§1 Introduction

1.1 The Problem

Paper 44 (DOI: 10.5281/zenodo.19247859) established the Reset-Slack Reduction and Theorem A (Conj.2 ⟹ Conj.1 + NI1 A-side); the \(M_\mathrm{spf}\)-T-S decomposition narrowed the attack surface to \(E[(\Delta M_\mathrm{spf}^-)^2]\). Paper 45 (DOI: 10.5281/zenodo.19275286) proved the Capacity-Allocation Law, providing Conjecture 1 with an independent route.

The sole remaining principal target: Conjecture 2 (\(E[(\Delta M^-)^2 \mid \Omega=k] \leq \mathrm{poly}(k)\)).

This paper reduces the growth bottleneck of Conjecture 2 to specific analytic number theory inputs (PMH-DS and PCF), and in the process discovers three groups of new regularities in the microscopic structure of the DP recurrence.

1.2 A Priori Principles

(P1) Mean drift is the cost of conditioning; fluctuation is the intrinsic quantity. \(\Omega=k\) is an increasingly selective condition that pulls shell means but does not change fluctuations.

(P2) Coarse bound conservation. Var declining (conditioning compresses diversity) and \((\mu-\mu^-)^2\) growing (conditioning separates shell from predecessor) are two sides of the same operation.

(P3) Ternary lock within the SPF skeleton. Fluctuations of \(M_\mathrm{spf} = f + \tilde{r}\) are compressed by the DP selection mechanism; \(f\) and \(\tilde{r}\) differences are complementary.

(P4) Centered factorization = subtracting the trend to see fluctuation. \(f(n) = \lambda\log n + \xi(n)\); the Euler product cares only about the fluctuation \(\xi\).

(P5) Fixed log budget + near-affine \(f\). \(\Omega=k\) conditioning fixes the log budget; \(f\) nearly reads only the total budget. \(\mathrm{Var}(f) = \lambda^2\mathrm{Var}(\theta) + \mathrm{Var}(\xi) = O(1)\).

(P6) Corr ≈ 0.89 from shared shell-depth \(\theta\). Centered residuals \(\xi\) are nearly uncorrelated.

(P7) \(\tilde{r}\) declining = cofactor dense self-averaging. At high \(\Omega\), the cofactor enters a dense regime; \(\mathrm{Var}(\tilde{r})\) contracts.

1.3 Notation

As in Paper 44 §1.2. Additional notation: \(f(m) := \sum_{q^a \| m} \rho_E(q^a)\) (additive part); \(\tilde{r}_\mathrm{spf}(m) := M_\mathrm{spf}(m) - f(m)\) (SPF remainder); \(\theta := \log N - \log n\) (shell-depth); \(\xi(n) := f(n) - \lambda\log n\) (centered residual); \(\lambda \approx 3.12\) is the candidate linearization constant (\(E[\rho_E(p)]/E[\log p]\)).

1.4 Main Results

(1) SPF Skeleton Reduction (theorem-level). \(M_\mathrm{spf} = f + \tilde{r}_\mathrm{spf}\), \(\Delta M_\mathrm{spf}^- = \Delta f^- + \Delta\tilde{r}_\mathrm{spf}^-\). \(E[(\Delta\tilde{r})^2]\) tame (declining, numerical \(O(1)\)); \(E[(\Delta f)^2 \mid \Omega=k] \leq \mathrm{poly}(k)\) is a sufficient condition for closing Conjecture 2.

(2) Analytic interface: three key findings. (A1) \(\mathrm{Var}(f) = O(1)\) and declining; (A2) growth comes entirely from \((\mu-\mu^-)^2\); (A3) \(\mathrm{Corr} \approx 0.89\), \(k\)-independent.

(3) \(\theta\) latent variable verification. \(\mathrm{Var}(f) = \lambda^2\mathrm{Var}(\theta) + \mathrm{Var}(\xi) - 2\lambda\mathrm{Cov}(\theta,\xi)\) (exact identity, all terms \(O(1)\)); \(\mathrm{Corr}(f,f^-) \approx \lambda^2\mathrm{Var}(\theta)/[\lambda^2\mathrm{Var}(\theta)+\mathrm{Var}(\xi)]\) (predicted/actual ratio ≈ 1.01).

(4) Prime-power structure (Proposition 3). \(\rho_E(p^a) \leq a\cdot\rho_E(p)\) (unconditional upper bound). Adding UBPD gives \(\rho_E(p^a) = a\cdot\rho_E(p) + O(a)\).

(5) Conditional Closure. UBPD + PMH-DS + PCF + RT \(\Rightarrow\) Conjecture 2 \(\Rightarrow\) H'.

§2 SPF Skeleton Reduction

2.1 Exact Decomposition

Define \(\tilde{r}_\mathrm{spf}(m) := M_\mathrm{spf}(m) - f(m)\).

Piecewise expansion of \(\tilde{r}_\mathrm{spf}\) (\(m = p^v \cdot b\), \(p = P^-(m)\), \((p,b)=1\)): \(\tilde{r}_\mathrm{spf} = r(m/p) - \varepsilon_\mathrm{spf}(m)\), where \(\varepsilon = 0\) (\(v=1\)) and \(\varepsilon = \rho_E(p^v) - \rho_E(p) - \rho_E(p^{v-1})\) (\(v \geq 2\)). \(\varepsilon\) is negligible (\(E[\varepsilon^2] < 0.03\) for \(k \geq 5\)).

2.2 Core Moment Table (\(N = 10^7\))

2.3 Three Findings

(F1) \(E[(\Delta\tilde{r})^2]\) is declining and tame. From 2.10 (\(k=6\) peak) to 1.18 (\(k=14\)). The cofactor (\(\Omega = k-1\)) enters a dense self-averaging regime at high \(k\) (P7). Decline is driven by \(\mathrm{Var}(\tilde{r} \mid \Omega=k)\) declining (1.08 → 0.41); \(\mathrm{Corr}(\tilde{r}, \tilde{r}^-) \approx 0\) (no correlation compression).

(F2) \(E[(\Delta f)^2]\) is growing — the sole bottleneck. From 2.44 to 8.86. Growth comes from mean drift (§3).

(F3) \(\mathrm{Cov}(\Delta f, \Delta\tilde{r})\) is strongly negative (Corr ≈ −0.50 to −0.70). Paper 18's anti-correlation engine replicated within the SPF skeleton (P3).

2.4 Reduction of Conjecture 2

\(E[(\Delta M_\mathrm{spf})^2] \leq 2\cdot E[(\Delta f)^2] + 2\cdot E[(\Delta\tilde{r})^2]\). Data shows \(E[(\Delta\tilde{r})^2] = O(1)\) (declining).

Relationship to Paper 18: Paper 18 ruled out "unconditional control of \(\mathrm{Var}(\Delta f)\) first." This paper first strips away \(\tilde{r}\) (tame), then controls \(E[(\Delta f)^2]\) conditioned on \(\Omega=k\). The target has changed; the logic has not reversed.

§3 Analytic Interface

3.1 Four-Quantity Decomposition

3.2 Data (\(N = 10^7\))

3.3 Three Key Findings

(A1) \(\mathrm{Var}(f \mid \Omega=k) = O(1)\) and declining (P5). From 11.4 to 8.6. Not \(O(k)\) — it is \(O(1)\) and decreasing. The \(\Omega=k\) conditioning fixes the log budget; \(f \approx \lambda\log n\) reads nearly only the total budget.

(A2) Growth comes entirely from \((\mu-\mu^-)^2\) (P1). \(\mu_\mathrm{shell}\) declines slightly; \(\mu_\mathrm{pred}\) rises slightly. The gap is linear in \(k\) (≈ \(-0.2k\)). \((\mu-\mu^-)^2 \approx 0.04k^2\). Mean drift is the cost of conditioning, not dynamical instability.

(A3) \(\mathrm{Corr}(f, f^-) \approx 0.89\), \(k\)-independent (P6). Comes from shared shell-depth \(\theta\).

(A4) Coarse bound \(2\mathrm{Var}+2\mathrm{Var}^-+(\mu-\mu^-)^2 \approx 40\), \(k\)-independent (P2). Var declining exactly compensates \((\mu-\mu^-)^2\) growing — two sides of conditioning in balance. (The value 40 is the coarse bound; the actual value with \(\mathrm{Corr} \approx 0.89\) compresses to \(\sim 0.6k\).)

3.4 \(\theta\) Latent Variable

\(f(n) = \lambda(\log N - \theta) + \xi(n)\), \(\quad \theta = \log N - \log n\), \(\quad \lambda \approx 3.12\).

Q2 model: \(\mathrm{Corr}(f,f^-) \approx \lambda^2\mathrm{Var}(\theta) / [\lambda^2\mathrm{Var}(\theta) + \mathrm{Var}(\xi)]\)

The 89% correlation comes almost entirely from shared \(\theta\). \(\mathrm{Cov}(\xi,\xi^-) \approx 0\) — centered residuals are nearly uncorrelated.

§4 Uniform Linearization (UL-weak)

4.1 Proposition 3 (Prime-Power Structure)

The following three results hold unconditionally for all primes \(p\) and positive integers \(a\):

(a) Successor dominance for primes. \(\rho_E(p) = \rho_E(p-1) + 1\). Proof: primes have no nontrivial factorization (\(M(p) = +\infty\)). Verified: 664,579 primes, zero failures.

(b) Prime-power sub-additivity. \(\delta_a(p) := \rho_E(p^a) - \rho_E(p) - \rho_E(p^{a-1}) \leq 0\). Proof: \(p \times p^{a-1}\) is a factorization of \(p^a\); by R1, \(\rho_E(p^a) \leq \rho_E(p) + \rho_E(p^{a-1})\). Verified: \(\delta \leq 0\) exact, \(\max(\delta) = 0\).

(c) Recursive expansion. \(\rho_E(p^a) = a\cdot\rho_E(p) + \sum_{j=2}^a \delta_j(p)\), \(\sum\delta \leq 0\). Upper bound: \(\rho_E(p^a) \leq a\cdot\rho_E(p)\).

On the O(a) lower bound (UBPD hypothesis): The upper bound \(\rho_E(p^a) \leq a\cdot\rho_E(p)\) is an unconditional theorem. The lower bound requires \(\sup_{p,a} |\delta_a(p)| < \infty\) (Uniform Bounded Prime-power Defect, UBPD). Data shows \(|\delta| \leq 7\) (\(N = 10^7\)), but UBPD as a global assertion is not yet proved.

4.2 UL-weak and UL-strong

UL-weak (\(\rho_E(p^a) \leq a\cdot\rho_E(p)\)) is an unconditional theorem. Adding UBPD gives \(\rho_E(p^a) = a\cdot\rho_E(p) + O(a)\).

UL-strong requires \(\rho_E(p) = \lambda\log p + O(1)\). Numerically very strong (\(\mathrm{Var}(\eta_1) = 0.62\), range [−4.83, +3.65]), but depends on the behavior of \(\rho_E(p-1)\). UL-strong is not the correct target for SCF — SCF requires the prime harmonic mean (PMH), not pointwise boundedness. UL-strong is neither necessary nor sufficient for PMH.

§5 Centered Factorization and Conditional Closure

5.1 Centered Factorization Architecture

By Proposition 3 (unconditional), \(\rho_E(p^a) \leq a\cdot\rho_E(p)\) and \(\delta_a \leq 0\). Adding UBPD, \(\rho_E(p^a) = a\cdot\rho_E(p) + O(a)\). Define \(\eta(p) := \rho_E(p) - \lambda\log p\), \(g_t(p^a) := e^{t\cdot(\rho_E(p^a) - \lambda a\log p)}\). The variable substitution \(w := s - \lambda t\) pulls the Euler product's singularity back to \(w=1\).

5.2 PMH-DS (Prime Mean Hypothesis, Dirichlet-Series Version)

PMH-DS is stronger than the partial-sum version but is precisely the analytic input required by Selberg-Delange coefficient extraction. Numerical support: \(\eta(p)\) is approximately Gaussian on primes, \(E[\eta] = 0.00\), \(\mathrm{Var}[\eta] = 0.62\).

5.3 SCF Theorem (Conditional on UBPD + PMH-DS)

Under UBPD and PMH-DS, Selberg-Delange analysis of the centered Euler product gives:

\(K_{x,k}(t) = \lambda t\log x + k\log\beta(t) + O(1)\)

Corollary (shell side only): \(\mu_{x,k} = \lambda\log x + k(\log\beta)'(0) + O(1)\), \(\mathrm{Var}(f \mid \Omega=k) = k(\log\beta)''(0) + O(1) = O(k)\).

Data is stronger: \(\mathrm{Var} = O(1)\), implying \((\log\beta)''(0) \approx 0\) — extra rigidity that SCF need not prove.

5.4 PCF

PCF is not almost independence (\(B_1 \neq 0\); data confirms \(\mu^-\) has \(k\)-slope). Literature interface: Goudout (conditioned shifted Erdős-Kac), Fouvry-Tenenbaum (shifted additive laws), Mangerel (bivariate Erdős-Kac), Verwee (weighted fibre framework). The path exists but is not yet directly covered.

5.5 Remainder Tameness

Data: \(E[(\Delta\tilde{r})^2]\) declines from 2.1 to 1.2 — far exceeding \(\mathrm{poly}(k)\) requirements. Listed as an explicit input to the conditional closure.

5.6 Conditional Closure Theorem

§6 Numerical Evidence

6.1 SPF Skeleton Identity Verification

\(M_\mathrm{spf} = f + \tilde{r}_\mathrm{spf}\): 8,670,842 samples, zero failures. \(\tilde{r}_\mathrm{spf} = r(m/P^-) - \varepsilon_\mathrm{spf}\): zero failures.

6.2 UL-weak Verification

Successor dominance: 664,579 primes, zero failures. \(\delta_a \leq 0\): exact across all samples. \(p \times p^{a-1}\) optimality: 100% at \(a=2,3\).

6.3 \(\theta\) Latent Variable Verification

\(\mathrm{Var}(f) = \lambda^2\mathrm{Var}(\theta) + \mathrm{Var}(\xi) - 2\lambda\mathrm{Cov}(\theta,\xi)\): exact (check = 0.0000 across all \(k\)). \(\mathrm{Corr}(f,f^-) \approx \lambda^2\mathrm{Var}(\theta)/[\lambda^2\mathrm{Var}(\theta)+\mathrm{Var}(\xi)]\): predicted/actual ratio ≈ 1.01.

6.4 \(\tilde{r}\) Decline Mechanism

Decline driven by \(\mathrm{Var}(\tilde{r})\); \(\mathrm{Corr} \approx 0\) (no compression effect).

§7 SAE Interpretation

7.1 Conditioning Bias vs Dynamical Fluctuation

The sole source of \(E[(\Delta f)^2]\) growth is \((\mu-\mu^-)^2\). Fluctuations (\(\mathrm{Var}(f)\), \(\mathrm{Var}(\xi)\)) are stable at \(O(1)\). The system has not deteriorated — the observation window has narrowed. As \(\Omega=k\) conditioning becomes more selective, shell and predecessor means separate further, but within-shell fluctuation shrinks.

This is the precise realization of SAE's intrinsic/observational quantity distinction within the DP recurrence.

7.2 Ternary Lock within the SPF Skeleton

\(\mathrm{Cov}(\Delta f, \Delta\tilde{r})\) is strongly negative — \(f\) and \(\tilde{r}\) differences are complementary. The fluctuation of \(M_\mathrm{spf}\) (chisel's output) is compressed because construct (\(f\)) and remainder (\(\tilde{r}\)) fluctuations cancel precisely. This is the replica of Paper 18's \(\mathrm{Cov}(\Delta f, \Delta r) \approx -\mathrm{Var}(\Delta f)\) at the SPF skeleton level.

7.3 Centered Factorization = Subtracting the Trend to See Fluctuation

\(f = \lambda\log n + \xi\). The Euler product cares only about \(\xi\) (fluctuation), not \(\lambda\log n\) (trend). Centering is "subtracting the construct's mean trend to see only the remainder's fluctuation."

§8 Complete H' Closure Chain

Proposition 3 (a)(b)(c)  (unconditional theorems)
+ UBPD  (uniform bounded prime-power defect, hypothesis)
+ PMH-DS  (prime Dirichlet-series analytic control, hypothesis)
⟹ SCF  (shell cumulant fibre)
+ PCF  (predecessor cumulant fibre, hypothesis)
+ RT  (remainder tameness, hypothesis)
⟹ E[(Δf)²|Ω=k] = O(k²)
⟹ E[(ΔM_spf)²] = O(poly(k))
⟹ Conjecture 2
⟹ Conj.1 + NI1 A-side     [Paper 44 Thm A]
⟹ NI1_poly                  [+ Paper 20 B-side]
⟹ NI2_poly easier           [Paper 44 Thm B]
⟹ Paper 42 chain
⟹ H'

Distance assessment:

§9 Open Questions

1. Proof of PMH-DS. Prime harmonic mgf asymptotics for \(\eta(p)\). Standard Mertens framework + Selberg-Delange.

2. Proof of PCF. Shifted weighted Sathe-Selberg on \(\Omega\)-shells.

3. \((\log\beta)''(0) \approx 0\) — why is \(\mathrm{Var}(f|\Omega=k) = O(1)\)? Possibly related to multinomial constraints under fixed log budget (Ford).

4. Precise constant of the coarse bound ≈ 40. May admit a more compact expression.

5. UL-strong as bonus conjecture. \(\rho_E(p) = \lambda\log p + O(1)\) — stronger than PMH and not a necessary condition for it.

Data Sources

Scripts: spf_skeleton.c, analytic_interface.c, theta_latent.c, q3_verify.c, ul_verify.c (C, gcc -O2). Data: \(\rho_E\) via DP min for \(n \leq 10^7+1\).

References

[1] H. Qin. ZFCρ Paper XLIV. DOI: 10.5281/zenodo.19247859.
[2] H. Qin. ZFCρ Paper XLV. DOI: 10.5281/zenodo.19275286.
[3] H. Qin. ZFCρ Paper XLII. DOI: 10.5281/zenodo.19226607.
[4] H. Qin. ZFCρ Paper XVIII. DOI: 10.5281/zenodo.19024385.
[5] H. Qin. ZFCρ Paper XX. DOI: 10.5281/zenodo.19027892.
[6] M. Verwee. Additive functions on shifted primes (2026). arXiv:2603.17682.
[7] É. Goudout. Lois de répartition des diviseurs (2018).
[8] A. Roy. Landau-Selberg-Delange for Ω (2025). arXiv:2511.15928.
[9] A. Gafni, N. Robles. Selberg-Delange for Ω-counting (2025). arXiv:2502.05298.
[10] R. Benatar. Short-interval Hildebrand-Tenenbaum (2024). arXiv:2408.16576.
[11] E. Fouvry, G. Tenenbaum. Shifted additive laws.
[12] K. Ford. Conditional prime-subset counts.
[13] H. Helfgott. Ω-shell shifted averages (2021).
[14] A. Mangerel. Bivariate Erdős-Kac (2016).

Acknowledgments

ChatGPT (Gongxihua): Proposal of the SPF Skeleton Reduction (\(M_\mathrm{spf} = f + \tilde{r}_\mathrm{spf}\) decomposition). Centered factorization architecture (variable substitution \(w = s - \lambda t\), correct definition of PMH, complete derivation of the SCF theorem). Conditional Closure Theorem (SCF + PCF ⟹ Conjecture 2). Analytic interface exact identity (four-quantity decomposition of \(E[(\Delta f)^2]\)). Demotion of UL-strong (PMH is the correct prime input for SCF). Proposal of the \(\theta\) latent variable (shared shell-depth explaining Corr ≈ 0.89).

Claude (Zilu): All numerical experiments (SPF skeleton verification, core moment table, analytic interface four quantities, \(\theta\) latent variable, Q3 verification, UL verification). Text drafting, working notes v1–v4. Unconditional proof of Proposition 3.

Claude (Thermodynamic thread): Identification of P1 (conditioning bias). Interpretation of P2 (coarse bound conservation). UL \(\eta = O(a)\) correction. Coarse bound ≈ 40 as "Corr=0 baseline" explanation. Canonical ensemble analogy for SCF.

Gemini (Zixia): Positioning of UL-weak as the "Archimedean fulcrum" of centered factorization. Emphasis on the rigidity constraint \((\log\beta)''(0) \approx 0\). Confirmation of selection bias vs dynamical fluctuation distinction.

Grok (Zigong): Series consistency audit (Paper 42–46 chain verification). Suggestion to label Conditional Closure Theorem with explicit hypotheses. Numerical precision confirmation.

Han Qin (author): A priori discovery of P1 ("mean drift is the cost of conditioning; fluctuation is the intrinsic quantity"). A priori audit methodology (P1–P7 vs Q1–Q3 distinction). Scope decision for Paper 46 (SPF skeleton + conditional closure). Acceptance of UL-strong → PMH demotion. Final text independently completed by the author.