Twin Counts in Short Intervals: Singular-Series Rigidity and the Density-Gradient Variance

We measure the variance of the number of twin centres in short intervals on the 6N skeleton — the Montgomery–Soundararajan question for the twin constellation — sieving all centres to 6N ≤ 6×10^8 (2,166,300 twin centres). Partitioning the centre axis into disjoint windows of length H, the Fano factor F(H) = Var/mean traces a clean crossover: sub-Poisson for small windows (dipping to F ≈ 0.92 near H ≈ 50), crossing F = 1 at H* ≈ 410, then rising steeply super-Poisson (F ≈ 37 at H = 10^5). Two cleanly separable mechanisms explain the whole curve. (i) The local pair correlation C(k) of twin centres at separation k is governed exactly by the Hardy–Littlewood singular series of the 4-tuple of offsets {−1, 1, 6k−1, 6k+1}; we derive the closed form C(k) = G · Π_{p|k} (p−2)/(p−4) · Π_{p|3k±1} (p−3)/(p−4) with generic constant G = Π_{p>3}(1 − 4/(p−2)^2) = 0.39688, and verify it against the data to a median of 1.7% across k ≤ 3000, reproducing every arithmetic spike. The integrated local correlation is negative — the four-fold killer constraint makes twin centres locally more regular than Poisson — and this rigidity is exactly the sub-Poisson dip. (ii) The super-Poisson rise is not local: it is the smooth deterministic decline of twin density across [1, X]. Density-detrending splits the variance into a local part F_local(H), which stays sub-Poisson and tracks the singular-series prediction to ≈1%, and a trend part F_trend(H) ∝ H (Var_trend ∝ H^2), which reaches 92% of the total at H = 10^5. The crossover H* marks where the global density gradient overtakes the local arithmetic rigidity. We state plainly what we do not resolve: the genuine Montgomery–Soundararajan lower-order term — the H·log(X/H) fluctuation at fixed density and the asymptotic Gaussianity — sits below resolution at reachable H and X, dominated by rigidity and by the deterministic trend; we bound where it must live and leave it open. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input, and no claim is made about the infinitude of twins. Part XXIX of "Arithmetic Geodynamics on the 6N Skeleton." Code and measured data: https://github.com/Ruqing1963/6N-shortinterval-variance