Monotonicity, Roughness Stability, and the Narrowing of the Relay Architecture / 单调性、粗糙度稳定性与接力架构的收窄

We substantially narrow the open inputs of Paper 16's relay engine for Conjecture H' (D(N) → 1). Four rigorous propositions are proved: (J) the predecessor-jump identity A(n) = j(n) − 1, reducing A-tail to the variance bound Var(G | Ω = k) = O(1); (C) roughness stability via Cauchy-Schwarz, giving O_k((ln ln X)^{-1/2}) control on conditioned shell means; (A) failure of fixed small-prime dominance for fixed k as X → ∞; (B) exact support cutoff at high k. One conditional closure is derived: Sub-lemma I-b under scale term monotonicity and shell variance. Three numerical observations are reported: (S) scale term X ↦ μ_k(X,p) is strictly decreasing across 18 points (10⁴ to 10⁷) with zero exceptions; (K) bridge term E[K_p] turns positive at k ≥ 6 (growing to +1.7 at k = 12), stable across five windows, overturning Paper 16 Observation 8; (E) prime-power correction ε_p(2) ∈ [−6, 2], confirming B-bound cannot be upgraded. The proof of D(N) → 1 now reduces to four principal relay inputs (scale term monotonicity, σ(G) = O(1), B-bound, I-a for fixed k), plus Lemma II and downstream SPF-positivity convergence. Bilingual edition: English original with Chinese translation.

Keywords

integer complexity, ρ-arithmetic, ZFCρ, insertion monotonicity, roughness stability, relay mechanism, bridge term, scale term, predecessor-jump identity, Cauchy-Schwarz, Sathe-Selberg

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Language

English

Related Identifiers (all "continues")

• 10.5281/zenodo.19013602 (Paper 16)
• 10.5281/zenodo.19007312 (Paper 15)
• 10.5281/zenodo.18991986 (Paper 13)
• 10.5281/zenodo.18914682 (Paper 1)