Counts Converge, Spacings Do Not: A Two-Level Decomposition of Hardy-Littlewood Deviations in Twin Prime Residue Classes

The Hardy-Littlewood conjecture predicts the asymptotic global density of twin primes, but makes no per-class prediction for the 15 valid residue classes modulo $210$. We decompose deviations from this conjecture into two independent levels. Using a Python segmented sieve enumerating all $3{,}424{,}502$ twin prime pairs $(p, p+2)$ up to $10^9$: (i) count-based deviations per class are at most 0.17% and follow the expected $O(1/\sqrt{N})$ statistical fluctuations, consistent with Hardy-Littlewood being asymptotically correct for counts; (ii) gap-spacing deviations, measured via the normalized ratio $\rho(p) = 2C_2\,g(p)/(\ln p)^2$, are 4-5% in magnitude, Bonferroni-significant in 14 of 15 classes ($|t|$ up to $24.2$), and persist across all $\log p$ sub-ranges with Spearman $r_S > 0.97$ between adjacent ranges. We rule out the Lemke Oliver-Soundararajan anti-persistence bias as an explanation: self-transition rates are uniformly suppressed below $1/15$ across all classes (Spearman $r_S = -0.04$, $p = 0.90$ against gap deviation), confirming our finding is independent of that effect. Two-stage PySR symbolic regression confirms Hardy-Littlewood at the per-instance level, then at the class level discovers the term $m_{70}/\delta_7$ as the strongest number-theoretic predictor, reducing regression loss by 76% over a constant baseline. The gap-spacing deviation is a persistent higher-order phenomenon not predicted by the standard singular series.