The Anti-Correlation Engine: Variance Structure of ρ_E and the Local Smoothness of Integer Complexity / 反相关引擎:ρE\rho_E ρE 的方差结构与整数复杂度的局部光滑性

We report the decisive M8 exploration results on the variance structure of the integer complexity function ρ_E. Five numerical discoveries reshape the proof landscape for Conjecture H' (D(N) → 1). (1) Shell deficiency drift: Var(d_ρ | Ω=k) grows as a(k) + b(k)·ln ln N with b(k) > 0 increasing faster than linearly in k — not O(1). (2) Local smoothness: Var(ρ_E(n-1) - ρ_E(n) | Ω=k) lies in [0.3, 1.35] with zero N-growth trend (seven-point scan). (3) Anti-correlation engine: in the decomposition ρ_E = f + r, Var(Δf) ≈ 10 and Var(Δr) ≈ 7, but Cov(Δf, Δr) ≈ −7.5 compresses Var(A) to ≈ 1. (4) Variance anatomy: Var(B) → 1/4, Var(A) saturates near 1.3. (5) Gaussian character: Var(G) ≤ 1.45, kurtosis ∈ [2.62, 2.92]. Three proof routes eliminated. Most leveraged remaining target: local Lipschitz property. Appendix documents four-AI parallel methodology and thermodynamic interface contributions. Bilingual edition.

Keywords

integer complexity, ρ-arithmetic, ZFCρ, anti-correlation, variance decomposition, local smoothness, deficiency, DP recursion, Turán-Kubilius, Erdős-Kac, local Lipschitz

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Language

English

Related Identifiers (all "continues")

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