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

# RG-SCP Validation Round 05
## RG-SCP 05 Claim / Source Audit
### Truth-Leakage Correction, Heuristic Classification, and Escape-Observable Audit of Prime Approximation, Skew Fields, and Counterexample Pressure Probing

**Validation series:** RG-SCP Validation  
**Validation phase:** V0 — Claim / Source Audit  
**Round:** 05  
**Source paper:** RG-SCP 05, Prime Approximation, Skew Fields, and Counterexample Pressure Probing, v0.1  
**Date:** 2026-09-28  
**Version:** v0.1  
**Authors:** Neo.K; Aletheia (GPT-5.6 Sol)  
**Document type:** Canonical Audit / Heuristic-Proof Boundary / Data-Role Audit  
**Status:** Fifth audit round; no new global Goldbach proof

---

## Abstract

This document conducts a claim/source audit of RG-SCP 05, Prime Approximation, Skew Fields, and Counterexample Pressure Probing.

The primary task of the fifth paper was never to prove Goldbach, but to establish a discovery layer:

$$
\boxed{
\text{approximation}
\to
\text{anomaly}
\to
\text{candidate theorem}
\to
\text{independent proof}.
}
$$

This round confirms that the overall direction is correct, but finds that the source paper also needs a dimension more important than 'exact / heuristic':

$$
\boxed{
\text{data role}.
}
$$

A quantity may be completely exact yet still unsuitable as input to an exploratory model, because it may already contain the target itself.

For example, the fourth round verified:

For even:

$$
N\ge10,
$$

we have:

$$
F_G(N)=1
\Longleftrightarrow
\Delta_6(N)>0
\land
N-3\notin\mathbb P.
$$

Thus, if a 'counterexample pressure model' uses:

$$
\Delta_6(N)
$$

and:

$$
B_3(N)
=
\mathbf1_{\{N-3\notin\mathbb P\}}
$$

simultaneously as features, it has effectively obtained:

$$
F_G(N)
$$

a complete exact encoding of this target.

On:

$$
6\mid N,
\quad
N\ge12,
$$

a stronger statement holds:

$$
\boxed{
F_G(N)=1
\Longleftrightarrow
\Delta_6(N)>0.
}
$$

Thus on this branch, using:

$$
\Delta_6(N)
$$

alone to 'predict Goldbach failure' is direct truth leakage.

This round therefore upgrades the fifth paper's original one-dimensional tag:

$$
\tau
\in
\{
\mathrm{exact},
\mathrm{heuristic},
\mathrm{mixed}
\}
$$

to a two-dimensional classification:

## Epistemic Status

$$
\boxed{
\mathrm{Exact}
\;|\;
\mathrm{Heuristic}
\;|\;
\mathrm{Mixed}
}
$$

## Data Role

$$
\boxed{
\mathrm{ProofInput}
\;|\;
\mathrm{DiscoveryProbe}
\;|\;
\mathrm{Diagnostic}
\;|\;
\mathrm{TruthLabel}
}
$$

If an exact quantity is a:

$$
\mathrm{TruthLabel},
$$

it cannot serve as input to a discovery model of the same target.

This round also classifies $\varepsilon_{\mathrm{full}}$ more strictly.

The source paper states:

- an exact table at small scales;
- regression at intermediate scales;
- an analytic asymptotic at large scales.

This round corrects these to:

### The Small-Scale Table

If its values are computed directly from known prime data and a preregistered finite-difference rule,

then it is:

$$
\boxed{
\text{exact finite measurement relative to that rule}.
}
$$

But it is not the exact value of an unspecified continuous derivative.

### Intermediate-Scale Regression

$$
[\mathrm H].
$$

### The Large-Scale Branch

If it is only a truncated approximation derived from a PNT-style asymptotic expansion, without a rigorous remainder theorem, it must still be labeled:

$$
[\mathrm H]
$$

or:

$$
[\mathrm{Mixed}],
$$

Using an analytic main term does not automatically admit it to the proof channel.

This round also corrects the term 'intrinsic arithmetic residual'.

$$
\Delta_\pi(x)
=
\pi(x)-\operatorname{Li}(x)
$$

and:

$$
\Delta_\psi(x)
=
\psi(x)-x
$$

have exact numerical values after they are defined.

But the residual still depends on the selected baseline:

$$
\operatorname{Li}(x),
\quad
x.
$$

Thus the more precise name should be:

$$
\boxed{
\text{standard exact arithmetic residual}
}
$$

rather than describing them as entirely baseline-free intrinsic invariants.

This round obtains an important strengthening for `escape prime`.

If:

$$
k\ge2,
$$

$$
M_k\mid N,
$$

and:

$$
N>6,
$$

then the reduced endpoint of any Goldbach representation must satisfy:

$$
p>p_k.
$$

Thus the source paper's definition:

$$
p_{\mathrm{esc}}(N;k)
=
\min
\left\{
p\in\mathbb P:
p_k<p\le N/2,
\quad
N-p\in\mathbb P
\right\}
$$

does not merely say that it 'may exist for a Goldbach-success state'.

On this branch:

$$
\boxed{
R_G(N)>0
\Longleftrightarrow
p_{\mathrm{esc}}(N;k)
\text{ exists}.
}
$$

Thus `escape prime existence` is an exact target-equivalent observable.

This means:

> `escape prime` and `escape depth` are well suited to truth / evaluation outputs, but must not be used as pre-target heuristic features of the same $N$.

This round also corrects surrogate / null-world semantics.

If a synthetic set:

$$
\widetilde{\mathbb P}
$$

is not the actual primes, one cannot continue using:

$$
F_G
$$

to denote its Goldbach target.

Instead define:

$$
R_{\mathcal Q}^{-}(N)
=
\#\left\{
q\in\mathcal Q:
q\le N/2,
\quad
N-q\in\mathcal Q
\right\},
$$

and:

$$
F_{\mathcal Q}(N)
=
\mathbf1_{\{R_{\mathcal Q}^{-}(N)=0\}}.
$$

Only then does a 'surrogate contradictory fiber' have clear semantics.

Finally, this round separates the promotion gate into two distinct gates:

### Discovery Promotion

Use held-out tests, null models, cross-model reconstruction, and so on to decide whether a pattern deserves investment in proof search.

### Proof Promotion

Consider only:

- the exact statement;
- a valid proof / certificate;
- assumptions;
- source provenance.

Once a theorem is proved rigorously, its mathematical validity does not depend on passing cross-AI reconstruction or an empirical held-out test.

The verdict of this round:

$$
\boxed{
\mathrm{PASS\ WITH\ TRUTH\mbox{-}LEAKAGE\ FIREWALL,\ HEURISTIC\ RECLASSIFICATION,\ AND\ ESCAPE\mbox{-}OBSERVABLE\ STRENGTHENING}.
}
$$

---

# 1. Scope of the Audit

This round audits only:

# RG-SCP 05
## Prime Approximation, Skew Fields, and Counterexample Pressure Probing

It does not reprove:

- the exact theorems of RG-SCP 01–04;
- all coefficients of the original $\varepsilon_{\mathrm{full}}$ project;
- the entire analytic-number-theory background of Skew Line;
- external PNT / explicit-formula theorems.

This round asks only:

> When the fifth paper integrates these into RG-SCP, are the proof boundaries clean?

---

# 2. The Correct Overall Position of the Fifth Paper

The canonical role of the fifth paper should be fixed as:

$$
\boxed{
\text{Discovery Infrastructure}
}
$$

not:

$$
\boxed{
\text{Proof Source by Default}.
}
$$

That is:

$$
\mathcal H^{\mathrm{disc}}
$$

may be very free, but:

$$
\mathcal K^{\mathrm{proof}}
$$

must remain closed.

---

# 3. The Original Exact / Heuristic Tag Is Insufficient

The source paper defines:

$$
\tau(P_i)
\in
\{
\mathrm{exact},
\mathrm{heuristic},
\mathrm{mixed}
\}.
$$

This is necessary, but insufficient.

Because:

$$
\Delta_6(N)
$$

is exact,

yet, if the target is:

$$
F_G(N),
$$

then on the:

$$
6\mid N
$$

branch it is almost the answer itself.

Thus:

$$
\boxed{
\text{exactness}
\neq
\text{legal discovery input}.
}
$$

---

# 4. Two-Dimensional Data Classification

This round proposes labeling every feature with two fields simultaneously.

## Axis A — Epistemic status

```text
exact
heuristic
mixed
```

## Axis B — Data role

```text
proof_input
discovery_probe
diagnostic
truth_label
```

---

# 5. Definitions of Data Roles

## 5.1 Proof Input

A quantity legitimately usable in a proof chain within a proved theorem, certificate, or exact assumption.

## 5.2 Discovery Probe

Does not directly encode the target; used to identify patterns or candidate theorems.

## 5.3 Diagnostic

May be used in post hoc analysis, but must not serve as leakage-free prediction input for the same target.

## 5.4 Truth Label

A quantity directly equivalent to, or almost directly defining, the target.

---

# 6. Truth leakage principle

If:

$$
T(N)
=
f(X_1(N),\ldots,X_m(N))
$$

has been proved exactly,

then, in a discovery experiment with:

$$
T
$$

as its prediction / anomaly target,

feeding:

$$
X_1,\ldots,X_m
$$

all to the model is:

$$
\boxed{
\text{target leakage}.
}
$$

Even if every:

$$
X_i
$$

is a legitimate exact arithmetic quantity.

---

# 7. $\Delta_6+B_3$ Is a Complete Target Encoding

The fourth round proved:

For even:

$$
N\ge10,
$$

$$
\boxed{
F_G(N)
=
\mathbf1_{\{\Delta_6(N)>0\}}
\cdot
\mathbf1_{\{N-3\notin\mathbb P\}}.
}
$$

Thus:

$$
\left(
\Delta_6(N),
B_3(N)
\right)
$$

cannot be an independent probe vector for 'predicting $F_G$'.

It is a target decomposition.

---

# 8. Stronger Leakage on the $6\mid N$ Branch

For:

$$
6\mid N,
\quad
N\ge12,
$$

$$
\boxed{
F_G(N)=1
\Longleftrightarrow
\Delta_6(N)>0.
}
$$

Thus, if the research target is:

$$
F_G,
$$

then:

$$
\Delta_6
$$

must be labeled:

```yaml
epistemic_status: exact
data_role: truth_label
```

not:

```yaml
data_role: discovery_probe
```

---

# 9. Canonical Correction C01 — Separating the Pressure Vector

The source paper:

$$
\mathbf H_G(N)
=
(
\Delta_6,
B_3,
\eta,
D_G,
\mathcal K,
\mathcal R
).
$$

This vector can serve as:

$$
\boxed{
\text{state description}
}
$$

but cannot serve as a whole:

$$
\boxed{
\text{counterexample prediction feature vector}.
}
$$

The canonical schema should be split into:

```yaml
truth_state:
  delta_6:
  boundary_3:
  goldbach_count:
  escape_exists:

exact_non_target_probes:
  wheel_residual:
  divisor_structure:
  residue_structure:

heuristic_probes:
  fitted_residual:
  skew_normalization:
  learned_features:
```

---

# 10. The Typing Problem of $D_G$

Before the fourth round it had already been noted:

$$
D_G(N)
$$

was originally the failure-state obstruction depth.

If:

$$
N
$$

is Goldbach-successful,

some complements are prime,

then:

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

is no longer a proper obstruction.

Thus, if the fifth paper stores:

$$
D_G(N),
$$

for a general successful dataset, its semantics must first be corrected.

---

# 11. Canonical Correction C02 — General-data depth

Proposed distinction:

## Failure obstruction depth

$$
D_G^{\mathrm{fail}}(N)
$$

Defined only when:

$$
F_G(N)=1
$$

the displayed failure condition holds.

## Composite-complement depth

For arbitrary $N$, define:

$$
\boxed{
D_{\mathrm{comp}}(N)
=
\max
\left(
\left\{
\operatorname{lpf}(N-p):
p\in\mathbb P_N^{-},
\quad
N-p\text{ composite}
\right\}
\cup
\{0\}
\right).
}
$$

This is an exact diagnostic usable on a general dataset.

The two must not share the same name.

---

# 12. Derivative Semantics of $\varepsilon(n)$

The source paper gives the idealized expression:

$$
m(n)
=
\frac{
d\log A(n)
}{
d\log n
}.
$$

But:

$$
n
$$

is a discrete index.

Without a specified interpolation,

this is not a uniquely defined exact derivative.

Thus the canonical exact observable should use a preregistered finite difference.

---

# 13. Canonical discrete slope

For example, fix a window rule:

$$
h(n)\in\mathbb N.
$$

Define:

$$
\boxed{
m_h(n)
=
\frac{
\log A(n+h(n))
-
\log A(n-h(n))
}{
\log(n+h(n))
-
\log(n-h(n))
}
}
$$

holds when the domain is admissible.

Then define:

$$
\boxed{
\varepsilon_h(n)
=
m_h(n)-1.
}
$$

Only this is an exact measurement on a finite dataset.

---

# 14. Audit of the Three Segments of $\varepsilon_{\mathrm{full}}$

## 14.1 Table segment

If the table is computed from:

- an exact prime list;
- a fixed finite-difference rule;
- deterministic arithmetic;

as listed above,

then a table entry is:

$$
\boxed{
[\mathrm T/F_{\mathrm{finite}}]
}
$$

Meaning:

> Exact finite measurement relative to the chosen measurement rule.

Not a 'continuous intrinsic derivative theorem'.

## 14.2 Regression segment

Every fitted coefficient:

$$
\beta_i
$$

and every formula obtained by regression on a finite dataset:

$$
[\mathrm H].
$$

## 14.3 Large-$n$ asymptotic segment

If a formula is inspired by known asymptotic reasoning, but the current document supplies no rigorous remainder bound:

$$
|\varepsilon(n)-\varepsilon_{\mathrm{asy}}(n)|
\le
E(n),
$$

then within RG-SCP it must still be labeled:

$$
\boxed{
[\mathrm H]
\text{ or }
[\mathrm{Mixed}].
}
$$

Including a PNT main term alone cannot justify labeling it a proof theorem.

---

# 15. Canonical Correction C03

The entire:

$$
\varepsilon_{\mathrm{full}}
$$

should be labeled uniformly:

```yaml
epistemic_status: mixed
data_role: discovery_probe
proof_role: none_by_default
```

Its table / regression / asymptotic components retain component-level provenance separately.

---

# 16. Prime trajectory residual

$$
\rho_p(n)
=
p_n-\widehat p(n).
$$

If:

$$
p_n
$$

is the ground truth, while:

$$
\widehat p
$$

is a fitted model,

then:

$$
\rho_p(n)
$$

can have an exactly computed value, but its meaning is model-relative.

Thus:

```yaml
epistemic_status: mixed
data_role: diagnostic_or_discovery_probe
baseline_dependent: true
```

---

# 17. $\Delta_\pi$ and $\Delta_\psi$

Define:

$$
\Delta_\pi(x)
=
\pi(x)-\operatorname{Li}(x),
$$

$$
\Delta_\psi(x)
=
\psi(x)-x.
$$

Their numerical values are exact once the baseline conventions are fixed.

Thus:

$$
[\mathrm T/F_{\mathrm{observable}}].
$$

---

# 18. Correcting the Term 'Intrinsic Residual'

Although:

$$
\pi(x)
$$

and:

$$
\psi(x)
$$

are canonical arithmetic functions,

the residual:

$$
\pi-\operatorname{Li}
$$

still chooses:

$$
\operatorname{Li}
$$

as its baseline.

Likewise:

$$
\psi-x
$$

chooses:

$$
x
$$

as its main term.

Thus this round proposes the name:

$$
\boxed{
\text{standard exact arithmetic residual}
}
$$

in place of:

$$
\boxed{
\text{intrinsic residual}
}
$$

to avoid the misunderstanding that it is 'entirely baseline-free'.

---

# 19. Exactness of a Residual Does Not Mean Goldbach Relevance

Even if:

$$
\Delta_\pi
$$

is exact,

one does not have:

$$
\boxed{
|\Delta_\pi(N-p)|\text{ large}
\Rightarrow
N-p\text{ composite}.
}
$$

Thus:

$$
\Delta_\pi(N-p)
$$

is usually:

```yaml
epistemic_status: exact
data_role: discovery_probe
target_implication: none
```

---

# 20. Baseline complement field

$$
B(x)=\frac1{\log x}
$$

as a prime-density scale is standard asymptotic intuition.

But:

$$
B(x)
$$

is not:

$$
\Pr(x\in\mathbb P).
$$

This round confirms that the source paper correctly prohibits the name `PrimeProbability`.

Retain:

$$
\boxed{
\text{density surrogate}
}
$$

or:

$$
\boxed{
\text{heuristic intensity}.
}
$$

---

# 21. Prime probability firewall

Any feature:

$$
s(x)
$$

without an established probability space and calibration theorem,

must not be written as:

$$
s(x)=\Pr(x\text{ prime}).
$$

Even if:

$$
0\le s(x)\le1.
$$

This is a research-engineering rule:

$$
[\mathrm F].
$$

---

# 22. Fitted prime-index pullback

$$
R_{\mathrm{fit}}(x)
$$

and:

$$
R_{N,\mathrm{fit}}(p)
$$

belong entirely to:

$$
[\mathrm H].
$$

The source paper's classification is PASS.

---

# 23. Skew pullback

If:

$$
K_N(p)
=
\Delta_\pi(N-p)
$$

or:

$$
\Delta_\psi(N-p),
$$

then the numerical value is exact.

But:

- aggregation;
- anomaly thresholds;
- spectral interpretation;

are usually heuristic.

Thus component-wise tags are needed, rather than classifying the entire `skew field` uniformly.

---

# 24. Exact observable / heuristic interpretation split

This round fixes the following pattern:

```text
observable:
  exact

normalization:
  maybe heuristic

anomaly threshold:
  heuristic

interpretation:
  heuristic until proved

theorem consequence:
  none by default
```

This applies to:

- $\Delta_\pi$;
- $\Delta_\psi$;
- $\eta_{M_k}$;
- $\omega(N)$;
- exact representation counts.

---

# 25. Shadow-gap skew

$$
K_\Delta(N)
=
\Delta_6(N)-B_\Delta(N)
$$

If:

$$
B_\Delta
$$

is an empirical baseline,

then:

$$
K_\Delta
$$

is mixed / heuristic.

More importantly:

if the target is Goldbach failure,

$$
\Delta_6
$$

may itself be truth-adjacent or even truth-equivalent.

Thus:

$$
K_\Delta
$$

cannot be treated as a leakage-free predictor.

---

# 26. Obstruction-depth skew

$$
K_D(N)
=
D_{\mathrm{comp}}(N)-B_D(N)
$$

can serve as a general diagnostic.

If one uses:

$$
D_G^{\mathrm{fail}},
$$

it is defined only after the target is known to be a failure, and all the more cannot serve as a failure-detector feature.

---

# 27. Wheel residual ratio

$$
\eta_{M_k}(N)
$$

is exact and does not require knowing complement primality.

Thus it can legitimately serve as an:

$$
\boxed{
\text{exact discovery probe}.
}
$$

This is a good example:

```yaml
epistemic_status: exact
data_role: discovery_probe
truth_leakage: low
```

---

# 28. Divisor structure

Quantities such as:

$$
\omega(N),
$$

$$
M_k\mid N,
$$

$$
N\bmod M_k
$$

also belong to:

$$
\boxed{
\text{exact discovery probes}.
}
$$

They do not directly include whether:

$$
N-p
$$

is prime.

---

# 29. Escape prime

The source paper defines:

$$
\boxed{
p_{\mathrm{esc}}(N;k)
=
\min
\left\{
p\in\mathbb P:
p_k<p\le N/2,
\quad
N-p\in\mathbb P
\right\}.
}
$$

when the set is nonempty.

Its value is exact.

But it directly uses:

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

Thus it is highly target-adjacent.

---

# 30. Escape Equivalence on the Primorial Branch

## Theorem R05-T01

If:

$$
k\ge2,
$$

$$
M_k\mid N,
$$

and:

$$
N>6,
$$

then:

$$
\boxed{
R_G(N)>0
\Longleftrightarrow
p_{\mathrm{esc}}(N;k)
\text{ exists}.
}
$$

---

# 31. Proof: Escape Existence Implies Success

By definition, if:

$$
p_{\mathrm{esc}}
$$

exists,

then:

$$
p_{\mathrm{esc}}
$$

and:

$$
N-p_{\mathrm{esc}}
$$

are both prime.

Thus:

$$
R_G(N)>0.
$$

---

# 32. Proof: Success Implies Escape Existence

Suppose:

$$
R_G(N)>0.
$$

Take a reduced representation:

$$
N=p+q,
$$

where:

$$
p\le q
$$

and:

$$
p\le N/2.
$$

Suppose instead:

$$
p\le p_k.
$$

Since:

$$
p
$$

is prime,

Thus:

$$
p
$$

is one of:

$$
p_1,\ldots,p_k
$$

the primes displayed above.

From:

$$
M_k\mid N,
$$

we obtain:

$$
p\mid N.
$$

Also:

$$
q=N-p,
$$

Thus:

$$
p\mid q.
$$

Since:

$$
q
$$

is prime,

the only possibility is:

$$
q=p.
$$

Therefore:

$$
N=2p.
$$

Moreover:

$$
k\ge2
$$

implies:

$$
6\mid M_k\mid N.
$$

Thus:

$$
6\mid2p,
$$

That is:

$$
3\mid p.
$$

Since $p$ is prime,

we have:

$$
p=3.
$$

Hence:

$$
N=6,
$$

and:

$$
N>6
$$

a contradiction.

Thus:

$$
p>p_k.
$$

Thus the defining escape set is nonempty.

This completes the proof.

---

# 33. Escape Existence Is a Truth Label

On the R05-T01 domain:

$$
\boxed{
\mathbf1_{\{p_{\mathrm{esc}}\text{ exists}\}}
=
\mathbf1_{\{R_G(N)>0\}}.
}
$$

Thus:

```yaml
escape_exists:
  epistemic_status: exact
  data_role: truth_label
```

It cannot serve as an input feature to a Goldbach-success detector.

---

# 34. Escape depth

If the escape prime exists:

$$
D_{\mathrm{esc}}(N;k)
=
\pi(p_{\mathrm{esc}})
-
\pi(p_k).
$$

Since:

$$
p_{\mathrm{esc}}>p_k,
$$

Thus:

$$
D_{\mathrm{esc}}\ge1.
$$

It exactly indicates:

> The index, after the cutoff, of the first prime input forming a representation.

Claim class：

$$
[\mathrm T/F_{\mathrm{observable}}].
$$

---

# 35. The Data Role of Escape Depth

It is defined only after success has occurred, and its finite / undefined status itself reveals the target,

Thus:

```yaml
escape_depth:
  epistemic_status: exact
  data_role: truth_label_or_posthoc_diagnostic
```

so it must not serve as a same-target preclassification feature.

---

# 36. Canonical Correction C04 — Separating Escape Fields

The source paper's heuristic state lists:

$$
D_{\mathrm{esc}}
$$

alongside ordinary probes.

Proposed revision:

```yaml
evaluation_truth:
  escape_exists:
  escape_prime:
  escape_depth:

discovery_inputs:
  wheel_residual:
  divisor_structure:
  nonleaking_skew:
  fitted_baseline_features:
```

---

# 37. Escape predictor

$$
\widehat p_{\mathrm{esc}}(N;k)
$$

can be a legitimate heuristic discovery output.

Its evaluation target is:

$$
p_{\mathrm{esc}}.
$$

But the training / prediction inputs must not include:

- the actual escape prime;
- the actual escape depth;
- the Goldbach representation count;
- the complete $\Delta_6+B_3$ truth encoding.

---

# 38. Windowed Goldbach subproblem

If a heuristic proposes:

$$
W(N),
$$

the genuine theorem target is:

$$
\boxed{
\exists p\in W(N)\cap\mathbb P,
\quad
N-p\in\mathbb P.
}
$$

Once this statement is independently proved, its heuristic origin does not affect theorem validity.

The source paper's direction on this point is correct.

---

# 39. The Discovery Gate and Proof Gate Should Be Separated

P1–P10 in the source paper contain a potentially confusing point:

As if a theorem must simultaneously pass:

- held-out tests;
- a null model;
- cross-AI checks;
- proof;
- formalization.

But mathematically, this is not the case.

---

# 40. Discovery Promotion Gate

Whether an empirical pattern deserves investment in proof search may be assessed through:

1. Held-out stability;
2. Matched controls;
3. Null-model discrimination;
4. Cross-seed stability;
5. Cross-model reconstruction;
6. Falsifiability;
7. Source novelty.

These belong to:

$$
[\mathrm H/F_{\mathrm{method}}].
$$

---

# 41. Proof Promotion Gate

For a statement to enter a theorem ledger, only the following are needed:

1. A precise statement;
2. Precise assumptions;
3. A valid proof;
4. Clear source provenance;
5. For a finite computational theorem, an adequate certificate / verifier.

Formalization provides a strong additional assurance, but cross-AI reconstruction is not a necessary condition for a mathematical theorem.

---

# 42. Canonical Correction C05

Change the promotion pipeline to two stages:

$$
\boxed{
\text{Discovery Gate}
}
$$

decides:

> Whether it is worth proving.

Then:

$$
\boxed{
\text{Proof Gate}
}
$$

decides:

> Whether it has already become a theorem.

The two cannot be merged into a single promotion score.

---

# 43. The Correct Status of Cross-AI Reconstruction

Cross-AI reconstruction can test:

- prompt dependence;
- rediscoverability;
- conceptual robustness.

However:

$$
\boxed{
\text{two AIs agree}
\not\Rightarrow
\text{theorem true}.
}
$$

Likewise:

$$
\boxed{
\text{other AI fails to rediscover}
\not\Rightarrow
\text{theorem false}.
}
$$

Thus it belongs only to:

$$
[\mathrm H].
$$

---

# 44. Null model semantics

If a surrogate set is established:

$$
\mathcal Q
\subseteq
\mathbb N,
$$

and it is not the actual:

$$
\mathbb P,
$$

its target should not be written as:

$$
F_G.
$$

---

# 45. Canonical surrogate target

For any:

$$
\mathcal Q,
$$

Define:

$$
\boxed{
R_{\mathcal Q}^{-}(N)
=
\#\left\{
q\in\mathcal Q:
2\le q\le N/2,
\quad
N-q\in\mathcal Q
\right\}.
}
$$

and:

$$
\boxed{
F_{\mathcal Q}(N)
=
\mathbf1_{\{R_{\mathcal Q}^{-}(N)=0\}}.
}
$$

Only when:

$$
\mathcal Q=\mathbb P
$$

do we have:

$$
F_{\mathcal Q}=F_G.
$$

---

# 46. Canonical Correction C06 — Surrogate world typing

Every surrogate experiment should record:

```yaml
world:
  carrier_set: Q
  target: F_Q
  relation_to_real_primes:
```

Do not label the surrogate target directly as:

$$
F_G.
$$

---

# 47. Approximate contradictory fiber

If a heuristic signature:

$$
J_H
$$

uses a tolerance:

$$
\|J_H(x)-J_H(y)\|\le\epsilon,
$$

while targets differ, one obtains only an:

$$
\boxed{
\text{empirical near-collision}.
}
$$

It cannot be called an exact contradictory fiber.

The source paper already cautions about the approximate / exact distinction; the direction is PASS.

---

# 48. An Exact Surrogate Collision Can Be Promoted to Formal Insufficiency

If two exact worlds / states are constructed:

$$
X,Y
$$

satisfying:

$$
\boxed{
J(X)=J(Y)
}
$$

However:

$$
\boxed{
T(X)\neq T(Y),
}
$$

then one can rigorously infer:

> The target $T$ cannot factor through the observation $J$.

That is:

$$
\boxed{
\nexists f,
\quad
T=f\circ J.
}
$$

Only this is a formal source-insufficiency certificate.

---

# 49. The Correct Conclusion of Null-Model Failure

If a model cannot distinguish the true world from a surrogate, at most one can say:

> Discrimination is insufficient for this model / probe family / chosen surrogate class.

One cannot directly infer:

$$
\boxed{
\text{the entire mathematical source class is forever insufficient}.
}
$$

unless there is an exact observation-factorization theorem.

---

# 50. Canonical Correction C07

The source paper:

> If a surrogate cannot be detected, the heuristic probe is not yet sensitive to the target.

Retain this.

But:

> This proves that these features are insufficient for Goldbach.

'Proves' is permissible only under exact-collision semantics.

For the empirical case, rewrite as:

$$
\boxed{
\text{evidence of insufficiency under tested surrogate class}.
}
$$

---

# 51. Density fit vs pair fidelity

The source paper:

$$
\text{one-point density fit}
\neq
\text{two-point additive sufficiency}.
$$

is correct as a methodological warning.

But to turn it into a theorem, the following must be specified:

- the fit metric;
- the tolerance;
- the model class;
- the target class.

Thus it is currently labeled:

$$
[\mathrm F/H].
$$

---

# 52. Skew fit vs pair fidelity

Likewise:

$$
\text{one-point skew fit}
\neq
\text{Goldbach-pair fidelity}.
$$

For now, this is:

$$
[\mathrm F/H].
$$

It cannot be treated as a formal impossibility theorem.

---

# 53. Goldbach fidelity

$$
\operatorname{GFid}(B)
$$

The source paper deliberately avoids defining a single scalar initially.

This decision is PASS.

Because:

- one-point counts;
- residues;
- pair correlation;
- representation counts;
- hard branches;

have different meanings.

They must not be weighted arbitrarily.

---

# 54. Pressure ridge / valley

Pressure surfaces, ridges, and valleys all belong to:

$$
[\mathrm H].
$$

unless a ridge equation is rewritten as an exact arithmetic theorem and proved independently.

---

# 55. Generation-preserving probe

This is design vocabulary of the series, not a standard theorem notion.

Classification:

$$
[\mathrm F].
$$

Turning it into a theorem in future requires defining:

> Exactly which event family / discrete structure the probe preserves.

This is not needed for now.

---

# 56. Pair-spectral target

The fifth paper proposes:

$$
r_2(N)
=
\sum_a
\mathbf1_{\mathbb P}(a)
\mathbf1_{\mathbb P}(N-a)
$$

as the genuine two-point target.

This direction is correct.

But note:

$$
r_2(N)
$$

is itself a representation count, and therefore only the target object.

New proof power must come from:

- a lower bound;
- a positivity certificate;
- a spectral invariant;
- a transform.

---

# 57. Exact / Heuristic / Role Matrix

This round proposes the canonical matrix:

| Object | Epistemic | Role if target = $F_G$ | Note |
|---|---|---|---|
| $N\bmod M_k$ | Exact | DiscoveryProbe | nonleaking |
| $M_k\mid N$ | Exact | DiscoveryProbe | nonleaking |
| $\omega(N)$ | Exact | DiscoveryProbe | nonleaking |
| $\eta_{M_k}(N)$ | Exact | DiscoveryProbe | no complement primality needed |
| $\Delta_\pi(x)$ | Exact | DiscoveryProbe / Diagnostic | baseline-relative |
| $\Delta_\psi(x)$ | Exact | DiscoveryProbe / Diagnostic | baseline-relative |
| fitted $\rho_p$ | Mixed | DiscoveryProbe | model-relative |
| $\varepsilon_{\mathrm{full}}$ | Mixed | DiscoveryProbe | no proof role by default |
| $\Delta_6(N)$ | Exact | TruthLabel on $6\mid N$; otherwise TruthAdjacent | target leakage risk |
| $B_3(N)$ | Exact | TruthAdjacent | with $\Delta_6$ fully determines target |
| $R_G^{-}(N)$ | Exact | TruthLabel | never probe same target |
| $p_{\mathrm{esc}}$ exists | Exact | TruthLabel on primorial branch | target-equivalent |
| $D_{\mathrm{esc}}$ | Exact | PosthocDiagnostic / TruthAdjacent | defined only on success |
| $D_G^{\mathrm{fail}}$ | Exact conditional | TruthLabel-adjacent | failure-only |
| $D_{\mathrm{comp}}$ | Exact | Diagnostic | safe if target protocol allows |

---

# 58. Leakage-free discovery feature set

If the objective is:

> To identify structure that may influence Goldbach representation,

safer initial inputs are:

$$
\boxed{
N\bmod M_k,
\quad
\omega(N),
\quad
\eta_{M_k}(N),
\quad
\text{small-factor residue profile},
\quad
\text{standard arithmetic residuals},
\quad
\text{fitted scale features}.
}
$$

Outputs / labels are stored separately:

$$
\boxed{
R_G^{-},
\quad
\Delta_6,
\quad
p_{\mathrm{esc}},
\quad
D_{\mathrm{esc}}.
}
$$

---

# 59. Time / range split

If exact target-derived quantities from the same $N$ enter the features,

the result is still leakage even when train/test sets are split by $N$ range.

Thus:

$$
\boxed{
\text{held-out split}
\neq
\text{leakage protection}.
}
$$

Feature semantics must be audited first.

---

# 60. Pre-registration PASS

The fifth paper requires preregistration of:

- the domain;
- the baseline;
- the threshold;
- the null model;
- the stopping rule;
- the promotion rule.

This is good research discipline.

Retain it.

---

# 61. Correction to MVE-1

MVE-1 in the source paper lists together:

- exact $R_G$;
- $\Delta_6$;
- $D_G$;
- wheels;
- skew.

The canonical form should split these into:

## Truth / Evaluation Table

```text
R_G^-
delta_6
boundary_3
escape_exists
escape_depth
```

## Leakage-Free Probe Table

```text
residue
divisor structure
wheel survivor ratios
standard skew residuals
fitted scale features
```

## Conditional Diagnostics

```text
failure obstruction depth
canonical obstruction statistics
```

---

# 62. MVE-2 primorial branch strengthening

For:

$$
k\ge2,
\quad
M_k\mid N,
\quad
N>6,
$$

do not compare only escape depth.

First verify the exact identity:

$$
\boxed{
R_G(N)>0
\Longleftrightarrow
p_{\mathrm{esc}}(N;k)\text{ exists}.
}
$$

This can serve as an E0 / E1 implementation control.

---

# 63. MVE-3 surrogate correction

If an 'artificially generated dual-hole surrogate' is not standard integer prime structure, do not use the genuine:

$$
\Delta_6
$$

unless the surrogate also explicitly defines a corresponding operator.

More safely:

define the world-relative:

$$
\Delta_{6,\mathcal Q}
$$

or use only:

$$
F_{\mathcal Q}.
$$

Avoid mixing a real-prime operator with a synthetic carrier.

---

# 64. MVE-4 exact vs empirical insufficiency

If:

$$
J_{\mathrm H}
$$

is only an approximate match, output:

```text
empirical_indistinguishability
```

If:

$$
J
$$

is an exact match while targets differ, output:

```text
formal_observation_insufficiency_witness
```

The two kinds of report must be separated.

---

# 65. MVE-5 candidate theorem extraction PASS

The fifth paper requires eventual output of an:

$$
\boxed{
\text{exact arithmetic statement}
}
$$

not a more accurate model.

This is one of the methodological points most worth retaining in the paper.

---

# 66. Stop Conditions audit

All S1–S6 can be retained as project-management / discovery stopping rules.

But they are not mathematical no-go theorems.

In particular:

> Failure of cross-AI reconstruction

can only lower discovery confidence, not rule out the possibility of a theorem.

---

# 67. Promotion Conditions audit

P1–P6 of the source paper are suited to:

$$
\boxed{
\text{Discovery Promotion}
}
$$

not conditions for theorem validity.

Among P7–P10:

- independent proof;
- formalization;
- a hidden-oracle audit;

are the items genuinely connected to the proof gate.

---

# 68. Priority of Formal Proof

If a candidate theorem already has a complete human proof, even if:

- its fit is unstable;
- another AI did not reconstruct it;
- its null-model performance is poor;

it may still be a theorem.

The validity of a theorem is determined by its proof.

The empirical pipeline is only discovery provenance.

---

# 69. Hidden oracle audit

This is a very important point of the fifth paper and should become a canonical rule.

Every candidate theorem / algorithm must ask:

> Are target primality / representation truths being used covertly in features, preprocessing, window selection, or normalization?

This is combined with the truth-leakage firewall.

---

# 70. Canonical proof-promotion pipeline

Proposed revision:

$$
\boxed{
\text{Discovery}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Exact Statement Extraction}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Oracle / Source Audit}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Independent Proof}
}
$$

$$
\Downarrow
$$

$$
\boxed{
\text{Certificate / Formalization}
}
$$

where:

- held-out tests;
- null models;
- AI reconstruction;

belong in the Discovery stage.

---

# 71. Canonical claim ledger — RG-SCP 05

| ID | Claim / Object | Class | Role | Status |
|---|---|---|---|---|
| R05-F01 | proof/discovery channel separation | F | research discipline | PASS |
| R05-F02 | heuristic anomaly does not imply proof | F | research discipline | PASS |
| R05-T01 | primorial branch success iff escape prime exists | T/E | TruthLabel | **AUDIT STRENGTHENING / PASS** |
| R05-T02 | escape depth exact once escape exists | T/F | PosthocDiagnostic | PASS |
| R05-T03 | $\Delta_\pi,\Delta_\psi$ exact after baseline definitions | T/F | Probe/Diagnostic | PASS |
| R05-H01 | $\varepsilon_{\mathrm{full}}$ as unified predictor | H/Mixed | DiscoveryProbe | KEEP |
| R05-H02 | fitted prime residual | H/Mixed | DiscoveryProbe | KEEP |
| R05-H03 | pressure ridge / valley | H | DiscoveryProbe | KEEP |
| R05-H04 | skew anomaly | H | DiscoveryProbe | KEEP |
| R05-F03 | null / surrogate benchmark framework | F/H | Falsification | PASS with typing correction |
| R05-O01 | heuristic yields theorem-worthy exact invariant | O | future | OPEN |
| R05-O02 | uniform escape-depth upper bound | O | future | OPEN |
| R05-O03 | pair-spectral source supplies positivity certificate | O | future | OPEN |

---

# 72. Canonical corrections ledger

## C01 — Add Data Role Axis

`exact / heuristic / mixed` is insufficient.

Add:

```text
proof_input
discovery_probe
diagnostic
truth_label
```

## C02 — General obstruction depth typing

For a general successful dataset use:

$$
D_{\mathrm{comp}}.
$$

Retain the failure-only object as:

$$
D_G^{\mathrm{fail}}.
$$

## C03 — $\varepsilon$ discrete semantics

The finite exact component must be bound to:

$$
h(n)
$$

and a finite-difference rule.

## C04 — Escape fields move to truth/evaluation

`escape_exists` is target-equivalent on the primorial branch.

It cannot be a same-target probe.

## C05 — Split Discovery Gate / Proof Gate

Empirical robustness is not theorem validity.

## C06 — Surrogate world target typing

Use:

$$
F_{\mathcal Q},
$$

rather than still calling a synthetic target:

$$
F_G.
$$

## C07 — Empirical vs formal insufficiency

Near-collision:

$$
[\mathrm H].
$$

Exact collision:

can produce a formal source-insufficiency theorem.

## C08 — Residual terminology

Replace `intrinsic residual` with:

```text
standard exact arithmetic residual
```

---

# 73. Source audit

RG-SCP 05 itself mainly introduces:

$$
\boxed{
\mathcal S_6
=
\text{heuristic approximation / skew source}.
}
$$

It assumes that:

$$
\mathcal S_6
$$

does not directly enter the proof chain.

Exact probes, in contrast, still derive from:

- $\mathcal S_0$: elementary;
- $\mathcal S_1$: factors;
- $\mathcal S_2$: quotients;
- standard analytic observables.

---

# 74. Source novelty verdict

The fifth paper does not claim:

$$
\mathcal S_6
$$

is itself a proof-complete source.

This point is PASS.

Genuine source novelty must be audited after:

$$
\boxed{
\text{heuristic}
\to
\text{new exact invariant}
}
$$

the transition displayed above.

---

# 75. Formalization priority

The fifth paper does not need large-scale Lean formalization.

Only a few firewalls / exact interfaces are proposed for formalization:

## F05-01

Primorial escape equivalence R05-T01.

## F05-02

World-relative representation count:

$$
R_{\mathcal Q}^{-}.
$$

## F05-03

Observation-factorization insufficiency lemma:

If:

$$
J(x)=J(y)
$$

However:

$$
T(x)\neq T(y),
$$

then there is no:

$$
T=f\circ J.
$$

The remaining heuristic pipeline need not be sent to a proof assistant.

---

# 76. What Must Not Enter the Synchronous Proof Core

The following go by default into:

$$
10\_\mathrm{ASYNC\_TEXT}
$$

or:

$$
30\_\mathrm{ASYNC\_EXPERIMENTS}.
$$

- fitted coefficients;
- pressure surfaces;
- ridges / valleys;
- anomaly thresholds;
- learned predictors;
- null-model result summaries;
- cross-AI rediscovery notes.

unless promoted to an exact theorem.

---

# 77. What Should Be Synchronized

The items that need to enter:

$$
00\_\mathrm{SYNC\_CORE}
$$

are:

- the truth-leakage schema;
- the exact/role taxonomy;
- the escape theorem;
- surrogate-target definitions;
- a promoted exact candidate statement;
- a proof / certificate;
- a source-ledger update.

---

# 78. Experiment schema correction

A canonical experiment record should distinguish at least:

```yaml
inputs:
  leakage_free_exact:
  heuristic:
  mixed:

truth:
  goldbach_count:
  delta_6:
  boundary_3:
  escape_exists:
  escape_depth:

posthoc_diagnostics:
  composite_depth:
  obstruction_profile:

surrogate:
  carrier_set:
  world_target:

promotion:
  discovery_gate:
  exact_statement:
  proof_gate:
```

---

# 79. Leakage audit checklist

Every experiment must answer before running:

```text
[ ] Does any feature directly test N-p primality?
[ ] Does any feature reconstruct R_G?
[ ] Does any pair of features exactly determine F_G?
[ ] Is escape existence included as an input?
[ ] Is delta_6 included on a 6|N target?
[ ] Is target-derived preprocessing used?
[ ] Is window selection based on the true representation?
```

If any answer is YES,

label:

```text
truth_leakage: true
```

---

# 80. Validation verdict

RG-SCP 05:

$$
\boxed{
\mathrm{PASS\ WITH\ TRUTH\mbox{-}LEAKAGE\ FIREWALL,\ HEURISTIC\ RECLASSIFICATION,\ AND\ ESCAPE\mbox{-}OBSERVABLE\ STRENGTHENING}
}
$$

Its greatest value is not a particular predictor, but:

$$
\boxed{
\text{separating discovery freedom from proof validity}.
}
$$

This round further separates:

$$
\boxed{
\text{exactness}
}
$$

and:

$$
\boxed{
\text{legal feature role}
}
$$

as well.

---

# 81. The Next Round

V0 Round 06 audits only:

# RG-SCP 06
## The Counterexample-Demand / Obstruction-Capacity Inequality

Focus:

1. Does $D_N(y)\le I_N(y)$ genuinely have nontrivial proof content?
2. Boundaries of the $R_G^{-}=D+\Omega-I$ identity;
3. Does $I_N(y)$ actually read the target factorization directly?
4. Is the raw AP capacity upper bound correct?
5. When $r\mid N$, domain details of $A_r(N)\le1$;
6. Deduplication of the primorial demand formula against R03;
7. Is the demand/capacity inequality circular?
8. Which capacity quantities can be proof inputs, and which are suitable only for finite verification?

---

# Appendix A: Canonical Data Taxonomy

```yaml
feature:
  name:

  epistemic_status:
    - exact
    - heuristic
    - mixed

  data_role:
    - proof_input
    - discovery_probe
    - diagnostic
    - truth_label

  target:
  truth_leakage:
  source_class:
  baseline_dependency:
  proof_dependency:
```

---

# Appendix B: Primorial Escape Theorem

Assume:

$$
k\ge2,
\quad
M_k\mid N,
\quad
N>6.
$$

Then:

$$
\boxed{
R_G(N)>0
\Longleftrightarrow
\exists p,
\quad
p_k<p\le N/2,
\quad
p,N-p\in\mathbb P.
}
$$

Hence:

$$
\boxed{
R_G(N)>0
\Longleftrightarrow
p_{\mathrm{esc}}(N;k)\text{ exists}.
}
$$

---

# Appendix C: Surrogate World

For arbitrary:

$$
\mathcal Q\subseteq\mathbb N,
$$

define:

$$
R_{\mathcal Q}^{-}(N)
=
\#
\left\{
q\in\mathcal Q:
2\le q\le N/2,
\quad
N-q\in\mathcal Q
\right\}.
$$

and:

$$
F_{\mathcal Q}(N)
=
\mathbf1_{\{R_{\mathcal Q}^{-}(N)=0\}}.
$$

Only when:

$$
\mathcal Q=\mathbb P
$$

do we write:

$$
F_G.
$$

---

# Appendix D: Promotion Pipeline

```text
DISCOVERY
  |
  | held-out / controls / nulls / cross-model robustness
  v
EXACT STATEMENT EXTRACTION
  |
  v
ORACLE + SOURCE AUDIT
  |
  v
INDEPENDENT PROOF
  |
  v
CERTIFICATE / FORMALIZATION
  |
  v
THEOREM LEDGER
```

---

# Appendix E: Commit Decision

```yaml
validation_round: V0-R05
decision: PASS_WITH_TRUTH_LEAKAGE_FIREWALL_HEURISTIC_RECLASSIFICATION_AND_ESCAPE_STRENGTHENING

strengthening:
  - primorial_success_iff_escape_exists_for_k_ge_2_N_gt_6
  - escape_depth_reclassified_as_exact_posthoc_truth_observable
  - exact_surrogate_target_definition

canonical_corrections:
  - add_data_role_axis
  - split_truth_from_probe_features
  - type_failure_obstruction_depth
  - discrete_epsilon_measurement
  - standard_exact_residual_not_intrinsic
  - split_discovery_gate_and_proof_gate
  - world_relative_surrogate_target
  - distinguish_empirical_from_formal_insufficiency

commit_to_sync_core: true
promote_new_goldbach_theorem: false
next_round: V0-R06
```