\ge3.
}
$$

Thus the Goldbach shift can be rewritten as:

# Product + Complement

rather than:

# Product in a Large Residue Class

---

# 32. Why Complement Promotion Is Potentially Useful

The fixed-$a$ reciprocity route places:

$$
A_X
$$

in a reciprocal exponential numerator.

The complement route instead hopes to:

> Retain, before dispersion:
> $$
> w=A_X-nv
> $$
> as an arithmetic variable,
> and write the sieve modulus condition directly as:
> $$
> q\mid w.
> $$

Then:

$$
A_X
$$

appears only in the additive conservation law:

$$
nv+w=A_X,
$$

and need not become the magnitude of a spectral numerator.

---

# 33. Complement Promotion Is Exact, Not a Heuristic

## Theorem SRBRD-21-T04

For:

$$
nv<A_X,
$$

$$
\boxed{
nv\equiv A_X\pmod q
}
$$

if and only if there exists a unique:

$$
w=A_X-nv>0
$$

such that:

$$
\boxed{
q\mid w.
}
$$

This is merely an algebraic equivalence.

---

# 34. But It Creates a New Shifted-Convolution Problem

After promotion, one must estimate:

$$
\boxed{
\sum_{n,v}
\alpha_n\beta_v
\mathbf1_{q\mid A_X-nv}.
}
$$

This is still a:

$$
\boxed{
\text{shifted prime-product divisor problem}.
}
$$

Thus complement promotion is not a proof.

It is merely:

> An alternate normal form avoiding fixed-$a$ spectral-frequency inflation.

---

# 35. Two Replacement Architectures

SRBRD-22 can compare:

### Route U — Uniform Residue

Extend Maynard-III uniform Type-II / Zhang-style technology downward to the low row.

### Route C — Complement Promotion

Retain:

$$
w=A_X-nv
$$

then use:

- divisor switching；
- bilinear / trilinear forms；
- Kloosterman fraction estimates；
- convolution-BV；
- maybe Yang/Pascadi technology。

The two may ultimately converge.

---

# 36. Spectral Barrier Is Stronger Than a Constant Barrier

SRBRD-19 / 20 described the problem as:

$$
\boxed{
\text{fixed-residue constant dependence}.
}
$$

SRBRD-21 corrects this:

$$
\boxed{
\text{the proof's internal scale itself changes with }|a|.
}
$$

This is a qualitatively stronger diagnosis.

---

# 37. Direct Error Calculation as a Formal Audit Test

For any future fixed-residue theorem,

first check whether there appears:

$$
\boxed{
e
\left(
\frac{
a\cdot(\text{frequency})
}{
\text{modulus}
}
\right)
}
$$

and, after reciprocity is applied, the following is discarded:

$$
\boxed{
\frac{
a\cdot(\text{frequency})
}{
\text{product of large variables}
}
}
$$

as a small error.

If so,

then, for:

$$
a\asymp X
$$

the size audit must be redone.

One cannot look only at the final theorem's:

$$
\ll_a.
$$

---

# 38. Dependency Classification — Final

| Dependency | Status for $a_X=X+3$ |
|---|---|
| $(q,a)=1$ | harmless |
| $a\bmod q$ only | harmless in principle |
| $\tau(a)^C$ | absorbable |
| $(\log a)^C$ | absorbable |
| $a$ in unit-modulus coefficient only | potentially harmless |
| $a/X$ small-phase assumption | **fatal to current proof** |
| spectral index range $Y\propto|a|$ | **fatal to current proof** |
| $a_q$ chosen per modulus | locally small, globally nonseparable |
| full residue supremum | safe |

---

# 39. Implication for the Literature Map

### Maynard-II

low-row strength：

$$
\boxed{
\text{excellent}
}
$$

but：

$$
\boxed{
\text{intrinsically fixed/small-}a
\text{ in current proof}.
}
$$

### Maynard-III

uniform residue：

$$
\boxed{
\text{genuine architecture replacement}.
}
$$

not merely improved constants.

### Yang/Pascadi

potential alternate convolution / Kloosterman technology：

$$
\boxed{
\text{candidate replacement}
}
$$

especially if formulated in complement or variable-residue form.

---

# 40. Could One Rescale the Ambient $X$?

No.

Maynard Type-II hypotheses require:

$$
NM\asymp X.
$$

SRBRD factors satisfy:

$$
nv\asymp X.
$$

One cannot artificially choose:

$$
X'=X^B
$$

to make：

$$
a_X=X
$$

a small power of：

$$
X',
$$

because then:

$$
NM\not\asymp X'.
$$

so this is not a valid workaround.

---

# 41. Could One Subtract a Multiple of $q$ Globally?

No single：

$$
k
$$

makes：

$$
A_X-kq
$$

small for all：

$$
q
$$

in a dyadic modulus family。

choosing：

$$
k=k(q)
$$

returns to the q-dependent representative problem of §19–22。

---

# 42. Minimal New Analytic Lemma

This round further renames SRBRD-20's target:

# GSHIFT-LOW

Require:

> low-row prime-product discrepancy at shift：
> $$
> A_X=X+3
> $$
> with Maynard-II-strength modulus factorability,
> proved by a method that is **modular in the shift** rather than **archimedean in $|A_X|$**.

This is currently the smallest and most precise missing analytic theorem.

---

# 43. Possible Formal Statement

```yaml
GSHIFT_LOW:

  scale:
    X:
      large: true

  shift:
    A_X:
      X + 3

  factors:
    N:
      X^(1/3+eta)
      to:
      X^(2/5-eta)

    M:
      asymptotic:
        X/N

  coefficients:
    alpha:
      prime_indicator_or_SW: true

    beta:
      prime_indicator_or_divisor_bounded: true

  moduli:
    coprime_to_A_X: true
    well_factorable_strength:
      target:
        Maynard_II_like

  conclusion:
    discrepancy:
      <= X/log(X)^A

  forbidden_proof_dependency:
    small_absolute_shift_over_X: true
```

---

# 44. Formal Theorem Queue

```yaml
SRBRD_21:

  T01:
    Bezout_dropped_fraction_scale:
      abs_a_log5_over_X

  T02:
    second_Bezout_error_scale:
      abs_a_logO1_over_X

  T03:
    spectral_frequency_range:
      Y_a:
        proportional_to_abs_a

  T04:
    complement_promotion_exact_equivalence

  R01:
    q_dependent_small_residue_repairs_local_size_but_breaks_separability
```

---

# 45. Acceptance Gate

Before SRBRD-21 is formalized, it requires:

1. Checking Maynard-II's fixed-$a$ declaration;
2. Checking Maynard-I's fixed-or-small-power statement;
3. Lemma 18.3 Bezout identity；
4. hidden $|a|$ restored error；
5. Lemma 14.5 / Maynard-II 6.9 hidden $|a|$ restored error；
6. Lemma 8.1:
   $$
   y=af\ell;
   $$
7. spectral $J(Y)$ dependence；
8. fixed proof use of:
   $$
   Z>Y;
   $$
9. residue-representative local rescue calculation；
10. q-switch nonseparability warning；
11. complement promotion exact identity；
12. no claim complement promotion solves estimate；
13. No Goldbach proof claim.

---

# 46. Verdict

This round can formally give:

$$
\boxed{
\mathrm{SRBRD\mbox{-}21\ FIXED\mbox{-}SHIFT\ SPECTRAL\ BARRIER\ PASS}.
}
$$

and corrects SRBRD-20's optimistic possibility:

$$
\boxed{
C(a_X)\le X^{o(1)}
}
$$

cannot be obtained directly using the original Maynard-II proof.

The reason is not merely the implied constant:

### Breakpoint 1

$$
\boxed{
\text{Bezout error}
\sim
|a|(\log X)^5/X.
}
$$

### Breakpoint 2

$$
\boxed{
\text{second reciprocal simplification}
\sim
|a|(\log X)^{O(1)}/X.
}
$$

### Breakpoint 3

$$
\boxed{
\text{spectral frequency}
Y
\propto
|a|.
}
$$

For:

$$
a_X=X+3,
$$

all three leave the original regime.

Thus:

$$
\boxed{
\text{LOW-ROW UNIFORMIZATION REQUIRES A NEW DISPERSION/SPECTRAL NORMAL FORM}.
}
$$

However:

$$
\boxed{
\mathrm{GOLDBACH\ STILL\ OPEN}.
}
$$

---

# 47. Recommendations for the Next Round

# SRBRD-22
## Goldbach-Shift Complement Promotion / Uniform Low-Row Replacement

The next round no longer tracks Maynard-II's hidden constants.

Directly compare the two replacement routes.

## Route U — Uniform-Residue Descent

Try extending Maynard-III's:

- variable-residue；
- Type-II；
- Kloosterman architecture；

toward:

$$
N\approx X^{1/3}\text{--}X^{2/5}
$$

as displayed above.

Question:

$$
\boxed{
\text{Which Maynard-III parameter inequality fails in the low row?}
}
$$

## Route C — Complement-Promoted Dispersion

From:

$$
nv+w=A_X
$$

and:

$$
q\mid w
$$

start,

without letting:

$$
A_X
$$

enter the reciprocal numerator.

Prioritize checking:

1. divisor switching in $w$；
2. dyadic $w$ ranges；
3. bilinear Kloosterman fraction form；
4. Yang convolution-BV compatibility；
5. Pascadi / Deshouillers–Iwaniec modern spectral bounds；
6. Whether the Maynard-II-strength well-factorable modulus level can be retained.

The actual target:

$$
\boxed{
\text{a low-row estimate modular in }A_X
\text{ rather than polynomial in }|A_X|.
}
$$

---

# Appendix A: Fixed-$a$ Dependency Ledger

```yaml
MaynardII:

  fixed_a_declaration:
    true

  Prop8_2:
    depends_on:
      - Lemma6_10
      - Lemma6_9
      - Lemma8_1

  Lemma6_10:
    source:
      MaynardI_Lemma18_3
    critical_step:
      Bezout
    hidden_error:
      abs(a)*log(X)^5/X

  Lemma6_9:
    source:
      MaynardI_Lemma14_5
    critical_step:
      reciprocal_phase_simplification
    hidden_error:
      abs(a)*log(X)^O(1)/X

  Lemma8_1:
    spectral_index:
      y:
        a*f*(h1*s2-h2*s1)
    range:
      proportional_to_abs_a: true

Goldbach:
  a_X:
    X+3
  exponent:
    1

verdict:
  direct_constant_tracking:
    FAIL
```

---

# Appendix B: Residue Representative Audit

```yaml
for_each_q:
  choose:
    a_q:
      congruent_to_A_X_mod_q
      abs_a_q <= q/2

local_phase:
  hidden_Bezout_error:
    <= q*log(X)^C/X

if_q_le_X^(3/5):
  local_error:
    <= X^(-2/5)*log(X)^C

problem:
  after_q_to_f_switch:
    q:
      function_of_n1_n2_d_f
    a_q:
      depends_on_q
    numerator:
      no_longer_separable: true

verdict:
  local_size_fix:
    true
  black_box_global_fix:
    false
```

---

# Appendix C: Complement Promotion

```yaml
Goldbach_shift:
  A_X:
    X+3

product:
  n*v:
    <= X

complement:
  w:
    A_X-n*v
    >= 3

equivalence:
  n*v == A_X mod q:
    iff:
      q divides w

additive_constraint:
  n*v + w:
    A_X

research_goal:
  keep_A_X_out_of_absolute_spectral_frequency: true
```

---

# Appendix D: Proof-Breakpoint Source Notes

```yaml
MaynardI_intro:
  Deshouillers_Iwaniec_route:
    requires:
      a_fixed_or_small_power_of_x: true

MaynardI_Lemma18_3:
  Bezout_residual_fraction:
    a*h/(n*q)
  fixed_a_written_error:
    log(X)^5/X

MaynardII_Lemma8_1:
  off_diagonal:
    ell:
      h1*s2-h2*s1
    y:
      a*f*ell

  DI_input:
    J:
      depends_on_Y: true

  later_simplification:
    requires:
      Z > Y
```

---

# Appendix E: Correction to SRBRD-20

```yaml
old_optimistic_target:
  trace_C_of_a:
    hope:
      C(X+3) <= X^o(1

status:
  rejected_for_current_proof: true

new_target:
  replace_fixed_a_reciprocity_chain: true

preferred_forms:
  - uniform_residue_low_row_dispersion
  - complement_promoted_shifted_convolution

unchanged:
  middle_high_MaynardIII_route: viable_shape
  modulus_factorization_gate: pending
  constant_crossing_gate: pending
```

---

# Appendix F: Commit Decision

```yaml
research_round: SRBRD-21

decision:
  explicit_fixed_a_breakpoints_found: true
  direct_constant_tracking_uniformization: false
  q_dependent_residue_local_fix_only: true
  complement_promotion_exact: true
  new_low_row_normal_form_required: true
  goldbach_proved: false

next:
  SRBRD-22_GoldbachShift_ComplementPromotion_UniformLowRow_Replacement
```

---

# References / Audited Sources

1. James Maynard, **Primes in arithmetic progressions to large moduli II: Well-factorable estimates**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1543; arXiv:2006.07088.

   Audited locations:
   - §4: $a$ is fixed and implied constants may depend on $a$.
   - Proposition 8.2: low-row well-factorable convolution theorem.
   - proof of Proposition 8.2: reduction through Lemmas 6.9, 6.10, and Lemma 8.1.
   - Lemma 8.1: off-diagonal variable $y=af(h_1s_2-h_2s_1)$ and Deshouillers–Iwaniec input with explicit $Y$ dependence.

2. James Maynard, **Primes in arithmetic progressions to large moduli I: Fixed residue classes**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1542; arXiv:2006.06572.

   Audited locations:
   - introduction: Kuznetsov / Deshouillers–Iwaniec route requires $a$ fixed or a small power of $x$.
   - Lemma 14.5: reciprocal exponential simplification with fixed-$a$ small error.
   - Lemma 18.3: Bezout reciprocity and the fixed-$a$ error written as $O((\log x)^5/x)$.

3. James Maynard, **Primes in arithmetic progressions to large moduli III: Uniform residue classes**, *Memoirs of the American Mathematical Society* 306 (2025), no. 1544; arXiv:2006.08250.

   Role in this round:
   - demonstrates that genuinely uniform-residue Type-II architecture exists;
   - supports replacing, rather than merely re-constant-tracking, the fixed-residue reciprocal/spectral chain.