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

# DMSD Phase 3G — Structured Null Escalation
## Source-Depth Transition and Baseline–Source Compatibility

**Date:** 2026-09-30  
**Status:** COMPLETE  
**Exact prime bound:** \(10^7\)  
**Primary partition:** 1000 bins  
**Scales:** \(L_4,L_5,L_6\)

---

# 1. Purpose

Phase 3F left two of the most stable finite diagnostics:

\[
\boxed{
\text{Residual Underdispersion}
}
\]

and:

\[
\boxed{
\operatorname{Corr}(Z_j,Z_{j+1})<0.
}
\]

But that round's null source only excluded:

\[
2,3,5,7,11,13.
\]

This round therefore no longer asks:

> What does the residual look like?

Instead, it asks:

> **If we add more known trial-divisibility sources to the null, at what layer will these residual diagnostics be absorbed, changed, or rendered invalid?**

---

# 2. Structured Source Escalation

For threshold \(y\), define the candidate source:

\[
C_y(n)
=
\mathbf 1[
n\text{ has no prime }p\le y\text{ as a divisor}
].
\]

That is, generalize:

\[
W_{30030}
\]

into a “divisibility filter for all primes \(\le y\).”

This round scans:

\[
y=
13,17,23,31,43,61,97,151,223,313,463,673,997,1499,2203,3137,
\]

retaining only the valid portion for each scale.

---

# 3. Candidate Precision

For interval \(B\), define:

\[
\operatorname{Prec}_y(B)
=
\frac{\#\{\text{primes in }B\}}
{\#\{\text{\(y\)-rough candidates in }B\}}.
\]

This is not a statistical estimate of classifier precision,
but an exact finite-domain ratio.

As \(y\) increases:

\[
\operatorname{Prec}_y(B)\uparrow.
\]

Eventually, if:

\[
y\ge \sqrt{\max B},
\]

every composite has at least one prime divisor \(\le y\),
so the candidate set closes to the prime set.

---

# 4. Trial-Division Closure

For:

\[
L_k=[10^k,10^{k+1}),
\]

as long as the divisibility source includes all primes:

\[
p\le \sqrt{10^{k+1}-1},
\]

every composite:

\[
n<10^{k+1}
\]

will be excluded.

Closure thresholds for this round:

### \(L_4\)

\[
y=313.
\]

### \(L_5\)

\[
y=997.
\]

### \(L_6\)

\[
y=3137.
\]

At that point:

\[
\boxed{
\text{candidate set}
=
\text{prime set}
}
\]

holds on the given finite interval.

This is merely a multiscale representation of classical trial-division closure,
not a new primality theorem.

---

# 5. Structured Null 1: Smooth-Weighted Candidate Binomial

For each bin \(j\), first compute the exact candidate count:

\[
C_j^{(y)}.
\]

Then use the smooth density:

\[
\frac1{\log x}
\]

to construct normalized expected mass:

\[
Q_j^{(y)}
\propto
C_j^{(y)}
\cdot
\frac{1}{|B_j|}
\int_{B_j}\frac{dx}{\log x}.
\]

Let the exact prime total be \(N\). Then:

\[
E_j=NQ_j^{(y)}.
\]

The conditional candidate success rate:

\[
p_j=\frac{E_j}{C_j^{(y)}}.
\]

Only if:

\[
0\le p_j\le1
\]

holds for all bins can we construct:

\[
X_j\sim\operatorname{Binomial}(C_j^{(y)},p_j).
\]

---

# 6. Structured Null 2: Fixed-Total Candidate Hypergeometric

Another control version uses only the candidate capacity:

\[
C_j^{(y)},
\]

and fixes the total prime count \(N\).

Select \(N\) positions from all candidate positions without replacement,
obtaining multivariate hypergeometric bin counts.

This null:

- preserves total count;
- preserves each bin's candidate capacity;
- does not add \(1/\log x\) smooth weighting.

It is a capacity-only control,
not a final prime model.

---

# 7. The first core result: Underdispersion is Source-Relative

## \(L_5\)

### \(y=13\)

candidate precision:

\[
0.399.
\]

observed RMS:

\[
0.616.
\]

The independent candidate null's expected RMS is approximately:

\[
0.775.
\]

Clearly underdispersed.

### \(y=313\)

candidate precision:

\[
0.819.
\]

observed RMS:

\[
0.401.
\]

independent-null expected RMS:

\[
0.424.
\]

ratio:

\[
\boxed{
0.945
}
\]

is already very close to 1.

The underdispersion that looked strong in the preceding round
therefore shrinks substantially after adding deeper divisibility sources.

---

# 8. This is even clearer in \(L_6\)

### \(y=13\)

precision:

\[
0.340.
\]

RMS ratio:

\[
\frac{0.554}{0.813}
\approx0.681.
\]

### \(y=313\)

precision:

\[
0.670.
\]

ratio:

\[
0.779.
\]

### \(y=673\)

precision:

\[
0.774.
\]

observed:

\[
0.502
\]

vs independent expectation:

\[
0.475.
\]

ratio:

\[
\boxed{
1.055.
}
\]

In other words, the original:

\[
\boxed{
\text{underdispersion}
}
\]

has not only disappeared, but has even crossed slightly to the other side at this source depth.

Therefore:

\[
\boxed{
\text{Residual Underdispersion}
}
\]

is not a fixed prime invariant;
it depends strongly on the null source class.

---

# 9. Lag-1 Negative Correlation is not a fixed invariant either

## \(L_5\)

### \(y=13\)

\[
\rho_1=-0.120.
\]

### \(y=223\)

\[
\rho_1=-0.098.
\]

### \(y=313\)

\[
\boxed{
\rho_1=+0.085.
}
\]

At this point:

\[
p_{\max}\approx0.929<1,
\]

so the smooth-weighted candidate null remains probabilistically feasible.

That is, negative lag-1 has already changed sign **before trial-division closure**.

---

# 10. The Source-Depth Transition in \(L_6\)

### \(y=313\)

\[
\rho_1=-0.137.
\]

### \(y=463\)

candidate precision:

\[
0.721.
\]

and:

\[
\boxed{
\rho_1=+0.080.
}
\]

At the same time:

\[
p_{\max}\approx0.801<1.
\]

Thus:

\[
\boxed{
\text{negative residual correlation}
}
\]

also disappears in \(L_6\) within the valid null range.

This means that the anti-correlation left by Phase 3F
cannot directly be elevated to a prime ontology invariant.

---

# 11. The difference in \(L_4\)

In \(L_4\):

\[
y=97
\]

still gives:

\[
\rho_1=-0.119.
\]

When \(y=151\),
the smooth-weighted model already has:

\[
p_{\max}>1.
\]

Thus, at this smaller scale,
the baseline family loses probabilistic admissibility
before entering the pronounced sign-flip region.

The three scales therefore do not share a universal transition threshold.

For now, we can only say:

\[
\boxed{
\text{source-depth transition is horizon/scale dependent}.
}
\]

---

# 12. The Baseline–Source Compatibility problem

This round unexpectedly reveals a methodological problem more important than the residual pattern itself.

For a shallow source:

```text
candidate filter is weak
```

so a:

\[
1/\log x
\]

smooth correction is very reasonable.

But as the source becomes deeper:

\[
C_y
\]

itself already includes increasingly more information about “why certain positions cannot be prime.”

Finally, near:

\[
y\approx\sqrt x
\]

the candidate set is almost the prime set.

Forcing an additional multiplication by:

\[
1/\log x
\]

then applies the density effect twice,
and can even lead to:

\[
\boxed{
p_j>1.
}
\]

This is a direct baseline failure certificate.

---

# 13. Baseline–Source Compatibility Principle

This round therefore proposes a methodological principle:

\[
\boxed{
\textbf{Baseline–Source Compatibility Principle}
}
\]

> **The applicability of a baseline correction depends on how much structure the current information source has already absorbed. When the source itself increasingly explains the effect described by the baseline, the original baseline must be recalibrated, weakened, or removed.**

That is:

\[
\boxed{
\text{Known Effect}
+
\text{Source Encoding Same Effect}
\neq
\text{Always Valid Combination}.
}
\]

Otherwise, there will be double-counting.

---

# 14. Source Saturation Boundary

We can view:

\[
y\to\sqrt b
\]

as a source saturation process.

When:

\[
y<\sqrt b
\]

composite candidates still survive.

When:

\[
y\ge\sqrt{b-1}
\]

and all required primes are included:

\[
\boxed{
\text{Source Closure}.
}
\]

At that point, a stochastic placement model is no longer needed:

\[
\text{candidate source itself}
\]

already determines prime membership exactly
on the given finite interval.

---

# 15. Source Depth Scan

In addition to four primary thresholds, this round
also performs a dense milestone scan.

## \(L_4\)

Within the smooth model's feasible range,
the RMS ratio is closest to the independent null approximately at:

\[
y=31,
\qquad
\text{ratio}\approx0.938.
\]

Lag-1 remains negative.

## \(L_5\)

Closest to a variance match:

\[
y=313,
\qquad
\text{ratio}\approx0.945.
\]

At the same time, lag-1 has already turned positive.

## \(L_6\)

Closest at:

\[
y=673,
\]

\[
\text{ratio}\approx1.055.
\]

Lag-1 changes sign earlier, at:

\[
y=463
\]

this threshold.

There is no shared universal \(y\).

---

# 16. Actual Source Attribution

In the preceding round, we could say:

> PNT+\(W_{30030}\)+independent candidates are insufficient.

In this round, we can go further:

> **As the divisibility source expands to higher primes, underdispersion and negative lag-1 both weaken significantly, and disappear at some scales before full closure.**

At least a substantial part of them can therefore be attributed to:

\[
\boxed{
\text{unmodeled higher-prime divisibility structure}.
}
\]

This is more conservative and more accurate than saying:

> “We have discovered a new prime anti-correlation.”

as a claim.

---

# 17. The role of the Capacity-Only Null

The fixed-total hypergeometric control shows:

Knowing only:

\[
\text{candidate capacity}
\]

is itself also insufficient to reproduce the exact residual at every source depth.

In particular, while the source is still shallow,
ignoring the smooth density gradient leaves strong systematic structure.

Therefore:

\[
\boxed{
\text{capacity source}
\neq
\text{density source}.
}
\]

The two must be distinguished.

But when the source reaches complete closure:

\[
C_j=O_j
\]

the capacity-only model degenerates to the exact histogram,
with RMS residual:

\[
0.
\]

This is only a closure identity.

---

# 18. This round revises Phase 3F's candidate propositions

Phase 3F's most stable candidates:

\[
\text{Residual Underdispersion}
\]

and:

\[
\text{Negative Lag-1 Residual Correlation}
\]

must now both be downgraded.

The more accurate status:

```text
Underdispersion:
  SOURCE-RELATIVE
  largely reduced by deeper divisibility sources

Negative lag-1:
  SOURCE-RELATIVE
  disappears before closure on L5/L6
```

Thus, at present:

\[
\boxed{
\text{No residual feature is sufficient to be elevated to a new prime invariant.}
}
\]

This is actually a good result.

---

# 19. A new Source-Depth axis emerges in DMSD

DMSD's original axes:

\[
\text{Scale}
\times
\text{Position}
\times
\text{Resolution}.
\]

We now also need to add:

\[
\boxed{
\text{Source Depth}.
}
\]

The residual must therefore be written as:

\[
R(
\text{scale},
\text{resolution},
\text{source class},
\text{source depth},
\text{baseline}
).
\]

Writing only:

\[
R
\]

is no longer enough.

---

# 20. Source Depth and computational cost

Greater source depth:

\[
y\uparrow
\]

means more divisibility information must be checked.

Eventually, reaching:

\[
y\approx\sqrt n
\]

means complete trial division.

Therefore:

\[
\boxed{
\text{residual reduction}
\leftrightarrow
\text{source / computational burden increase}.
}
\]

This reconnects with SDRPC's earlier proof-burden perspective:

> More source information can eliminate ambiguity / residuals, but it is not free.

---

# 21. The actual result of this round

Phase 3G did not discover a new prime law.

Instead, it successfully further attributes the two most stable residual features of Phase 3F
to:

\[
\boxed{
\text{source insufficiency artifacts}
}
\]

At least on the current finite exact scales,
deeper divisibility sources are sufficient to:

- substantially shrink underdispersion;
- bring the variance ratio close to 1;
- make negative lag-1 correlation disappear or change sign;
- ultimately eliminate candidate ambiguity entirely at trial-division closure.

This matches exactly what this research line cared about most from the outset:

\[
\boxed{
\text{Do not mistake residuals caused by insufficient sources
for new mathematical ontology.}
}
\]

---

# 22. Next step

We should not keep increasing source depth now,
because this will ultimately reach:

\[
\sqrt n
\]

trial-division closure.

A more valuable next step is:

# DMSD Phase 3H — Source-Efficiency Frontier

Study:

\[
\boxed{
\text{how much source burden}
\rightarrow
\text{eliminates how much residual / ambiguity}
}
\]

For example:

- candidate precision gain;
- residual RMS reduction;
- lag-1 disappearance;
- compute cost;
- source depth;
- closure distance.

That is, construct a Pareto frontier for:

\[
\boxed{
\text{Information Gain}
/
\text{Source Burden}
}
\]

these quantities.

This will truly connect DMSD with SDRPC's earlier proof-burden approach.

---

# 23. Conclusion

The most important aspect of Phase 3G is not a deeper wheel.

Rather, it is obtaining:

\[
\boxed{
\text{Residual Structure is Source-Relative}.
}
\]

and:

\[
\boxed{
\text{Baseline must remain compatible with source depth}.
}
\]

The previously most stable underdispersion / negative lag-1
are not structures that remain fixed under source escalation.

Thus, the most reasonable conclusion at present is not to propose a new prime conjecture,
but:

> **First study the efficiency frontier among source depth, computational burden, and residual elimination.**

---

```text
DMSD Phase 3G: COMPLETE
Validation: PASS

Next:
DMSD Phase 3H — Source-Efficiency Frontier
```