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 ~ 10^5) 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) = sumq>3 1{q|N}, the mean shift between twin centres and ordinary integers has the convergent closed form <omega>_twin - <omega>_ord = sumq>3 (1/(q-2) - 1/q) = 0.2604, cutoff-independent (each term ~ 2/q^2), 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.