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§1. Introduction

§1.1. Context

Paper 18 (DOI: 10.5281/zenodo.19024385) identified the local Lipschitz property as the most leveraged target for $\sigma(G) = O(1)$, discovered the anti-correlation engine, and eliminated three proof routes. Two questions remained: (i) can B-bound be reduced to a precise algebraic condition? (ii) what is the microstructural mechanism that keeps $\mathrm{Var}(A) = O(1)$?

The present paper addresses both.

§1.2. Notation

As in Papers 17–18. Additionally: $v_{P^-}(n)$ denotes the exponent of $P^-(n)$ in the factorization of $n$. $\varepsilon_p(a) = \rho_E(p^a) - \rho_E(p^{a-1}) - \rho_E(p)$ for $a \geq 2$ (prime-power correction).

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§2. The Three-Point Law for $B$

§2.1. Empirical Distribution

At $N = 10^7$, the distribution of $B(n) = \rho_E(n) - \rho_E(n/P^-) - \rho_E(P^-) - 2$ on the $\Omega = k$ shell:

$B$	$k = 8$	$k = 10$	$k = 12$
−3	0.04%	0.00%	0.00%
−2	13.28%	12.21%	11.95%
−1	69.58%	74.23%	75.72%
0	17.10%	13.55%	12.32%

$P(B \in \{-2, -1, 0\}) > 99.9\%$ for $k \geq 10$. (Cross-validated by independent implementations at $N = 10^7$; see Appendix A for details.)

Corresponding moments:

$k$	$E[B]$	$\mathrm{Var}(B)$
8	−0.963	0.304
10	−0.987	0.258
12	−0.996	0.243

§2.2. The Three-Point Structure

From the data: $\lambda \approx 0.76$, $\alpha \approx 0.12$, $\beta \approx 0.12$, trending toward $\alpha \approx \beta$ (symmetric wings).

Algebraic consequence. If $B$ concentrates on $\{-2, -1, 0\}$ with weights $(\alpha, \lambda, \beta)$:

$$E[B] = -2\alpha - \lambda = -1 - \alpha + \beta$$

$$\mathrm{Var}(B) = \alpha + \beta - (\beta - \alpha)^2$$

If $\alpha = \beta$ (symmetric): $E[B] = -1$, $\mathrm{Var}(B) = 2\alpha$. For $\mathrm{Var}(B) = 1/4$: $\alpha = 1/8$.

Current data ($k = 12$): $\alpha = 0.12$, $\beta = 0.12$, $\mathrm{Var} = 0.24 \approx 1/4$. The data suggest convergence toward $(\alpha, \lambda, \beta) \approx (1/8, 3/4, 1/8)$, which would imply $\mathrm{Var}(B) \to 1/4$.

§2.3. Correction to Paper 18

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§3. The B-Bound Reduction Theorem

§3.1. Decomposition of $B$

$$B(n) = R_k(n) + E_k(n) - 2$$

where $R_k(n) = r(n) - r(n/P^-(n))$ and $E_k(n) = \varepsilon_{P^-(n)}(n)$ (the prime-power correction at $P^-$; zero when $v_{P^-} = 1$).

Bounds: $R_k \in [-(2k-4), 2(k-1)]$ (from $0 \leq r(m) \leq 2(\Omega(m) - 1)$). So for fixed $k$, $\mathrm{Var}(R_k) = O(k^2)$.

§3.2. Conditional Theorem

Remark. The hypothesis $E[\varepsilon_{P^-}^2] < \infty$ is the precise algebraic bottleneck. Paper 17 showed $\varepsilon_p(2) \in [-6, 2]$ pointwise, but the Sathe-Selberg weighted average on $\Omega$-shells remains to be controlled.

§3.3. Status of the $\varepsilon_p$ Second Moment

Numerically, $\mathrm{Var}(B) \in [0.24, 0.30]$ for $k \geq 8$, consistent with bounded shell-averaged second moment of $\varepsilon_{P^-}$, though not by itself proving it. The theoretical challenge is to prove uniform boundedness as $N \to \infty$ for fixed $k$, which requires controlling the Sathe-Selberg measure of primes $p$ with extreme $\varepsilon_p$ values.

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§4. Stratification by $v_{P^-}$

§4.1. Data

$v$	Share ($k=8$)	$E[B \mid v]$	$\mathrm{Var}(B \mid v)$
1	7.87%	−0.975	0.120
2	12.00%	−1.058	0.135
3	17.25%	−0.872	0.170
≥4	62.87%	−0.968	0.392

(Cross-validated by independent implementations at $N = 10^7$.)

§4.2. The Dominance of High Multiplicity

At $k = 8$, $v_{P^-} \geq 4$ accounts for 63% of the shell and contributes approximately 81% of $\mathrm{Var}(B)$ (by the law of total variance decomposition).

This overturns the early-series emphasis on squarefree ($v = 1$) behavior. Paper 16's narrative that "$> 95\%$ of composites have $v_{P^-} = 1$" is correct only for $k \leq 5$; at $k = 8$, $v = 1$ is already a 7.9% minority.

Structural interpretation. At high $k$, most integers on the $\Omega = k$ shell are "smooth" (dominated by small primes with high multiplicity). The $B$ value then depends on how the DP handles repeated prime factors — specifically, whether the prime-power correction $\varepsilon_p$ adds or subtracts from the additive baseline.

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§5. DP Self-Correction

§5.1. Large-Jump Reset

On the $\Omega = 8$ shell at $N = 10^7$:

Condition	$E[j(n+1)]$
Unconditional	2.515
After $j(n) \geq 3$	0.113

A 95% regression. After a large jump, the system nearly always enters a non-jump state ($j \approx 0$) at the next step.

Mechanism. When $j(n)$ is large: $\rho_E(n) = M_n \ll S_n$, so $S_{n+1} = M_n + 1$ is "low." For $n+1$ to also jump, $M_{n+1}$ must be even lower than $S_{n+1} = M_n + 1$. Since $M_{n+1}$ depends on $(n+1)$'s factorization, which shares no prime factors with $n$ ($\gcd(n, n+1) = 1$), there is no structural reason for $M_{n+1}$ to be systematically low. The large jump has been "forgotten."

§5.2. Shell-Order Decorrelation

On the $\Omega = 8$ shell, ordering elements $n_1 < n_2 < \ldots$ by magnitude (note: consecutive shell elements $n_i, n_{i+1}$ are typically NOT consecutive integers):

Lag $\ell$	$\mathrm{Corr}(G(n_i), G(n_{i+\ell}))$
1	−0.037
2	≈ 0
3+	≈ 0

The $G$ values on the shell are nearly decorrelated in shell order. The slight negative autocorrelation at lag 1 may reflect residual local structure among nearby shell elements.

§5.3. Implications for Local Lipschitz

The self-correction mechanism provides the structural explanation for why $\mathrm{Var}(A) = O(1)$:

• Large deviations in $A$ (large jumps) are immediately corrected at the next integer
• The data are consistent with weak dependence together with one-step self-correction
• This suppresses variance accumulation rather than proving its impossibility

This does not constitute a proof of Local Lipschitz, but it identifies the precise DP mechanism that would need to be formalized.

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§6. Predecessor-Target Cancellation

§6.1. Data

On the $\Omega = 8$ shell at $N = 10^7$:

$$\mathrm{Var}(\rho_E(n)) = 18.93, \quad \mathrm{Var}(\rho_E(n-1)) \approx 18.93$$

$$\mathrm{Corr}(\rho_E(n), \rho_E(n-1)) = 0.9676$$

$$\mathrm{Var}(A) = \mathrm{Var}(\rho_E(n-1) - \rho_E(n)) = 1.226$$

Check: $\mathrm{Var}(A) = 2 \times 18.93 \times (1 - 0.9676) = 2 \times 18.93 \times 0.0324 = 1.227$. ✓

§6.2. The 97% Cancellation

Two quantities with variance $\approx 19$ produce a difference with variance $\approx 1.2$. The correlation coefficient $0.9676$ means that 96.76% of the fluctuation in $\rho_E(n)$ is shared with $\rho_E(n-1)$. Only the residual 3.24% contributes to $\mathrm{Var}(A)$.

This cancellation is the "macro" manifestation of the anti-correlation engine discovered in Paper 18: the $O(\ln N)$ variances of $S_n$ and $M_n$ cancel to $O(1)$ because the DP couples them through the shared history.

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§7. $A$-Tail Structure

§7.1. Tail Probabilities

$k$	$P(A \geq 0)$	$P(A \geq 1)$	$P(A \geq 2)$	$P(A \geq 3)$	$P(A \geq 4)$	$P(A \geq 5)$
4	0.716	0.321	0.062	0.004	0.000	0.000
8	0.975	0.824	0.496	0.181	0.035	0.003
12	0.997	0.962	0.805	0.485	0.194	0.041

§7.2. Short-Tail Character

Successive tail ratios at $k = 8$:

$$\frac{P(A \geq 2)}{P(A \geq 1)} = 0.60, \quad \frac{P(A \geq 3)}{P(A \geq 2)} = 0.36, \quad \frac{P(A \geq 4)}{P(A \geq 3)} = 0.20, \quad \frac{P(A \geq 5)}{P(A \geq 4)} = 0.09$$

The ratios decrease — faster than geometric decay. This is consistent with short-tail / exponential-type behavior on the tested range, not a power law.

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§8. The Revised Proof Landscape

§8.1. What M9 Establishes

Proved (conditional):

• B-bound Reduction: $E[\varepsilon_{P^-}^2 \mid \Omega = k] < \infty \implies \mathrm{Var}(B) = O_k(1)$

Discoveries (numerical, strong evidence):

• Three-point law: $B \in \{-2, -1, 0\}$ with $P(B = -1) \approx 0.76$
• Data suggest $\mathrm{Var}(B) \to 1/4$ via three-point convergence toward $(1/8, 3/4, 1/8)$
• DP self-correction: 95% regression after large jumps
• $G$ shell-order autocorrelation $-0.037$ at lag 1, $\approx 0$ at lag $\geq 2$
• 97% predecessor-target cancellation
• $A$-tail consistent with exponential-type decay on tested range, incompatible with power law
• $v_{P^-} \geq 4$ dominates $\mathrm{Var}(B)$ at high $k$

Corrections:

• Paper 18 "Bernoulli near $\{0, -1\}$" → three-point $\{-2, -1, 0\}$
• Paper 16 "squarefree dominance" → $v \geq 4$ dominance at $k \geq 8$

§8.2. Updated Open Inputs for $D(N) \to 1$

Input	Status	M9 impact
$\sigma(G) = O(1)$	Local Lipschitz (open)	Self-correction + tail structure identified as mechanism
B-bound	Reduced to $E[\varepsilon_p^2]$ control (conditional theorem)	Reduced to $E[\varepsilon_p^2]$ control
$E[\varepsilon_p^2]$ control	Open	New input identified by M9
Scale term monotonicity	Numerical	Unchanged
I-a for fixed $k$	Open	Unchanged
Lemma II	Numerically benign	Unchanged
SPF-positivity convergence	Numerical (downstream)	Unchanged

§8.3. Proof Dependency Graph (Updated)

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Appendix A. Methodological Note

All mathematical claims, citations, and computations in this paper were independently checked and are the author's responsibility. AI systems were used for literature search, exploratory computation, and drafting support.

A.1. Four-AI Contributions to M9

ChatGPT predicted the three-point law $\{-2, -1, 0\}$ from the algebraic constraint $E[B] = -0.9$, $\mathrm{Var}(B) = 0.25$ (ruling out two-point Bernoulli), proved the B-bound Reduction theorem, identified the $\varepsilon_p$ second-moment bottleneck, and independently computed the $B$ distribution and $A$ tail/moment tables that were used as the primary data source.

Gemini proposed the initial PMF computation and the "two-state collapse" hypothesis (subsequently corrected: a code-level offset error produced $\{0, -1\}$ instead of $\{-2, -1, 0\}$; the error was caught by cross-validation with ChatGPT and Grok). Gemini's M8 discovery of the anti-correlation engine remains the foundational mechanism result.

Grok provided the decisive self-correction data (Task 1: $j(n+1) = 0.113$ after large jumps), $G$ autocorrelation (Task 2: lag-1 $= -0.037$), predecessor-target correlation (Task 3: 0.9676), and served as the independent tiebreaker confirming the three-point law.

Claude analyzed the DP recursion structure (Route B: self-correction via coprimality of $n$ and $n+1$; Route A: Gaussian model circularity; Route C: $I_{\mathrm{spf}}$ blocked), and identified the self-correction mechanism as the most promising formalization path.

A.2. Role of the Thermodynamic Interface

The thermodynamic interface (documented in Paper 18 Appendix A.2) continued to inform M9 through its emphasis on fluctuation-dissipation structure: the DP self-correction mechanism corresponds to a "dissipative restoring force" that returns the system to equilibrium after large fluctuations, a standard feature of systems obeying fluctuation-dissipation relations.

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References

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