The variance of twin primes across residue classes: a conditional asymptotic and numerical evidence

Let T_c(x) count the twin-prime centers n ≤ x (both n−1 and n+1 prime) with n ≡ c (mod q), over the q−2 admissible classes, and let X² be the usual chi-squared statistic of the T_c about their mean, per degree of freedom. Numerically, X² ≈ 0.5–0.7 for 31 ≤ q ≤ 1009 and 10^8 ≤ x ≤ 10^10, well below the Poisson value 1 and below the value ≈ 0.93 obtained by thinning the primes at random to twin density, while the primes themselves give X² ≈ log q / log x → 0 (Hooley's conjecture). We show that the Hardy–Littlewood prime 4-tuple conjecture, applied in the manner of Montgomery and Soundararajan, predicts X² → 1/2 for every fixed q, with corrections of relative size 2 log q / log x. The mechanism is a double pole: the singular-series sum Σ_{h≤H} S({−1,1,h−1,h+1})/(2C_2)² = H − (1/(8C_2)) log² H + … carries a log² H correction where the prime pair correlation carries (1/2) log H, and the twin density 1/log² x removes exactly two logarithms. The exact second-order prediction, computed from the singular series for all h ≤ x, agrees with computation at x = 10^7, 10^8, 10^9 within sampling noise and decreases toward 1/2 at the predicted rate. The statement appears to be an unrecorded corollary of known work; the ingredients are in Montgomery–Soundararajan (2004) and Kuperberg (2025). We also record, as context, a sequence of numerical experiments on a primorial-wheel formulation of the twin prime problem.