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

# RG-SCP Validation Round 06
## RG-SCP 06 Claim / Source Audit
### Canonical Capacity Reconstruction, Incidence Decentering, and AP Boundary Audit of Counterexample Demand–Obstruction Capacity Inequalities

**Validation series:** RG-SCP Validation  
**Validation stage:** V0 — Claim / Source Audit  
**Round:** 06  
**Corresponding original draft:** RG-SCP 06, Counterexample Demand–Obstruction Capacity Inequalities, v0.1  
**Date:** 2026-09-28  
**Version:** v0.1  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Nature:** Canonical Audit / Demand-Capacity Core / Source-Role Audit  
**Status:** Sixth audit round; no new global Goldbach proof is claimed

---

## Abstract

This document performs a claim/source audit of RG-SCP 06, Counterexample Demand–Obstruction Capacity Inequalities.

This round specifically checks:

1. how much independent proof content $D_N(y)\le I_N(y)$ actually has;
2. whether $R_G^{-}=D+\Omega-I$ is genuine compression or a target identity;
3. whether exact incidence $I_N(y)$ already reads too much complement factorization;
4. whether “capacity” should be defined as incidence mass or as the composite-survivor cardinality actually coverable;
5. whether the raw arithmetic-progression capacity upper bound is correct;
6. whether, when $r\mid N$, the capacity column can be strengthened from $\le1$ to an exact formula;
7. whether self-suppression on the small-factor demand side can also be strengthened;
8. whether the primorial demand formula duplicates RG-SCP 03;
9. whether the conditions of the Brun–Titchmarsh interface are correct;
10. whether demand / capacity closure has circularity or truth leakage.

The most important conclusion of this round is:

> **The genuinely canonical capacity in the sixth article should not be the multiplicity-weighted incidence $I_N(y)$, but the “number of composite survivors actually covered by deep proper factors.”**

For:

$$
2\le y<\frac N2,
$$

Define the survivor set:

$$
\mathbb H_y(N)
=
\left\{
p\in\mathbb P_N^{-}:
\gcd(N-p,P(y))=1
\right\},
$$

and demand:

$$
D_N(y)
=
|\mathbb H_y(N)|.
$$

Then define the reduced Goldbach-success set:

$$
\boxed{
\mathbb G_N^{-}
=
\left\{
p\in\mathbb P_N^{-}:
N-p\in\mathbb P
\right\}.
}
$$

From:

$$
y<N/2
$$

and:

$$
N-p\ge N/2,
$$

every prime complement automatically avoids:

$$
P(y).
$$

Thus:

$$
\boxed{
\mathbb G_N^{-}
\subseteq
\mathbb H_y(N).
}
$$

Next, define the canonical composite-survivor set:

$$
\boxed{
\mathbb C_y(N)
=
\left\{
p\in\mathbb H_y(N):
N-p\text{ composite}
\right\}.
}
$$

Then there is an exact partition completely free of multiplicity:

$$
\boxed{
\mathbb H_y(N)
=
\mathbb G_N^{-}
\bigsqcup
\mathbb C_y(N).
}
$$

Therefore:

$$
\boxed{
D_N(y)
=
R_G^{-}(N)
+
C_N^{\mathrm{can}}(y),
}
$$

where:

$$
\boxed{
C_N^{\mathrm{can}}(y)
=
|\mathbb C_y(N)|.
}
$$

Thus:

$$
\boxed{
R_G^{-}(N)
=
D_N(y)
-
C_N^{\mathrm{can}}(y).
}
$$

This is the canonical demand–capacity identity that this round recommends promoting for RG-SCP 06.

Then, using the least-factor partition:

$$
B_r(N;y)
=
\left\{
p\in\mathbb H_y(N):
N-p\text{ composite},
\quad
\operatorname{lpf}(N-p)=r
\right\},
$$

This gives:

$$
\boxed{
\mathbb C_y(N)
=
\bigsqcup_{\substack{
r\in\mathbb P\\
r>y
}}
B_r(N;y),
}
$$

Therefore:

$$
\boxed{
C_N^{\mathrm{can}}(y)
=
\sum_{r>y}
|B_r(N;y)|.
}
$$

Goldbach failure now takes its cleanest form:

$$
\boxed{
F_G(N)=1
\Longleftrightarrow
D_N(y)
=
C_N^{\mathrm{can}}(y)
}
$$

for any:

$$
2\le y<N/2.
$$

Thus, the genuine proof target should be:

$$
\boxed{
\text{Demand Lower Bound}
>
\text{Canonical Coverage Capacity Upper Bound}.
}
$$

That is, if one can prove:

$$
D_N(y)\ge L_N(y),
$$

$$
C_N^{\mathrm{can}}(y)\le U_N^{\mathrm{can}}(y),
$$

and:

$$
\boxed{
L_N(y)>U_N^{\mathrm{can}}(y),
}
$$

we immediately have:

$$
R_G^{-}(N)>0.
$$

The original draft’s:

$$
I_N(y)
$$

is not wrong; it is exact **deep factor-incidence mass**.

whereas:

$$
\Omega_N(y)
$$

is exact **incidence-overlap surplus**.

This round proves:

$$
\boxed{
I_N(y)
=
C_N^{\mathrm{can}}(y)
+
\Omega_N(y).
}
$$

Therefore:

$$
R_G^{-}
=
D+\Omega-I
$$

is merely:

$$
R_G^{-}
=
D-C^{\mathrm{can}}
$$

with:

$$
C^{\mathrm{can}}
=
I-\Omega
$$

substituted back, giving an equivalent expansion.

Thus:

- $D+\Omega-I$ should be retained as an audit identity;
- $I$ should be renamed `incidence mass`;
- genuine canonical capacity should use $C_N^{\mathrm{can}}$ instead.

This also explains precisely:

$$
D_N(y)>I_N(y)
$$

what it actually detects.

From:

$$
R_G^{-}
=
D+\Omega-I,
$$

we have:

$$
\boxed{
D>I
\Longleftrightarrow
R_G^{-}>\Omega.
}
$$

Thus, $D>I$ is not a general Goldbach-success characterization.

It triggers only when:

> The Goldbach representation count exceeds the incidence-overlap surplus

this condition holds.

For example, in the finite example:

$$
N=68,
\quad
y=2
$$

one can have:

$$
D=10,
\quad
I=10,
\quad
\Omega=2,
\quad
R_G^{-}=2.
$$

Therefore:

$$
R_G^{-}>0
$$

but:

$$
D>I
$$

does not hold.

Thus, the implication in RG-SCP-06.1 is correct, but weak as a direct detector.

Genuine proof value comes from:

> Establishing a rigorous upper bound from independent structure rather than exactly computing $I$ or $C^{\mathrm{can}}$.

This round also strengthens the self-suppression AP column.

The original draft only has:

$$
r\mid N
\Longrightarrow
A_r(N)\le1.
$$

In fact, this can be written exactly as:

$$
\boxed{
A_r(N)
=
\mathbf1_{\{r\le x_r(N)\}}
}
$$

where:

$$
x_r(N)
=
\min
\left(
N/2,
N-r^2
\right).
$$

Because:

$$
r\le x_r(N)
$$

is equivalent to:

$$
N\ge r^2+r,
$$

Thus:

$$
\boxed{
r\mid N
\Longrightarrow
A_r(N)
=
\mathbf1_{\{N\ge r(r+1)\}}.
}
$$

The small-factor demand side can also be strengthened.

If:

$$
\ell\le y<N/2,
$$

and:

$$
\ell\mid N,
$$

Then:

$$
\boxed{
S_\ell(N)=\{\ell\},
}
$$

rather than merely knowing:

$$
|S_\ell(N)|\le1.
$$

Both exact sharpenings should enter the formal core.

Finally, the external analytic interface audit confirms that the Brun–Titchmarsh form cited by the original draft:

$$
\pi(x;q,a)
\le
\frac{2x}{
\varphi(q)\log(x/q)
}
$$

under:

$$
\gcd(a,q)=1,
\quad
q<x
$$

has the correct form under these standard applicability conditions.

But it should still remain in:

$$
\boxed{
\mathcal S_5
=
\text{External Analytic Source}
}
$$

rather than being incorporated into the elementary core.

The verdict of this round:

$$
\boxed{
\mathrm{PASS\ WITH\ CANONICAL\mbox{-}CAPACITY\ RECONSTRUCTION,\ INCIDENCE\ DEMOTION,\ AND\ SUPPRESSION\ SHARPENING}.
}
$$

---

# 1. Audit Scope

This round audits only:

# RG-SCP 06
## Counterexample Demand–Obstruction Capacity Inequalities

This round does not audit RG-SCP 07 minimality transport in advance.

It also retains V0-R05’s two-dimensional data classification:

## Epistemic Status

```text
exact
heuristic
mixed
```

## Data Role

```text
proof_input
discovery_probe
diagnostic
truth_label
```

---

# 2. Reduced prime domain

Retain:

$$
\boxed{
\mathbb P_N^{-}
=
\left\{
p\in\mathbb P:
2\le p\le N/2
\right\}.
}
$$

Define:

$$
\boxed{
R_G^{-}(N)
=
\#\left\{
p\in\mathbb P_N^{-}:
N-p\in\mathbb P
\right\}.
}
$$

we have:

$$
F_G(N)=1
\Longleftrightarrow
R_G^{-}(N)=0.
$$

---

# 3. Cutoff and Small-Factor Product

For:

$$
2\le y<N/2,
$$

Define:

$$
\boxed{
P(y)
=
\prod_{\substack{
\ell\in\mathbb P\\
\ell\le y
}}
\ell.
}
$$

Canonical formalization recommends letting:

$$
y
$$

take only natural-number values or prime cutoff indices,

to avoid unnecessary real-valued cutoff coercions.

---

# 4. Survivor set

Define:

$$
\boxed{
\mathbb H_y(N)
=
\left\{
p\in\mathbb P_N^{-}:
\gcd(N-p,P(y))=1
\right\}.
}
$$

and:

$$
\boxed{
D_N(y)
=
|\mathbb H_y(N)|.
}
$$

These use only:

- prime input $p$;
- divisibility by small primes.

They do not need to know:

$$
N-p
$$

whether this ultimately is prime.

Therefore:

```yaml
D_N(y):
  epistemic_status: exact
  data_role: proof_input_or_discovery_probe
  truth_leakage: low
```

---

# 5. Prime Complements Always Survive

## Theorem R06-T01

If:

$$
2\le y<N/2,
$$

and:

$$
p\in\mathbb P_N^{-},
$$

and:

$$
N-p\in\mathbb P,
$$

Then:

$$
\boxed{
p\in\mathbb H_y(N).
}
$$

---

## Proof

Because:

$$
p\le N/2,
$$

Thus:

$$
N-p\ge N/2>y.
$$

If some:

$$
\ell\le y
$$

prime divides:

$$
N-p,
$$

Because:

$$
N-p
$$

is itself prime,

can only be:

$$
\ell=N-p,
$$

but:

$$
\ell\le y<N-p,
$$

a contradiction.

Therefore:

$$
\gcd(N-p,P(y))=1.
$$

This completes the proof.

---

# 6. Goldbach-success set

Define:

$$
\boxed{
\mathbb G_N^{-}
=
\left\{
p\in\mathbb P_N^{-}:
N-p\in\mathbb P
\right\}.
}
$$

By R06-T01:

$$
\boxed{
\mathbb G_N^{-}
\subseteq
\mathbb H_y(N).
}
$$

and:

$$
|\mathbb G_N^{-}|
=
R_G^{-}(N).
$$

---

# 7. Composite survivor set

Define:

$$
\boxed{
\mathbb C_y(N)
=
\left\{
p\in\mathbb H_y(N):
N-p\text{ composite}
\right\}.
}
$$

Then every survivor complement:

$$
N-p\ge2
$$

is either prime or composite.

Thus:

$$
\boxed{
\mathbb H_y(N)
=
\mathbb G_N^{-}
\bigsqcup
\mathbb C_y(N).
}
$$

Claim class:

$$
[\mathrm T].
$$

---

# 8. Canonical coverage capacity

Define:

$$
\boxed{
C_N^{\mathrm{can}}(y)
=
|\mathbb C_y(N)|.
}
$$

It is called:

# Canonical Composite-Survivor Capacity

It represents:

> The number of left vertices within survivor demand actually covered by deep proper-factor structure.

---

# 9. Canonical Demand–Capacity Identity

## Theorem R06-T02

For:

$$
2\le y<N/2,
$$

we have:

$$
\boxed{
D_N(y)
=
R_G^{-}(N)
+
C_N^{\mathrm{can}}(y).
}
$$

Thus:

$$
\boxed{
R_G^{-}(N)
=
D_N(y)
-
C_N^{\mathrm{can}}(y).
}
$$

This is the canonical identity that this round recommends promoting.

---

# 10. Failure characterization

By R06-T02:

$$
\boxed{
F_G(N)=1
\Longleftrightarrow
D_N(y)
=
C_N^{\mathrm{can}}(y).
}
$$

whereas:

$$
R_G^{-}(N)>0
$$

is equivalent to:

$$
\boxed{
D_N(y)
>
C_N^{\mathrm{can}}(y).
}
$$

This is the most direct demand / actual-coverage relation.

---

# 11. Note: $C_N^{\mathrm{can}}$ Is Target-Adjacent

Although:

$$
C_N^{\mathrm{can}}(y)
$$

is exact,

its definition directly asks:

$$
N-p
$$

whether it is composite.

Also:

$$
R_G^{-}
=
D-C^{\mathrm{can}}.
$$

Thus:

```yaml
C_can:
  epistemic_status: exact
  data_role: truth_label_adjacent_or_posthoc_diagnostic
```

One cannot treat the exact:

$$
C_N^{\mathrm{can}}
$$

itself as a leakage-free discovery input.

Genuine proof value comes from:

$$
\boxed{
C_N^{\mathrm{can}}(y)
\le
U_N^{\mathrm{can}}(y)
}
$$

an independent upper bound of this form.

---

# 12. Canonical least-factor cells

For:

$$
r>y,
$$

Define:

$$
\boxed{
B_r(N;y)
=
\left\{
p\in\mathbb H_y(N):
N-p\text{ composite},
\quad
\operatorname{lpf}(N-p)=r
\right\}.
}
$$

Adding:

$$
N-p\text{ composite}
$$

is necessary typing.

Otherwise, a prime complement:

$$
q=N-p
$$

also has:

$$
\operatorname{lpf}(q)=q,
$$

and would wrongly enter a “proper obstruction cell.”

---

# 13. Canonical cell partition

For distinct:

$$
r
$$

the cells:

$$
B_r(N;y)
$$

are disjoint.

whereas:

$$
p\in\mathbb C_y(N)
$$

when this holds,

its least factor:

$$
r=\operatorname{lpf}(N-p)
$$

satisfying:

$$
r>y.
$$

Thus:

$$
\boxed{
\mathbb C_y(N)
=
\bigsqcup_{\substack{
r\in\mathbb P\\
r>y
}}
B_r(N;y).
}
$$

Therefore:

$$
\boxed{
C_N^{\mathrm{can}}(y)
=
\sum_{r>y}
|B_r(N;y)|.
}
$$

---

# 14. Canonical closure theorem

## Theorem R06-T03

If there exist rigorous bounds:

$$
D_N(y)\ge L_N(y),
$$

and:

$$
C_N^{\mathrm{can}}(y)
\le
U_N^{\mathrm{can}}(y),
$$

and:

$$
\boxed{
L_N(y)>U_N^{\mathrm{can}}(y),
}
$$

Then:

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

---

## Proof

From:

$$
R_G^{-}
=
D-C^{\mathrm{can}},
$$

we obtain:

$$
R_G^{-}
\ge
L-U^{\mathrm{can}}
>
0.
$$

This completes the proof.

---

# 15. Canonical branch closure

If, for a family:

$$
\mathcal F
$$

one can prove:

$$
L_N(y_N)
>
U_N^{\mathrm{can}}(y_N)
$$

for every:

$$
N\in\mathcal F,
$$

Then:

$$
\boxed{
\mathcal F
\cap
\mathcal C_G
=
\varnothing.
}
$$

This is RG-SCP 06’s cleanest family-level target.

---

# 16. Deep incidence multiplicity

The original draft defines:

$$
m_y(N,p)
=
\#\left\{
r\in\mathbb P:
y<r,
\quad
r^2\le N-p,
\quad
r\mid(N-p)
\right\}.
$$

This round replaces the real square root with square form.

For a survivor:

- prime complement: $m_y=0$;
- composite complement: $m_y\ge1$.

exact.

---

# 17. Deep incidence mass

Define:

$$
\boxed{
I_N(y)
=
\sum_{p\in\mathbb H_y(N)}
m_y(N,p).
}
$$

This round recommends renaming it:

# Deep Factor-Incidence Mass

rather than calling it canonical capacity.

Reason:

If the same composite survivor has several deep prime divisors,

it is counted repeatedly.

---

# 18. Overlap surplus

Define:

$$
\boxed{
\Omega_N(y)
=
\sum_{p\in\mathbb H_y(N)}
\max
\left(
m_y(N,p)-1,
0
\right).
}
$$

exact.

---

# 19. Incidence–Coverage Identity

## Theorem R06-T04

we have:

$$
\boxed{
I_N(y)
=
C_N^{\mathrm{can}}(y)
+
\Omega_N(y).
}
$$

---

## Proof

For each:

$$
p\in\mathbb H_y(N),
$$

If the complement is prime:

$$
m_y=0,
$$

and contributes neither to:

$$
C^{\mathrm{can}}
$$

nor to:

$$
\Omega.
$$

If the complement is composite:

$$
m_y\ge1,
$$

it contributes to:

$$
C^{\mathrm{can}}
$$

the amount:

$$
1,
$$

For:

$$
\Omega
$$

it contributes:

$$
m_y-1.
$$

Sum:

$$
m_y.
$$

Summing element by element gives the result.

This completes the proof.

---

# 20. Source of the Original Identity

From:

$$
R_G^{-}
=
D-C^{\mathrm{can}}
$$

and:

$$
C^{\mathrm{can}}
=
I-\Omega,
$$

we immediately obtain:

$$
\boxed{
R_G^{-}
=
D+\Omega-I.
}
$$

Thus, the original draft’s RG-SCP-06.4 is entirely correct.

But it is now repositioned as:

$$
\boxed{
\text{derived audit identity}.
}
$$

rather than a canonical capacity definition.

---

# 21. The Actual Status of $D\le I$

If there is failure:

$$
R_G^{-}=0,
$$

Then:

$$
D=C^{\mathrm{can}}.
$$

Also:

$$
I=C^{\mathrm{can}}+\Omega.
$$

Thus:

$$
\boxed{
I=D+\Omega\ge D.
}
$$

Therefore:

$$
F_G(N)=1
\Longrightarrow
D_N(y)\le I_N(y)
$$

PASS.

But it is actually a weak corollary of the canonical partition with incidence multiplicity added.

---

# 22. What $D>I$ Actually Means

From:

$$
R_G^{-}
=
D+\Omega-I,
$$

we have:

$$
D-I
=
R_G^{-}-\Omega.
$$

Thus:

$$
\boxed{
D>I
\Longleftrightarrow
R_G^{-}>\Omega.
}
$$

Thus, the original draft’s:

$$
D>I
\Longrightarrow
R_G^{-}>0
$$

is correct,

but captures only:

$$
\boxed{
\text{representation count exceeds overlap surplus}
}
$$

success states of this kind.

It is not a general success characterization.

---

# 23. Finite Example: Success Without Triggering $D>I$

Take:

$$
N=68,
\quad
y=2.
$$

The reduced survivor domain has:

$$
D_{68}(2)=10.
$$

where:

$$
68-5=63
$$

has deep factors:

$$
3,7,
$$

whereas:

$$
68-23=45
$$

has deep factors:

$$
3,5.
$$

Thus, the overlap surplus:

$$
\Omega_{68}(2)=2.
$$

One obtains:

$$
I_{68}(2)=10.
$$

At the same time, the reduced Goldbach representations include:

$$
68=7+61
$$

and:

$$
68=31+37.
$$

Thus:

$$
R_G^{-}(68)=2.
$$

The exact identity:

$$
2
=
10+2-10.
$$

But:

$$
D>I
$$

is:

$$
10>10,
$$

does not hold.

This shows:

$$
\boxed{
D>I
}
$$

can serve only as a sufficient detector,

not as a canonical capacity criterion.

---

# 24. Truth-Leakage Status of Exact $I$

Exactly computing:

$$
I_N(y),
$$

usually requires knowing the deep prime factors of survivor complements.

This is already very close to complete factorization truth.

whereas:

$$
R_G^{-}
=
D+\Omega-I
$$

further shows:

$$
(I,\Omega,D)
$$

together directly reconstruct the target.

Thus, the exact:

$$
I_N(y)
$$

should be labeled:

```yaml
epistemic_status: exact
data_role: posthoc_diagnostic_or_truth_adjacent
```

rather than:

```yaml
data_role: leakage_free_discovery_probe
```

---

# 25. The Genuine Proof Input Is an Upper Bound

Important distinction:

$$
I_N(y)
$$

its exact value is not proof compression.

But if an external theorem that does not read complement factorization yields:

$$
\boxed{
I_N(y)\le U_N(y),
}
$$

Then:

$$
U_N
$$

can be a legitimate proof input.

Similarly:

$$
C_N^{\mathrm{can}}(y)
\le
U_N^{\mathrm{can}}(y)
$$

is the capacity theorem genuinely worth pursuing.

---

# 26. Incidence decomposition

For:

$$
r>y,
$$

Define:

$$
C_r^{\mathrm{inc}}(N;y)
=
\#\left\{
p\in\mathbb H_y(N):
r^2\le N-p,
\quad
r\mid(N-p)
\right\}.
$$

Then:

$$
\boxed{
I_N(y)
=
\sum_{r>y}
C_r^{\mathrm{inc}}(N;y).
}
$$

To avoid confusion with canonical capacity:

$$
C_N^{\mathrm{can}}
$$

this quantity,

this round recommends renaming the original draft’s:

$$
C_r(N;y)
$$

rename it:

$$
\boxed{
C_r^{\mathrm{inc}}(N;y).
}
$$

---

# 27. Proper-factor range

$$
r^2\le N-p
$$

is equivalent to:

$$
p\le N-r^2.
$$

Also:

$$
p\le N/2.
$$

Thus:

$$
\boxed{
x_r(N)
=
\min
\left(
N/2,
N-r^2
\right).
}
$$

Then:

$$
C_r^{\mathrm{inc}}(N;y)
=
\#
\left\{
p\in\mathbb H_y(N):
p\le x_r(N),
\quad
p\equiv N\pmod r
\right\}.
$$

PASS.

---

# 28. Raw AP column

Define:

$$
\boxed{
A_r(N)
=
\#\left\{
p\in\mathbb P:
2\le p\le x_r(N),
\quad
p\equiv N\pmod r
\right\}.
}
$$

Removing:

$$
p\in\mathbb H_y
$$

this restriction,

we have:

$$
\boxed{
C_r^{\mathrm{inc}}(N;y)
\le
A_r(N).
}
$$

exact.

---

# 29. Raw AP sum upper bound

Therefore:

$$
\boxed{
I_N(y)
\le
\sum_{\substack{
r\in\mathbb P\\
y<r\\
r^2\le N-2
}}
A_r(N).
}
$$

exact.

This is a legitimate upper envelope,

Because:

$$
A_r
$$

requires only:

- left prime distribution;
- residue;
- proper-factor range;

and does not need to determine:

$$
N-p
$$

whether it is prime.

---

# 30. Data Role of the AP Envelope

Thus:

```yaml
A_r(N):
  epistemic_status: exact
  data_role: proof_input
  target_leakage: low
```

If an analytic theorem is then used to upper-bound:

$$
A_r(N),
$$

this is likewise legitimate.

---

# 31. Divisor column strengthening

The original draft:

$$
r\mid N
\Longrightarrow
A_r(N)\le1.
$$

can be strengthened.

Because:

$$
p\equiv N\pmod r
$$

and:

$$
r\mid N
$$

implies:

$$
r\mid p.
$$

The prime:

$$
p
$$

can only satisfy:

$$
p=r.
$$

Thus:

$$
A_r(N)=1
$$

if and only if:

$$
r
$$

lies in the admissible range:

$$
2\le r\le x_r(N).
$$

Therefore:

$$
\boxed{
A_r(N)
=
\mathbf1_{\{r\le x_r(N)\}}
}
$$

For:

$$
r\mid N.
$$

---

# 32. A Simpler Exact Divisor-Column Formula

Because:

$$
x_r(N)
=
\min(N/2,N-r^2),
$$

For:

$$
r\ge2,
$$

the condition:

$$
r\le x_r(N)
$$

is equivalent to:

$$
N\ge r^2+r.
$$

Because:

$$
r^2+r\ge2r.
$$

Thus:

$$
\boxed{
r\mid N
\Longrightarrow
A_r(N)
=
\mathbf1_{\{N\ge r(r+1)\}}.
}
$$

This is Strengthening S01.

---

# 33. Non-divisor AP column

If:

$$
r\nmid N,
$$

Because $r$ is prime:

$$
\gcd(N,r)=1.
$$

Thus:

$$
\boxed{
A_r(N)
=
\pi
\left(
x_r(N);
r,
N\bmod r
\right).
}
$$

exact.

---

# 34. Brun–Titchmarsh interface

If:

$$
r\nmid N
$$

and:

$$
r<x_r(N),
$$

the standard Brun–Titchmarsh upper bound can be connected:

$$
\boxed{
\pi(x_r;r,N\bmod r)
\le
\frac{
2x_r
}{
(r-1)\log(x_r/r)
}.
}
$$

This theorem belongs to:

$$
\boxed{
\mathcal S_5
=
\text{External Analytic Number Theory}.
}
$$

This round only checks interface compatibility,

and does not incorporate its proof into the RG-SCP elementary core.

---

# 35. $x_r\le r$ fallback

If:

$$
x_r(N)\le r,
$$

a fixed residue class:

$$
\bmod r
$$

in:

$$
[1,x_r]
$$

has at most one integer representative.

Thus:

$$
A_r(N)\le1.
$$

PASS.

But if:

$$
r\mid N,
$$

the preceding exact indicator should be preferred.

---

# 36. Canonical AP envelope

One can define:

$$
\boxed{
U_r^{\mathrm{AP}}(N)
=
\begin{cases}
\mathbf1_{\{N\ge r(r+1)\}},
& r\mid N,\\
\pi(x_r;r,N\bmod r),
& r\nmid N.
\end{cases}
}
$$

This is tighter than uniformly writing the original draft’s divisor case as:

$$
1
$$

this value.

If the second branch is replaced by a rigorous analytic upper bound,

a computable symbolic envelope is obtained.

---

# 37. Small-factor demand classes

For:

$$
\ell\le y,
$$

Define:

$$
S_\ell(N)
=
\left\{
p\in\mathbb P_N^{-}:
\ell\mid(N-p)
\right\}.
$$

Then:

$$
\boxed{
\mathbb H_y(N)
=
\mathbb P_N^{-}
\setminus
\bigcup_{\ell\le y}
S_\ell(N).
}
$$

PASS.

---

# 38. Demand exact complement form

$$
\boxed{
D_N(y)
=
\pi(N/2)
-
\left|
\bigcup_{\ell\le y}
S_\ell(N)
\right|.
}
$$

exact.

---

# 39. Demand union-bound lower envelope

From:

$$
\left|
\bigcup_{\ell\le y}
S_\ell
\right|
\le
\sum_{\ell\le y}
|S_\ell|,
$$

we have:

$$
\boxed{
D_N(y)
\ge
\pi(N/2)
-
\sum_{\ell\le y}
|S_\ell(N)|.
}
$$

PASS.

---

# 40. The Small-Factor Self-Cell Can Be Strengthened

If:

$$
\ell\le y<N/2
$$

is prime,

and:

$$
\ell\mid N,
$$

Then:

$$
p\equiv N\pmod\ell
$$

among primes $p$, can only give:

$$
p=\ell.
$$

whereas:

$$
\ell<N/2
$$

guarantees:

$$
\ell\in\mathbb P_N^{-}.
$$

Also:

$$
\ell\mid(N-\ell).
$$

Therefore:

$$
\boxed{
S_\ell(N)=\{\ell\}.
}
$$

rather than merely:

$$
|S_\ell|\le1.
$$

This is Strengthening S02.

---

# 41. General Demand Lower Envelope

If:

$$
V_\ell(N)\ge|S_\ell(N)|
$$

are rigorous upper bounds,

Then:

$$
\boxed{
L_N(y)
=
\max
\left(
0,
\pi(N/2)
-
\sum_{\ell\le y}
V_\ell(N)
\right)
}
$$

satisfying:

$$
D_N(y)\ge L_N(y).
$$

PASS.

---

# 42. Primorial demand formula

If:

$$
M_k\mid N,
$$

and:

$$
p_k\le N/2,
$$

then, by V0-R03:

$$
\boxed{
\mathbb H_{p_k}(N)
=
\left\{
p\in\mathbb P:
p_k<p\le N/2
\right\}.
}
$$

Thus:

$$
\boxed{
D_N(p_k)
=
\pi(N/2)-k.
}
$$

exact.

---

# 43. Deduplicating Theorem Ownership

The arithmetic geometry of this formula already belongs to:

$$
\boxed{
\text{RG-SCP 03}.
}
$$

The new content of RG-SCP 06 is:

> Connecting this exact demand to the capacity upper-bound architecture.

Thus, the ledger should state:

```text
R03:
  primorial hard-sector theorem

R06:
  use hard-sector cardinality as demand
  compare against canonical/AP capacity
```

---

# 44. The Same Applies to M6 Demand

If:

$$
6\mid N,
$$

and:

$$
N\ge12,
$$

Then:

$$
\boxed{
D_N(3)
=
\pi(N/2)-2.
}
$$

This likewise derives from R03 hard-sector geometry.

R06 is responsible only for the capacity interpretation.

---

# 45. Canonical primorial closure target

For:

$$
M_k\mid N,
$$

The genuinely clean target should be written as:

$$
\boxed{
\pi(N/2)-k
>
U_N^{\mathrm{can}}(p_k).
}
$$

rather than giving priority to:

$$
\pi(N/2)-k>I_N(p_k).
$$

because the exact:

$$
I_N
$$

carries multiplicity and reads factor truth.

---

# 46. Connecting the Raw AP Envelope to Canonical Capacity

Because:

$$
B_r(N;y)
\subseteq
\left\{
p\in\mathbb P:
p\le x_r,
\quad
p\equiv N\pmod r
\right\},
$$

Thus:

$$
\boxed{
|B_r(N;y)|
\le
A_r(N).
}
$$

Therefore:

$$
\boxed{
C_N^{\mathrm{can}}(y)
\le
\sum_{r>y}
A_r(N).
}
$$

Thus, the raw AP sum can still directly upper-bound canonical capacity.

This ensures that canonical reconstruction does not discard the original draft’s AP route.

---

# 47. Why the Canonical Route Is Cleaner

Original route:

$$
D
\le
I
\le
\sum_r A_r.
$$

Canonical route:

$$
R
=
D-C^{\mathrm{can}},
$$

and:

$$
C^{\mathrm{can}}
\le
\sum_r A_r.
$$

If:

$$
D>\sum_rA_r,
$$

still implies:

$$
R>0.
$$

But repeated factor incidences need not be called capacity.

---

# 48. Canonical cell upper bound

More ideally:

$$
\boxed{
|B_r(N;y)|
\le
U_r^{\mathrm{sieve}}(N;y).
}
$$

Then:

$$
\boxed{
C_N^{\mathrm{can}}(y)
\le
\sum_{r>y}
U_r^{\mathrm{sieve}}(N;y).
}
$$

If:

$$
L_N(y)
>
\sum_{r>y}
U_r^{\mathrm{sieve}}(N;y),
$$

then failure can be excluded.

This retains the original draft’s most important future target.

---

# 49. Classification of “$B_r$ Is Usually Much Smaller than $A_r$”

exact:

$$
B_r\subseteq A_r.
$$

but:

> $B_r$ is usually much smaller than $A_r$

is an empirical / heuristic expectation.

Thus:

$$
[\mathrm H].
$$

Do not write “thus usually much smaller” in theorem wording.

---

# 50. Parity-barrier wording

The original draft states:

> This is where the parity problem appears.

This round recommends changing it to:

> Sharp upper bounds on canonical least-factor capacity may be close to sieve parity-type limitations, but RG-SCP 06 has not proved the two completely equivalent.

Thus, classify this as:

$$
[\mathrm F/H_{\mathrm{caution}}].
$$

Current failure to close cannot automatically be attributed to the parity barrier.

---

# 51. The Actual Defects of the Raw AP Sum

The original draft says the raw AP sum is loose because of overlap.

Correct.

After canonical reconstruction, however, one can state more precisely:

$$
\sum_r A_r
$$

has two forms of slack:

## S1 — Survivor slack

$$
A_r
$$

does not require:

$$
p\in\mathbb H_y.
$$

## S2 — Least-factor slack

$$
A_r
$$

requires only:

$$
r\mid(N-p),
$$

does not require:

$$
r=\operatorname{lpf}(N-p).
$$

Thus, even without discussing “counting the same p multiple times,”

the raw column itself is already larger than the canonical cell.

---

# 52. Overlap Is Not the Only Bottleneck

Thus, capacity compression should be decomposed as:

$$
\boxed{
A_r
\to
C_r^{\mathrm{inc}}
\to
B_r.
}
$$

The first step adds:

$$
p\in\mathbb H_y.
$$

The second step adds:

$$
r=\operatorname{lpf}(N-p).
$$

This is clearer than merely saying “subtract overlap.”

---

# 53. Inclusion–exclusion route

The original draft’s Chinese-remainder overlap analysis is exact.

If:

$$
r,s\mid(N-p),
$$

for distinct primes:

$$
r\neq s,
$$

Then:

$$
rs\mid(N-p),
$$

Thus:

$$
p\equiv N\pmod{rs}.
$$

PASS.

But this still belongs to:

$$
\mathcal S_1+\mathcal S_2.
$$

There is no source escape.

---

# 54. Matching analogy audit

The original draft correctly warns:

One obstruction prime:

$$
r
$$

can cover many:

$$
p.
$$

Thus, the obstruction graph cannot directly be treated as capacity-one matching.

PASS.

---

# 55. Local capacity inequality

If, for:

$$
S\subseteq\mathbb H_y(N)
$$

there is a rigorous coverage bound for each right obstruction class:

$$
u_r(S),
$$

Then:

$$
|S|
\le
\sum_r u_r(S)
$$

is a necessary condition for failure.

This is a valid template.

But the definition of $u_r(S)$ must:

- not use target truth circularly;
- upper-bound actual canonical coverage;
- handle overlap clearly.

---

# 56. Truth-Leakage Audit: $D$ vs $I$ vs $A$

## $D_N(y)$

Uses only small factors:

$$
\boxed{
\text{safe exact proof/discovery input}.
}
$$

## $I_N(y)$

Reads deep factorization:

$$
\boxed{
\text{truth-adjacent posthoc diagnostic}.
}
$$

## $\Omega_N(y)$

Likewise depends on deep factorization:

$$
\boxed{
\text{truth-adjacent posthoc diagnostic}.
}
$$

## $C_N^{\mathrm{can}}(y)$

Is directly equal to:

$$
D-R,
$$

Thus:

$$
\boxed{
\text{truth-label adjacent}.
}
$$

## $A_r(N)$

Uses only left primes and AP residues:

$$
\boxed{
\text{legitimate proof input}.
}
$$

---

# 57. Legitimate Form of a Capacity Theorem

The best proof route is not:

> Compute $C^{\mathrm{can}}$ exactly and find it small.

instead:

> Without reading complement primality, prove from an independent source:

$$
\boxed{
C_N^{\mathrm{can}}(y)
\le
U_N^{\mathrm{can}}(y).
}
$$

This is genuine proof compression.

---

# 58. Cutoff trade-off PASS

Increasing:

$$
y
$$

shrinks:

$$
D_N(y),
$$

while also shortening the available deep-factor range.

This trade-off exists exactly.

But “the optimal cutoff is in the middle” is only a hypothesis.

---

# 59. Data Role of Cutoff Optimization

If one uses the exact target-derived:

$$
I,
\Omega,C^{\mathrm{can}}
$$

to select, for every $N$, the best:

$$
y,
$$

and then reports model performance,

target-dependent cutoff leakage may result.

Thus, a heuristic cutoff selector should use only leakage-free features.

At the proof stage, one can choose any:

$$
y(N)
$$

provided that its definition and proof are independent.

---

# 60. Canonical Correction C01 — Capacity rename

Old:

```text
I_N(y) = capacity
```

New:

```text
I_N(y) = deep_factor_incidence_mass
C_can_N(y) = canonical_coverage_capacity
```

---

# 61. Canonical Correction C02 — Primary identity

Old primary:

$$
R=D+\Omega-I.
$$

New primary:

$$
\boxed{
R=D-C^{\mathrm{can}}.
}
$$

The old identity is retained as:

$$
\boxed{
C^{\mathrm{can}}=I-\Omega.
}
$$

---

# 62. Canonical Correction C03 — Least-factor cell typing

$$
B_r
$$

explicitly requires:

$$
N-p\text{ composite}
$$

or:

$$
r^2\le N-p.
$$

to avoid treating a prime complement as an obstruction cell.

---

# 63. Canonical Correction C04 — AP divisor column

Old:

$$
r\mid N
\Rightarrow
A_r\le1.
$$

New:

$$
\boxed{
A_r
=
\mathbf1_{\{N\ge r(r+1)\}}.
}
$$

---

# 64. Canonical Correction C05 — Small-factor suppression cell

Old:

$$
\ell\mid N
\Rightarrow
|S_\ell|\le1.
$$

New:

$$
\boxed{
S_\ell=\{\ell\}
}
$$

under:

$$
\ell\le y<N/2.
$$

---

# 65. Canonical Correction C06 — Data roles

exact:

$$
I,\Omega,C^{\mathrm{can}}
$$

are no longer treated as leakage-free discovery probes.

The upper/lower theorem bounds:

$$
L,U
$$

are the main body of the proof architecture.

---

# 66. Canonical Correction C07 — R03 ownership

The primorial demand formula belongs to the R03 theorem ledger.

R06 claims only:

$$
\boxed{
\text{demand-capacity use}.
}
$$

---

# 67. Canonical Correction C08 — External analytic isolation

Brun–Titchmarsh:

$$
\mathcal S_5.
$$

does not enter:

```text
RGSCP/Core
```

but enters:

```text
RGSCP/Interfaces
RGSCP/External
```

---

# 68. Canonical theorem ledger — RG-SCP 06

| ID | Claim | Class | Source | Status |
|---|---|---|---|---|
| R06-T01 | every reduced prime complement survives cutoff $y<N/2$ | T | S0 | **AUDIT ADDITION / PASS** |
| R06-T02 | $R_G^{-}=D-C^{\mathrm{can}}$ | T/E | S0+S1 | **CANONICAL RECONSTRUCTION / PASS** |
| R06-T03 | $L>U^{\mathrm{can}}$ implies Goldbach success | T template | bound-dependent | **CANONICAL CLOSURE / PASS** |
| R06-T04 | $I=C^{\mathrm{can}}+\Omega$ | T/E | S1 | **AUDIT ADDITION / PASS** |
| R06-T05 | $R_G^{-}=D+\Omega-I$ | T/E | S1 | PASS, derived |
| R06-T06 | failure implies $D\le I$ | T | S1 | PASS, weak corollary |
| R06-T07 | $D>I\iff R_G^{-}>\Omega$ | T/E | S1 | **AUDIT INTERPRETATION / PASS** |
| R06-T08 | incidence decomposition $\sum_rC_r^{inc}$ | T | S1 | PASS |
| R06-T09 | $C_r^{inc}\le A_r$ | T | S0+S1 | PASS |
| R06-T10 | divisor AP column exact indicator | T | S0+S1 | **STRENGTHENED / PASS** |
| R06-T11 | $\ell\mid N,\ell\le y<N/2\Rightarrow S_\ell=\{\ell\}$ | T | S0+S1 | **STRENGTHENED / PASS** |
| R06-T12 | primorial demand formula | T | S0+S2 | PASS, owned by R03 |
| R06-H01 | raw AP sum likely too loose | H/F | diagnostic | KEEP |
| R06-H02 | parity-type bottleneck interpretation | H/F | external caution | KEEP |
| R06-O01 | nontrivial canonical capacity upper bound | O | future | OPEN |
| R06-O02 | family crossing $L>U^{can}$ | O | future | OPEN |

---

# 69. Formalization priority

The R06 core is recommended to change to:

## F06-01

survivor contains all reduced Goldbach inputs.

## F06-02

canonical partition:

$$
H=G\bigsqcup C.
$$

## F06-03

$$
R=D-C^{\mathrm{can}}.
$$

## F06-04

least-factor cell partition of $C^{\mathrm{can}}$.

## F06-05

$$
I=C^{\mathrm{can}}+\Omega.
$$

## F06-06

incidence decomposition.

## F06-07

raw AP upper envelope.

## F06-08

divisor-column exact indicator.

## F06-09

small-factor self-cell exact theorem.

## F06-10

generic:

$$
L>U^{\mathrm{can}}
\Rightarrow
R>0.
$$

---

# 70. Not Prioritized for Formalization

There is no need for now to put:

- Brun–Titchmarsh proof;
- heuristic capacity density;
- cutoff phase diagram;
- parity interpretation;

into the core.

---

# 71. Correction of the Formal File Layout

Recommended:

```text
RGSCP/Core/
  Survivor.lean
  CanonicalCoverage.lean
  LeastFactorCells.lean
  IncidenceAudit.lean
  APColumns.lean
  Closure.lean

RGSCP/Interfaces/
  PrimeCountLowerBound.lean
  APUpperBound.lean
  CanonicalCellUpperBound.lean

RGSCP/External/
  BrunTitchmarsh.lean
  ExplicitPNT.lean
  SieveBounds.lean
```

---

# 72. First experiment control

E0 does not merely verify:

$$
R=D+\Omega-I.
$$

It should simultaneously verify:

$$
\boxed{
R=D-C^{\mathrm{can}}
}
$$

and:

$$
\boxed{
C^{\mathrm{can}}=I-\Omega.
}
$$

Stop if any of the three is inconsistent.

---

# 73. AP implementation control

If:

$$
r\mid N,
$$

the program should verify:

$$
A_r(N)
=
\begin{cases}
1,&N\ge r(r+1),\\
0,&N<r(r+1).
\end{cases}
$$

If there occurs:

$$
A_r>1,
$$

it must be a bug.

---

# 74. Small-factor implementation control

If:

$$
\ell\le y<N/2,
$$

and:

$$
\ell\mid N,
$$

the program must obtain:

$$
S_\ell(N)=\{\ell\}.
$$

This is stronger than the original:

$$
|S_\ell|\le1
$$

and is suitable for an invariant test.

---

# 75. The Capacity-Slack Experiment Should Be Rewritten

If the original E3 only examines:

$$
U/I
$$

this is not ideal.

More directly:

$$
\boxed{
\operatorname{CanonicalCapacitySlack}
=
\frac{
U_N^{\mathrm{can}}(y)
}{
C_N^{\mathrm{can}}(y)+1
}.
}
$$

but:

$$
C_N^{\mathrm{can}}
$$

may only be computed in the truth/evaluation channel.

The proof probe produces only:

$$
U_N^{\mathrm{can}}.
$$

---

# 76. Proof-compression verdict

Genuine proof compression in the sixth article does not lie in:

$$
D\le I.
$$

Nor in the exact:

$$
R=D+\Omega-I.
$$

What is genuinely worth retaining is:

$$
\boxed{
\text{leakage-free demand lower bound}
}
$$

against:

$$
\boxed{
\text{independently proved canonical coverage upper bound}.
}
$$

Namely:

$$
\boxed{
L_N(y)>U_N^{\mathrm{can}}(y).
}
$$

---

# 77. Source novelty verdict

The R06 exact core still primarily uses:

$$
\mathcal S_0
+
\mathcal S_1
+
\mathcal S_2.
$$

AP upper bounds begin connecting to:

$$
\mathcal S_5.
$$

There is no source-class escape.

But it adds an important architecture:

$$
\boxed{
\text{lower-bound source}
\quad\text{vs}\quad
\text{upper-bound source}
}
$$

which can be supplied by different mathematical tools.

---

# 78. Circularity verdict

Canonical closure:

$$
L>U^{\mathrm{can}}
$$

has no circularity in itself,

provided that:

- the proof of $L$ does not read $R_G$;
- the proof of $U^{\mathrm{can}}$ does not directly enumerate complement primality;
- the cutoff is not selected post hoc from target truth and then left undisclosed;
- assumptions are fully recorded.

Thus, a proof certificate must carry source provenance.

---

# 79. External analytic audit

The form of the original draft’s Brun–Titchmarsh interface:

$$
\pi(x;q,a)
\le
\frac{
2x
}{
\varphi(q)\log(x/q)
}
$$

under:

$$
\gcd(a,q)=1,
\quad
q<x
$$

can be retained under these standard conditions.

For a prime modulus:

$$
q=r,
$$

$$
\varphi(r)=r-1.
$$

But the formal core retains only an abstract interface:

```text
APUpperBound
```

The specific analytic theorem is cited separately.

---

# 80. Validation verdict

RG-SCP 06:

$$
\boxed{
\mathrm{PASS\ WITH\ CANONICAL\mbox{-}CAPACITY\ RECONSTRUCTION,\ INCIDENCE\ DEMOTION,\ AND\ SUPPRESSION\ SHARPENING}
}
$$

The most important canonical reconstruction:

$$
\boxed{
R_G^{-}(N)
=
D_N(y)
-
C_N^{\mathrm{can}}(y).
}
$$

The most important reinterpretation:

$$
\boxed{
I_N(y)
\text{ is incidence mass, not canonical coverage capacity}.
}
$$

The most important exact sharpening:

$$
\boxed{
r\mid N
\Longrightarrow
A_r(N)
=
\mathbf1_{\{N\ge r(r+1)\}},
}
$$

and:

$$
\boxed{
\ell\mid N,
\quad
\ell\le y<N/2
\Longrightarrow
S_\ell(N)=\{\ell\}.
}
$$

The genuine closure target changes to:

$$
\boxed{
L_N(y)
>
U_N^{\mathrm{can}}(y).
}
$$

---

# 81. Next Round

V0 Round 07 audits only:

# RG-SCP 07
## Minimal Counterexamples, Shift Transport, and Demand–Capacity Closure

Focus:

1. whether the lift-blocking theorem merely reindexes failure;
2. whether smooth shifts genuinely add constraint rank;
3. the deep-obstruction theorem for $M_k\mid h$;
4. the transport graph definition and matching lower bound;
5. whether $\nu_k(N_*)\le U$ is genuinely stronger than R06 ordinary demand;
6. whether the right-degree AP bound is correct;
7. whether the Hall route has hidden assumptions;
8. whether minimality-derived demand can be cleanly received by the canonical capacity framework.

---

# Appendix A: Canonical Demand–Capacity Core

```text
H_y(N):
  prime inputs whose complements avoid all primes <= y

D_N(y):
  |H_y(N)|

G_N^-:
  survivor inputs whose complements are prime

C_y(N):
  survivor inputs whose complements are composite

Partition:
  H_y(N) = G_N^- disjoint_union C_y(N)

Therefore:
  R_G^-(N) = D_N(y) - C_can_N(y)
```

---

# Appendix B: Canonical Least-Factor Capacity

```text
B_r(N;y):
  p in H_y(N)
  N-p composite
  lpf(N-p) = r

Then:
  C_y(N) = disjoint_union_{r>y} B_r(N;y)

  C_can_N(y)
    = sum_{r>y} |B_r(N;y)|
```

---

# Appendix C: Incidence Audit

```text
m_y(N,p):
  number of deep proper prime factors

I_N(y):
  sum_p m_y(N,p)

Omega_N(y):
  sum_p max(m_y(N,p)-1,0)

Identity:
  I_N(y) = C_can_N(y) + Omega_N(y)

Therefore:
  R_G^-
    = D_N(y) + Omega_N(y) - I_N(y)
```

---

# Appendix D: AP Suppression

For prime:

$$
r\mid N,
$$

raw proper-factor AP column:

$$
A_r(N)
=
\mathbf1_{\{N\ge r(r+1)\}}.
$$

For prime:

$$
\ell\le y<N/2,
\quad
\ell\mid N,
$$

small-factor screening class:

$$
S_\ell(N)=\{\ell\}.
$$

---

# Appendix E: Canonical Closure

If:

$$
D_N(y)\ge L_N(y),
$$

and:

$$
C_N^{\mathrm{can}}(y)
\le U_N^{\mathrm{can}}(y),
$$

then:

$$
R_G^{-}(N)
\ge
L_N(y)-U_N^{\mathrm{can}}(y).
$$

Hence:

$$
\boxed{
L_N(y)>U_N^{\mathrm{can}}(y)
\Longrightarrow
R_G^{-}(N)>0.
}
$$

---

# Appendix F: Source / Data Role

```yaml
D_N_y:
  status: exact
  role: proof_input
  source: S0+S2

C_can_N_y:
  status: exact
  role: truth_adjacent_diagnostic
  source: S1

I_N_y:
  status: exact
  role: truth_adjacent_diagnostic
  source: S1

Omega_N_y:
  status: exact
  role: truth_adjacent_diagnostic
  source: S1

A_r_N:
  status: exact
  role: proof_input
  source: S0+prime_left_domain

U_can:
  status: theorem_bound
  role: proof_input
  source: depends_on_bound
```

---

# Appendix G: Commit Decision

```yaml
validation_round: V0-R06
decision: PASS_WITH_CANONICAL_CAPACITY_RECONSTRUCTION_INCIDENCE_DEMOTION_AND_SUPPRESSION_SHARPENING

canonical_reconstruction:
  - R_equals_D_minus_Ccanonical
  - canonical_capacity_is_composite_survivor_count
  - least_factor_cells_partition_canonical_capacity
  - incidence_equals_capacity_plus_overlap

strengthening:
  - divisor_AP_column_exact_indicator
  - small_factor_self_cell_exact
  - D_greater_I_iff_R_greater_Omega

canonical_corrections:
  - rename_I_to_incidence_mass
  - demote_D_plus_Omega_minus_I_to_audit_identity
  - type_B_r_with_composite_condition
  - isolate_external_analytic_bounds
  - deduplicate_primorial_demand_with_R03
  - apply_truth_leakage_roles

commit_to_sync_core: true
promote_new_goldbach_theorem: false
next_round: V0-R07
```