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

# SRBRD-10
## Uniform Moment Envelope / Dual-Polynomial Closure
### Signed Moment Providers, Phase-Free Hypercell Brackets, Exact First Moment, and the CRT Phase-Discrepancy Barrier

**Parent program:** RG-SCP / Reverse Goldbach Structural Closure Program  
**Branch:** SRBRD — Symmetric Root-Bounded Relation Domain  
**Round:** SRBRD-10  
**Date:** 2026-09-30  
**Version:** v0.1  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Nature:** Uniform Bound Interface / Signed Moment Envelope / Barrier Audit  
**Prerequisites:** SRBRD-00 through SRBRD-09  
**Status:** Establishes a uniform moment-provider contract and identifies CRT phase discrepancy as the main remaining bottleneck; uniform Goldbach closure has not yet been obtained

---

# 0. Task of This Round

SRBRD-09 has written the fixed-$m$ non-cover certificate as:

$$
\boxed{
|\mathcal O_m|
\le
\sum_{j=1}^{s}
\lambda_j B_j(m)
}
$$

provided that the polynomial:

$$
P(k)
=
\sum_{j=1}^{s}
\lambda_j
\binom{k}{j}
$$

In:

$$
k=0,\ldots,\kappa(m)
$$

satisfies, on this domain:

$$
P(0)=0,
$$

$$
P(k)\ge1
\quad
(k\ge1).
$$

For fixed-$m$, one can compute exactly:

$$
B_j(m).
$$

But a uniform Goldbach proof cannot rely only on:

> Computing all moments exactly for every $m$.

Thus, the question of SRBRD-10 is:

> Can each moment $B_j$ be replaced by a proof-admissible uniform upper / lower envelope while keeping the polynomial certificate sound?

The answer has two parts:

1. **The signed moment-envelope contract can be established completely.**
2. **The most natural phase-free envelopes currently remain too loose, and the main gap is concentrated in CRT phase-hit upper bounds for positive-coefficient moments.**

Thus, this round is a:

$$
\boxed{
\text{Uniformization Interface PASS}
}
$$

and at the same time a:

$$
\boxed{
\text{CRT Phase-Discrepancy Barrier IDENTIFIED}.
}
$$

---

# 1. Signed Moment Envelope Principle

Suppose that for every:

$$
j=1,\ldots,s
$$

there are rigorous bounds:

$$
\boxed{
L_j(m)\le B_j(m)\le U_j(m).
}
$$

Given polynomial coefficients:

$$
\lambda_j.
$$

define the signed envelope:

$$
\boxed{
\mathcal U_P(m)
=
\sum_{\lambda_j\ge0}
\lambda_jU_j(m)
+
\sum_{\lambda_j<0}
\lambda_jL_j(m).
}
$$

---

# 2. Signed Envelope Theorem

## Theorem SRBRD-10-T01

If:

$$
P(k)
=
\sum_j
\lambda_j\binom{k}{j}
$$

In:

$$
0\le k\le\kappa(m)
$$

majorizes, on this domain:

$$
\mathbf1_{\{k\ge1\}},
$$

and:

$$
L_j\le B_j\le U_j,
$$

then:

$$
\boxed{
|\mathcal O_m|
\le
\mathcal U_P(m).
}
$$

proof：

SRBRD-09：

$$
|\mathcal O_m|
\le
\sum_j\lambda_jB_j.
$$

If:

$$
\lambda_j\ge0,
$$

substitute:

$$
B_j\le U_j.
$$

If:

$$
\lambda_j<0,
$$

substituting:

$$
L_j\le B_j
$$

yields:

$$
\lambda_jB_j
\le
\lambda_jL_j.
$$

Thus, the theorem holds.

---

# 3. Uniform Non-Cover Contract

## Corollary SRBRD-10-C01

If:

$$
\boxed{
\mathcal U_P(m)
<
|\Omega_m|,
}
$$

then:

$$
\boxed{
\operatorname{Res}(m)\neq\varnothing,
}
$$

Thus:

$$
\boxed{
R_G^{-}(2m)>0.
}
$$

This is the interface actually required by a uniform dual-polynomial theorem.

---

# 4. Provider Polarity

This theorem separates moment providers into two classes.

### Positive Coefficient

If:

$$
\lambda_j>0,
$$

we need:

$$
\boxed{
B_j\le U_j.
}
$$

### Negative Coefficient

If:

$$
\lambda_j<0,
$$

we need:

$$
\boxed{
L_j\le B_j.
}
$$

Thus, not all moments must be estimated in the same direction.

---

# 5. Exact First Moment

The first moment:

$$
B_1(m)
=
\sum_{r\in Q_m}
|P_{m,r}|.
$$

SRBRD-01 has already given the exact cardinality of each prime-group cell.

For an odd prime:

$$
r\in Q_m,
$$

Define:

$$
L_r(m)
=
\operatorname{OddCount}
\left(
r,
\left\lfloor\frac mr\right\rfloor
\right),
$$

and:

$$
R_r(m)
=
\operatorname{OddCount}
\left(
\max
\left(
r,
\left\lceil\frac mr\right\rceil
\right),
\left\lfloor
\frac{2m-3}{r}
\right\rfloor
\right).
$$

---

# 6. Exact Prime-Group Cardinality

SRBRD-01：

If:

$$
r\mid m,
$$

then:

$$
O_{L,r}\subseteq O_{R,r},
$$

Thus:

$$
\boxed{
|P_{m,r}|=R_r(m).
}
$$

If:

$$
r\nmid m,
$$

then the two sides are disjoint:

$$
\boxed{
|P_{m,r}|
=
L_r(m)+R_r(m).
}
$$

Therefore:

## Theorem SRBRD-10-T02

$$
\boxed{
B_1(m)
=
\sum_{r\in Q_m}
\left[
R_r(m)
+
\mathbf1_{\{r\nmid m\}}L_r(m)
\right].
}
$$

This is a completely elementary exact formula.

---

# 7. Exact $B_1$ Is Not Truth Leakage

Computing:

$$
B_1
$$

requires only:

- witness primes;
- factor divisibility;
- the root interval;
- center divisibility.

It does not need to know:

$$
m-d
$$

or:

$$
m+d
$$

whether they are prime.

Thus:

$$
\boxed{
B_1
}
$$

is a legitimate ProofInput.

---

# 8. Canonical Oriented Hypercell Recalled

For:

$$
U\subseteq Q_m,
$$

$$
|U|=j,
$$

and a canonical orientation:

$$
\alpha,
$$

SRBRD-08 gives:

$$
\boxed{
K_m(U,\alpha)
=
\left\{
x:
a_{U,\alpha}\le x\le b_{U,\alpha},
\quad
x\equiv c_{U,\alpha}\pmod{2P(U)}
\right\}.
}
$$

Let:

$$
\boxed{
M_U=2P(U)
}
$$

and the interval length:

$$
\boxed{
\ell_{U,\alpha}
=
\max
\left(
0,
b_{U,\alpha}-a_{U,\alpha}+1
\right).
}
$$

---

# 9. Residue-Free AP Bracket

For any modulus:

$$
M,
$$

and any residue class,

within a consecutive-integer interval of length:

$$
\ell
$$

as displayed above,

the number of occurrences of that residue always lies between:

$$
\left\lfloor\frac{\ell}{M}\right\rfloor
$$

and:

$$
\left\lceil\frac{\ell}{M}\right\rceil.
$$

Thus:

## Theorem SRBRD-10-T03

$$
\boxed{
\left\lfloor
\frac{
\ell_{U,\alpha}
}{
2P(U)
}
\right\rfloor
\le
|K_m(U,\alpha)|
\le
\left\lceil
\frac{
\ell_{U,\alpha}
}{
2P(U)
}
\right\rceil.
}
$$

This does not require knowing the CRT residue:

$$
c_{U,\alpha}.
$$

---

# 10. Hypercell Box Bounds

Canonical orientations are disjoint.

Thus, define:

$$
\boxed{
L_H^{\mathrm{box}}(m,U)
=
\sum_{\alpha}
\left\lfloor
\frac{
\ell_{U,\alpha}
}{
2P(U)
}
\right\rfloor,
}
$$

$$
\boxed{
U_H^{\mathrm{box}}(m,U)
=
\sum_{\alpha}
\left\lceil
\frac{
\ell_{U,\alpha}
}{
2P(U)
}
\right\rceil.
}
$$

then:

## Theorem SRBRD-10-T04

$$
\boxed{
L_H^{\mathrm{box}}(m,U)
\le
|H_m(U)|
\le
U_H^{\mathrm{box}}(m,U).
}
$$

---

# 11. Per-Orientation Uncertainty Is At Most One

For any:

$$
\ell,
M,
$$

we have:

$$
\left\lceil\frac{\ell}{M}\right\rceil
-
\left\lfloor\frac{\ell}{M}\right\rfloor
\in\{0,1\}.
$$

Thus:

## Corollary SRBRD-10-C02

$$
\boxed{
0
\le
U_H^{box}-L_H^{box}
\le
|\mathcal A_m^{\mathrm{nonempty-box}}(U)|.
}
$$

That is:

> Phase-free uncertainty differs by at most one lattice hit per orientation.

---

# 12. Moment Box Envelopes

Define:

$$
\boxed{
L_j^{\mathrm{box}}(m)
=
\sum_{\substack{
U\subseteq Q_m\\
|U|=j
}}
L_H^{box}(m,U),
}
$$

$$
\boxed{
U_j^{\mathrm{box}}(m)
=
\sum_{\substack{
U\subseteq Q_m\\
|U|=j
}}
U_H^{box}(m,U).
}
$$

From:

$$
B_j
=
\sum_{|U|=j}|H_m(U)|,
$$

we obtain:

## Theorem SRBRD-10-T05

$$
\boxed{
L_j^{box}(m)
\le
B_j(m)
\le
U_j^{box}(m).
}
$$

This is the first set of fully uniform moment envelopes requiring no exact CRT phase at all.

---

# 13. Phase-Uncertainty Budget

Define:

$$
\boxed{
A_j^{\mathrm{box}}(m)
=
\sum_{\substack{
U\subseteq Q_m\\
|U|=j
}}
|\mathcal A_m^{\mathrm{nonempty-box}}(U)|.
}
$$

then:

## Corollary SRBRD-10-C03

$$
\boxed{
0
\le
U_j^{box}-L_j^{box}
\le
A_j^{box}.
}
$$

Thus, uniform moment uncertainty has been precisely located as:

$$
\boxed{
\text{orientation-level CRT phase-hit uncertainty}.
}
$$

---

# 14. All-Right Uniform Lower Provider

All primes allow canonical right orientation:

- center-divisor prime: right is forced;
- noncenter prime: right is allowed.

Thus:

$$
\boxed{
\alpha_R(r)=R
\quad
\forall r\in U
}
$$

is always an admissible canonical orientation.

If:

$$
\max(U)^2\le m,
$$

then the right root bound is automatically satisfied for:

$$
y=m+d\ge m
$$

as displayed above.

Left-endpoint interval:

$$
3\le x\le m.
$$

Length:

$$
m-2.
$$

Therefore:

## Theorem SRBRD-10-T06

If:

$$
\max(U)^2\le m,
$$

then:

$$
\boxed{
|H_m(U)|
\ge
\left\lfloor
\frac{
m-2
}{
2P(U)
}
\right\rfloor.
}
$$

---

# 15. Uniform Right-Seed Moment Lower Bound

Let:

$$
Q_m^{\sqrt{}}
=
\left\{
r\in\mathbb P_{\mathrm{odd}}:
r^2\le m
\right\}.
$$

then:

## Corollary SRBRD-10-C04

$$
\boxed{
B_j(m)
\ge
L_j^{R}(m)
}
$$

where:

$$
\boxed{
L_j^R(m)
=
\sum_{\substack{
U\subseteq Q_m^{\sqrt{}}\\
|U|=j
}}
\left\lfloor
\frac{
m-2
}{
2P(U)
}
\right\rfloor.
}
$$

This is a very simple uniform negative-moment provider.

---

# 16. General Seed-Family Lower Bound

More generally,

for any chosen family:

$$
\mathcal S_j(m)
\subseteq
\{U\subseteq Q_m:|U|=j\},
$$

we have:

$$
\boxed{
B_j(m)
\ge
\sum_{U\in\mathcal S_j}
L_H^{box}(m,U).
}
$$

a uniform proof can therefore select only:

- the first few primes;
- structured prime blocks;
- center-divisor-related sets;

as negative-moment seeds.

There is no need to compute all hypercells exactly.

---

# 17. Positive-Moment Uniform Provider

For:

$$
\lambda_j>0,
$$

the most direct provider is:

$$
\boxed{
B_j(m)\le U_j^{box}(m).
}
$$

This can also be combined with the SRBRD-03 local-occupancy-cap upper bound:

$$
\boxed{
U_j^{cap}(m)
=
\sum_{d\in\Omega_m}
\binom{\kappa_m(d)}{j}
}
$$

by taking the minimum:

$$
\boxed{
U_j^{+}(m)
=
\min
\left(
U_j^{box}(m),
U_j^{cap}(m)
\right).
}
$$

---

# 18. Negative-Moment Uniform Provider

For:

$$
\lambda_j<0,
$$

one can take:

$$
\boxed{
L_j^{-}(m)
=
\max
\left(
L_j^{box}(m),
L_j^R(m),
L_j^{bal}(m),
L_j^{seed}(m)
\right),
}
$$

provided that each term has separately been proved to satisfy:

$$
\le B_j
$$

one can safely take the maximum.

---

# 19. Uniform Signed Envelope

For a polynomial:

$$
P(k)
=
\sum_{j=1}^{s}\lambda_j\binom{k}{j},
$$

Define:

$$
\boxed{
\mathcal U_P^{env}(m)
=
\lambda_1B_1(m)
+
\sum_{\substack{
j\ge2\\
\lambda_j>0
}}
\lambda_jU_j^{+}(m)
+
\sum_{\substack{
j\ge2\\
\lambda_j<0
}}
\lambda_jL_j^{-}(m).
}
$$

where:

$$
B_1
$$

uses the exact T02 formula.

---

# 20. Uniform Envelope Closure Theorem

## Theorem SRBRD-10-T07

If:

1. $P$ majorizes the occupancy indicator on:
   $$
   0,\ldots,\kappa(m);
   $$
2. all moment providers are rigorous;
3. 
   $$
   \boxed{
   \mathcal U_P^{env}(m)
   <
   |\Omega_m|;
   }
   $$

then:

$$
\boxed{
R_G^{-}(2m)>0.
}
$$

This is the formal closure contract of SRBRD-10.

---

# 21. Volume–Phase Decomposition

T03 can also be written in a more precise analytic form.

For an orientation:

$$
\boxed{
|K_m(U,\alpha)|
=
\frac{
\ell_{U,\alpha}
}{
2P(U)
}
+
\varepsilon_{U,\alpha}(m),
}
$$

where:

$$
\boxed{
|\varepsilon_{U,\alpha}(m)|<1.
}
$$

$\varepsilon$ is determined entirely by the CRT lattice phase relative to the interval endpoints.

---

# 22. Moment Volume Term

Define:

$$
\boxed{
V_j(m)
=
\sum_{\substack{
|U|=j\\
\alpha
}}
\frac{
\ell_{U,\alpha}
}{
2P(U)
}.
}
$$

and the phase discrepancy:

$$
\boxed{
\Delta_j^{phase}(m)
=
\sum_{\substack{
|U|=j\\
\alpha
}}
\varepsilon_{U,\alpha}(m).
}
$$

then:

## Theorem SRBRD-10-T08

$$
\boxed{
B_j(m)
=
V_j(m)
+
\Delta_j^{phase}(m).
}
$$

and:

$$
\boxed{
|\Delta_j^{phase}(m)|
<
A_j^{box}(m).
}
$$

---

# 23. Sparse-Lattice Interpretation

If, for some orientation:

$$
2P(U)>\ell_{U,\alpha},
$$

then:

$$
\left\lfloor
\frac{\ell}{2P}
\right\rfloor=0,
$$

$$
\left\lceil
\frac{\ell}{2P}
\right\rceil=1.
$$

Thus:

$$
|K_m(U,\alpha)|
\in\{0,1\}.
$$

In this case:

$$
\boxed{
\text{The entire moment uncertainty is whether
the unique CRT lattice point hits the interval.}
}
$$

---

# 24. Phase-Hit Variable

On a sparse orientation, define:

$$
\boxed{
\eta_{U,\alpha}(m)
=
\mathbf1_{\{
K_m(U,\alpha)\neq\varnothing
\}}.
}
$$

then:

$$
\boxed{
|K_m(U,\alpha)|
=
\eta_{U,\alpha}(m).
}
$$

Thus, at high product / high moment order:

$$
B_j
$$

gradually becomes:

$$
\boxed{
\text{a sum of many CRT phase-hit bits}.
}
$$

---

# 25. This Is the Main Uniform Barrier

Root geometry has already supplied:

- an exact interval;
- an exact modulus;
- an exact occupancy cap;
- exact load feasibility.

But a uniform positive-moment upper bound must still answer:

> Among many distinct prime-union CRT lattices, how many residue phases actually hit their respective short intervals?

Merely using:

$$
\lceil\ell/M\rceil
$$

treats each orientation as potentially having one hit,

causing a huge overcount.

---

# 26. Finite Audit: $m=1000$

exact：

$$
|\Omega|=499,
$$

$$
B_1=908,
$$

$$
B_2=656,
$$

$$
B_3=259.
$$

Phase-free envelopes:

$$
\boxed{
573\le B_2\le781,
}
$$

$$
\boxed{
88\le B_3\le1250.
}
$$

For degree-4:

$$
\kappa=6,
$$

$$
a=3,
$$

coefficients：

$$
1,
-\frac89,
\frac23,
-\frac13.
$$

Use the envelopes:

- $B_1$ exact;
- $B_2$ lower:
  $$
  573;
  $$
- $B_3$ upper:
  $$
  1250;
  $$
- $B_4$ lower:
  $$
  0;
  $$

we obtain:

$$
\boxed{
\mathcal U_P^{env}
=
1232
>
499.
}
$$

Thus, the phase-free uniform envelope is completely unable to achieve closure.

---

# 27. Finite Audit: $m=5000$

exact：

$$
|\Omega|=2499,
$$

$$
B_1=5495,
$$

$$
B_2=5206,
$$

$$
B_3=2870,
$$

$$
B_4=944.
$$

Phase-free:

$$
\boxed{
4841\le B_2\le5621,
}
$$

$$
\boxed{
1525\le B_3\le10871,
}
$$

$$
\boxed{
52\le B_4\le78261.
}
$$

degree-4：

$$
\kappa=7,
\quad
a=3,
$$

coefficients：

$$
1,
-\frac{37}{42},
\frac{9}{14},
-\frac27.
$$

Using:

- $B_1$ exact;
- $L_2=4841$;
- $U_3=10871$;
- $L_4=52$;

we obtain:

$$
\boxed{
\mathcal U_P^{env}
=
\frac{172283}{21}
\approx8203.952
>
2499.
}
$$

With exact moments, however, the same polynomial actually gives:

$$
\frac{52165}{21}
\approx2484.048
<
2499.
$$

Thus, almost the entire gap comes from:

$$
\boxed{
\text{positive }B_3\text{ phase-free upper bound being too loose}.
}
$$

---

# 28. Negative-Moment Side Is Already Relatively Tight

At $m=5000$:

$$
B_2=5206,
$$

while the phase-free lower bound is:

$$
4841.
$$

relative capture：

$$
\approx93\%.
$$

However:

$$
B_3=2870,
$$

phase-free upper：

$$
10871,
$$

is approximately:

$$
\approx3.79
$$

times the exact value.

Thus, the current uniform gap does not mean that all moment bounds are poor.

The main issue is:

$$
\boxed{
\text{positive-coefficient upper moments lacking phase-hit compression}.
}
$$

---

# 29. Degree-2 Exact No-Go Controls

Even if a degree-2 polynomial uses exact:

$$
B_1,B_2,
$$

the selected controls still fail.

Degree-2 cap majorant:

$$
P_\kappa^{(2)}
=
\binom{k}{1}
-
\frac2\kappa
\binom{k}{2}.
$$

Thus, the exact upper bound is:

$$
B_1-\frac2\kappa B_2.
$$

controls：

### $m=1000$

$$
908-\frac26(656)
\approx689.33
>
499.
$$

### $m=5000$

$$
5495-\frac27(5206)
\approx4007.57
>
2499.
$$

### $m=100000$

$$
139168-\frac29(176344)
\approx99980.44
>
49999.
$$

Thus:

$$
\boxed{
\text{A good }B_2\text{ lower bound alone does not suffice for uniform closure.}
}
$$

Positive higher moments are genuinely unavoidable.

---

# 30. Generic Aggregate Barrier

If all we know is:

$$
n=|\Omega|,
$$

$$
B_1=S,
$$

$$
0\le\mu_d\le\kappa,
$$

And:

$$
n\le S\le\kappa n,
$$

we can always construct an abstract occupancy histogram:

$$
\mu_d\ge1
$$

for every：

$$
d,
$$

and:

$$
\sum_d\mu_d=S.
$$

Therefore:

## Theorem SRBRD-10-T09

Using only:

$$
|\Omega|,
\quad
B_1,
\quad
\kappa
$$

cannot in general exclude a complete cover.

Uniform closure necessarily requires:

$$
\boxed{
\text{arithmetically constrained higher-moment information}.
}
$$

---

# 31. Phase-Free Hypercell Bracket Is Essentially Sharp Without Residue Information

Fix:

$$
U,\alpha,
$$

if all we know is:

- interval length:
  $$
  \ell;
  $$
- modulus:
  $$
  M;
  $$

but not the residue phase,

the count of a residue class in the interval can indeed be:

$$
\left\lfloor\frac{\ell}{M}\right\rfloor
$$

or:

$$
\left\lceil\frac{\ell}{M}\right\rceil.
$$

Thus:

$$
\boxed{
\text{T03 cannot be uniformly improved from length/modulus alone}.
}
$$

Reducing the gap requires using:

$$
\boxed{
\text{the shared structure of CRT residue phases across distinct hypercells}.
}
$$

---

# 32. This Is a Sieve-Like Barrier

Each sparse hypercell is like:

> A modulus:
> $$
> 2P(U)
> $$
> with a specified residue, asking whether it falls within a specified short interval.

To upper-bound many positive-moment hypercells,

one must control:

$$
\boxed{
\text{simultaneous lattice hits for many distinct product moduli}.
}
$$

This starts to approach:

- large-sieve-type counting;
- residue spacing;
- CRT packing;
- discrepancy;

rather than only a single factorization identity.

This round first isolates the interface,

without pretending to have overcome it.

---

# 33. Phase-Discrepancy Provider

Define:

$$
\boxed{
\operatorname{PhaseUpper}_j(m,\Delta)
:
\Longleftrightarrow
\Delta_j^{phase}(m)\le\Delta.
}
$$

then:

$$
\boxed{
B_j(m)
\le
V_j(m)+\Delta.
}
$$

If one can prove:

$$
\Delta
\ll
A_j^{box},
$$

the positive-moment upper bound can be substantially reduced.

---

# 34. Signed Phase Combination

For a polynomial certificate:

$$
P,
$$

what is actually required is not necessarily for each:

$$
\Delta_j
$$

to be small separately.

Define:

$$
\boxed{
\Delta_P^{phase}(m)
=
\sum_j
\lambda_j
\Delta_j^{phase}(m).
}
$$

then:

$$
\sum_j\lambda_jB_j
=
\sum_j\lambda_jV_j
+
\Delta_P^{phase}.
$$

Thus, a future theorem can directly bound:

$$
\boxed{
\Delta_P^{phase}
}
$$

instead of each moment discrepancy separately.

This may exploit cancellation between distinct orders.

---

# 35. Uniform Dual Certificate Rewritten

Thus, the actual uniform target can be written as:

$$
\boxed{
\sum_j\lambda_jV_j(m)
+
U_{\Delta,P}(m)
<
|\Omega_m|,
}
$$

where:

$$
\boxed{
\Delta_P^{phase}(m)
\le
U_{\Delta,P}(m).
}
$$

If this holds, non-cover is achieved.

This is the most focused version after SRBRD-10.

---

# 36. Why Exact CRT Is Not the Desired Uniform Provider

SRBRD-08 can already compute each of the following exactly:

$$
\eta_{U,\alpha}.
$$

But if a uniform proof merely:

> Computes all phase bits one by one,

that is only finite verification,

not symbolic compression.

Thus, what SRBRD-11 seeks is:

$$
\boxed{
\text{collective phase-hit upper theorem}.
}
$$

---

# 37. Candidate Phase-Compression Sources

Subsequent work can study:

### A. Modulus Packing

The distinct:

$$
2P(U)
$$

grow rapidly,

and the number of lattices that can simultaneously hit short intervals is constrained.

### B. Shared Prime Moduli

Many hypercells share prime factors,

so their CRT residues are not independent.

### C. Center Congruence

All residues are generated by the same:

$$
m
$$

through:

$$
0,
2m,
1
$$

CRT conditions.

### D. Nested Prime-Unions

If:

$$
U\subset V,
$$

then:

$$
2P(U)\mid2P(V).
$$

A higher-order lattice is a refinement of a lower-order lattice.

### E. Cofactor Packing

Each hit corresponds to:

$$
m\pm d
=
cP(S)
$$

type cofactor states.

Finite endpoint capacity can be used to constrain simultaneous hits.

---

# 38. Positive Moment as Lattice-Hit Sum

In exact form:

$$
\boxed{
B_j
=
\sum_{\substack{
|U|=j\\
\alpha
}}
\operatorname{APCount}
\left(
a_{U,\alpha},
b_{U,\alpha};
2P(U),
c_{U,\alpha}
\right).
}
$$

Thus, the positive-moment upper-bound problem has essentially been converted entirely into:

$$
\boxed{
\text{finite family of CRT arithmetic-progression hit counts}.
}
$$

The prime predicate is no longer in the formula.

---

# 39. Uniform Moment-Envelope Hierarchy

We can now define:

### Level 0 — Cap Upper

$$
B_j\le U_j^{cap}.
$$

### Level 1 — Box Upper

$$
B_j\le U_j^{box}.
$$

### Level 2 — Phase-Compressed Upper

$$
B_j\le V_j+U_{\Delta,j}.
$$

### Level 3 — Exact CRT

$$
B_j
\text{ exact}.
$$

The research target is:

> Obtain dual crossing at Level 2.

---

# 40. Negative Moment Hierarchy

On the negative-coefficient side:

### Level 0

Balanced generic lower bound.

### Level 1

All-right lower bound:

$$
L_j^R.
$$

### Level 2

Box lower bound:

$$
L_j^{box}.
$$

### Level 3

Selected exact CRT-seed lower bound.

### Level 4

Exact moment.

Current finite audits show that the negative side is already much closer to exact than the positive side.

---

# 41. New Gate

After SRBRD-10, the question should no longer be:

> Can we write an even more elegant polynomial?

There are already many polynomials that succeed with fixed exact moments.

The actual gate is:

$$
\boxed{
\text{Can we prove far fewer positive-moment CRT phase hits than the phase-free box upper bound?}
}
$$

---

# 42. Formal Theorem Queue

```yaml
SRBRD_10:
  T01: signed_moment_envelope_theorem
  T02: exact_B1_cell_formula
  T03: residue_free_AP_floor_ceil_bracket
  T04: hypercell_box_bounds
  T05: moment_box_envelopes
  T06: all_right_uniform_hypercell_lower_bound
  T07: uniform_signed_envelope_closure
  T08: volume_phase_moment_decomposition
  T09: aggregate_B1_cap_barrier
```

---

# 43. Acceptance Gate

Before SRBRD-10 enters the formal core, the following are needed:

1. the exact $B_1$ theorem;
2. the AP floor/ceil theorem;
3. a disjoint sum of canonical orientations;
4. hypercell box lower / upper bounds;
5. the moment box sum;
6. the all-right orientation theorem;
7. signed-polarity substitution;
8. volume-phase decomposition;
9. sparse-hit representation;
10. finite audit verified with exact rationals;
11. the generic aggregate barrier;
12. the phase-free sharpness statement;
13. no claim of uniform Goldbach closure.

---

# 44. Verdict

This round should give two verdicts.

## Positive

$$
\boxed{
\mathrm{SRBRD\mbox{-}10\ UNIFORM\ MOMENT\ ENVELOPE\ INTERFACE\ PASS}
}
$$

Because:

- exact $B_1$;
- uniform moment lower providers;
- uniform moment upper providers;
- signed polynomial substitution;

have formed a complete rigorous interface.

## Negative / Gate

$$
\boxed{
\mathrm{PHASE\mbox{-}FREE\ UNIFORM\ CLOSURE\ FAILS}
}
$$

The finite controls clearly show:

> Without CRT phase-hit compression, the positive-moment upper envelopes are so loose that they destroy the polynomial crossing already obtained from exact moments.

Thus:

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

---

# 45. Suggestion for the Next Round

# SRBRD-11
## CRT Phase-Hit Discrepancy / Lattice-Packing Closure

The next round should attack only one thing:

$$
\boxed{
\Delta_j^{phase}
}
$$

or directly:

$$
\boxed{
\Delta_P^{phase}.
}
$$

Priority routes:

1. nested product moduli;
2. shared-prime lattice refinement;
3. simultaneous-hit packing;
4. center-generated CRT residues;
5. endpoint cofactor capacity;
6. a large-sieve / spacing-type deterministic inequality;
7. first obtain a symbolic phase upper bound for fixed $j=3,4,5,6$;
8. then feed it back into the degree-4 / degree-6 dual polynomial.

If one can obtain here:

$$
\boxed{
U_{\Delta,P}
\ll
A_j^{box},
}
$$

only then is there a chance to genuinely turn the fixed-$m$ certificate of SRBRD-09 into a family theorem.

---

# Appendix A: Signed Provider Contract

```yaml
Polynomial:
  coefficients:
    lambda_j: rational
  majorant:
    P(k) >= indicator(k>=1)
    for:
      k: 0..kappa(m)

MomentProvider:
  if_lambda_positive:
    need:
      B_j <= U_j
  if_lambda_negative:
    need:
      L_j <= B_j

UnionUpper:
  sum_positive:
    lambda_j*U_j
  sum_negative:
    lambda_j*L_j
```

---

# Appendix B: Hypercell Box Contract

```yaml
oriented_cell:
  modulus:
    2*product(U)
  interval_length:
    ell_U_alpha

lower:
  floor(ell/(2P))

upper:
  ceil(ell/(2P))

uncertainty:
  per_orientation:
    <= 1
```

---

# Appendix C: Selected Provider Audit

### $m=1000$

```yaml
Omega: 499
exact:
  B1: 908
  B2: 656
  B3: 259

box:
  B2:
    lower: 573
    upper: 781
  B3:
    lower: 88
    upper: 1250

degree4_envelope_upper:
  value: 1232

status:
  crossing: false
```

### $m=5000$

```yaml
Omega: 2499
exact:
  B1: 5495
  B2: 5206
  B3: 2870
  B4: 944

box:
  B2:
    lower: 4841
    upper: 5621
  B3:
    lower: 1525
    upper: 10871
  B4:
    lower: 52
    upper: 78261

degree4_exact_moment_upper:
  exact: 52165/21
  decimal: 2484.047619...
  crossing: true

degree4_box_envelope_upper:
  exact: 172283/21
  decimal: 8203.952380...
  crossing: false

diagnosis:
  main_gap:
    positive_B3_phase_free_upper
```

---

# Appendix D: Volume–Phase Contract

```yaml
moment:
  B_j:
    equals:
      V_j + Delta_phase_j

volume:
  V_j:
    sum:
      interval_length/(2*product(U))

phase:
  Delta_phase_j:
    sum:
      orientation_AP_rounding_phase
  trivial_abs_bound:
    < number_of_nonempty_orientation_boxes

uniform_goal:
  prove_collective_phase_upper:
    much_smaller_than_trivial_orientation_count
```

---

# Appendix E: Commit Decision

```yaml
research_round: SRBRD-10

decision:
  signed_moment_envelope_interface: true
  exact_first_moment_provider: true
  residue_free_lower_providers: true
  residue_free_upper_providers: true
  phase_free_uniform_crossing: false
  uniform_goldbach_closure: false

main_barrier:
  CRT_phase_hit_discrepancy_for_positive_moments

next:
  SRBRD-11_CRT_PhaseHit_Discrepancy_LatticePacking_Closure
```