The Factor Statistics of Twin Centres on the 6N Skeleton: Each Prime Divides with Probability 1/(q-2), not 1/q

Part XII of the 6N twin-prime project. Throughout Parts I-XI the factor count omega>₃ (N) was used as a given stratum. Here we ask what distribution it follows for the sparse set of twin centres themselves. The answer is a single clean replacement: a prime q>3 divides a twin centre with probability 1/ (q-2), against 1/q for an ordinary integer. A twin centre N is constrained: 6N-1 and 6N+1 must both be prime, so modulo q>3 the residue of N must avoid the two forbidden classes dead (q) = +-6^{-1 mod q}, leaving q-2 admissible residues. The residue 0 (i. e. q|N) is one of these, and no admissible class is locally favoured, giving P (q|N | N twin) = 1/ (q-2). We verify this per prime on the 23, 988, 173 twin centres of S10 (and S9): for small q the measured probability equals 1/ (q-2) to within 0. 4% (0. 1% for the smallest, most frequent q), tightening from S9 to S10. Large q are sampling-limited (P ~ 1/q gives only tens of divisible centres at q ~ 10⁵) and are not a deviation from the law. This is the microscopic root of the Part I enrichment factor prod (q-1) / (q-3): the boosted probability gives a relative density factor P (q|N, twin) /P (q-not-N, twin) / (1/q) / (1-1/q) = (q-1) / (q-3), so the macroscopic enrichment is exactly the product of the single-prime probability boosts 1/q -> 1/ (q-2). Since omega>₃ (N) = sumₐ>₃ 1q|N, the mean shift between twin centres and ordinary integers has the convergent closed form ₜwin - ₒrd = sumₐ>₃ (1/ (q-2) - 1/q) = 0. 2604, cutoff-independent (each term ~ 2/q²), unlike the individual means which diverge as lnln. The measured finite-N shift grows with the shell (+0. 2220 on S9, +0. 2270 on S10), approaching the closed form as the large-q truncation recedes. We do NOT claim a normal law. At the accessible scale lnln (6N) ~ 3 the distribution has not reached its Erdos-Kac limit: the mean (2. 70) and variance (1. 04) are far apart, and the skewness (+0. 26) and excess kurtosis (-0. 29) are nonzero, for ordinary integers as much as for twins. Whether omega>₃ over twin centres tends to a normal law with Bernoulli parameter 1/ (q-2), and what governs its constrained variance, is left as an open problem. No claim is made about the infinitude of twin primes or any prime k-tuple conjecture. This is a measured, factor-resolved account of the arithmetic of the twin skeleton.