The Insertion Identity, Variance Splitting, and the Relay Engine of Conjecture H' / 插入恒等式、方差分裂与猜想 H' 的接力引擎

We establish the algebraic and analytic engine underlying the compositeness discount identified in Paper 15. Three structural results are proved unconditionally: (1) the exact insertion identity G_spf(pm) = j(m) + K_p(m), giving the recursion μ_{k+1}(x) = E[j(m)] + E[K_p(m)] and reducing the Unbounded Mean Gain conjecture to two sub-problems; (2) the variance splitting decomposition G_spf = A + B + 1 with A ≥ −1 and B ≤ 0, yielding Var(G_spf) = O_k(1) under numerically verified hypotheses; (3) the bridge term identity K_p = A(pm) + B(pm) − A(m). The bridge lower bound E[K_p] ≥ −C_2 is numerically observed (C_2 ≈ 1 at N = 10⁷) but remains an open input. The conjecture E[K_p] > 0 is false (p = 2 dominates at 88% weight). The insertion measure exhibits positive bias ≈ +0.31, confirmed to N = 10⁷. A two-parameter shell mean identity reduces the insertion bias to low-prime dominance plus controlled shell means, verified in a finite prime alphabet toy model via Chebyshev rearrangement. A finite-window recursion and Bridge Corollary give a conditional path to μ_spf(k) → ∞. At N = 10⁷: E[G_spf | Ω = k] grows linearly (slope 0.26, R² = 0.995), Var ∈ [1.21, 1.83], and the relay mechanism is quantitatively confirmed. The paper includes a methodological note documenting a four-AI parallel exploration protocol (Claude, ChatGPT, Gemini, Grok) with cross-validation, error correction, and reproducible workflow. Chinese translation included.

Keywords

integer complexity, ρ-arithmetic, ZFCρ, insertion identity, variance splitting, relay mechanism, compositeness discount, bridge term, Chebyshev bound, Sathe-Selberg formula, AI-assisted mathematics, parallel exploration protocol

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Language

English

Related Identifiers (all "continues")

• 10.5281/zenodo.19007312 (Paper 15)
• 10.5281/zenodo.18914682 (Paper 1)
• 10.5281/zenodo.18927658 (Paper 2)
• 10.5281/zenodo.18963539 (Paper 9)
• 10.5281/zenodo.18991986 (Paper 13)