Conditional Harmonic Memory in Consecutive Prime Gaps

Abstract

Let (g_n=p_{n+1}-p_n) denote consecutive prime gaps near a large height (X), with local scale (L=\log X). This paper develops a unified framework for two complementary forms of arithmetic memory. First, in the coordinates (s=g_n+g_{n+1}) and (t=g_n-g_{n+1}), a model that exactly matches the empirical marginal distribution of (s) and weights each admissible split by the Hardy–Littlewood triple singular series remains systematically too broad in the conditional (t)-direction. Across seven numerical bands containing 489,720,096 adjacent-gap pairs, the conditional variance ratio follows (1-\kappa/L) to first order, with (\kappa=0.351\pm0.009). The identity (\operatorname{Cov}(g_n,g_{n+1})=[\operatorname{Var}(s)-\operatorname{Var}(t)]/4) localizes the residual dependence inside the conditional splitting law. Second, under an explicitly stated quantitative Hardy–Littlewood/mod-Poisson count expansion and a square-tail uniform-integrability condition, an exact Palm inversion yields, for every fixed (k), the conditional asymptotic law[\operatorname{Cov}(g_n,g_{n+k})=-\frac{L}{2k}+o(L),\qquad-\rho_kL\longrightarrow\frac{1}{2k}.]The harmonic coefficient arises from the discrete identity (\Delta^2(mH_m)=1/m), rather than from curve fitting. An exact centered-singular-series transform identifies the contribution of every factorial rank to the count law, while the same Palm calculation produces all fixed block moments and predicts the parameter-free third-cumulant ratio (R_m^{(3)}(L)\to3). A segmented-sieve computation over 1,825,728,368 prime gaps at six heights tests both predictions. At (10^{15}), ranks (3,4,5) fit (q_k=C/k) with (C=0.49519\pm0.00222), and a joint fixed-limit fit of the third-cumulant statistic for (m=1,2) gives (\chi^2=6.975) on ten degrees of freedom, with (p=0.728). Exact identities, conditional theorems, model consequences, and finite computations are explicitly distinguished throughout the manuscript.