Conditional Harmonic Memory in Consecutive Prime Gaps

Let (g_n=pₙ₊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+gₙ₊₁) and (t=g_n-gₙ₊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-κ/L) to first order, with (κ=0.351±0.009). The identity (Cov(g_n,gₙ₊₁)=[Var(s)-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[Cov(g_n,gₙ₊ₖ)=-L/2k+o(L),-ρ_kL1/2k.]The harmonic coefficient arises from the discrete identity (Δ^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)→3). A segmented-sieve computation over 1,825,728,368 prime gaps at six heights tests both predictions. At (10¹⁵), ranks (3,4,5) fit (q_k=C/k) with (C=0.49519±0.00222), and a joint fixed-limit fit of the third-cumulant statistic for (m=1,2) gives (χ^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.