A Closed Form for the Conditional Twin-Gap Distribution on the 6N Skeleton: CRT Survival Factors and an omega-Independent Tail Constant

Part VII of the 6N twin-prime project. Part VI reduced the conditional twin-gap preference r (d|omega) to the right-centre survival P (N+d twin|omega) through an omega-independent bridge constant C (d), and left one problem: a closed form for the absolute survival in terms of per-prime quantities, a naive independent product having overstated it. We solve it. The absolute survival factors as P (N+d twin | N twin, omega) = K * prodₐ>₃ fq (d, N), where fq is the pure Chinese-Remainder conditional safety of the right centre at q: with dead (q) = +-6^{-1 mod q}, when q|N the left centre sits at 0 and the right centre at d, so fq = 1 if d mod q is q-safe and 0 otherwise; when q does not divide N, fq is the fraction of admissible nonzero residues r for which r+d is q-safe. K is an omega-independent tail constant from the primes beyond the working set 5. . 47, fixed once from the omega-merged ratio. There are no fitted parameters. On the 23, 988, 173 twin centres of S10 the form matches the measured survival to within 0. 5% for omega ₒmega. The rise of 42 to 1. 55 and the collapse of 210 to 0. 41 are therefore computed quantities from the centre's factorisation rather than described ones. The only departures above 0. 5% are at omega = 6, where the uniform-residue approximation in the q-not-N branch of fq is least accurate: real twin-centre residues carry the Part I enrichment bias rather than exact uniformity. We identify the omega = 6 residual as this enrichment correction, report it as the precision boundary of the present closed form, and do not absorb it into a fitted term; quantifying it is the natural refinement. No claim is made about the infinitude of twin primes or any prime k-tuple conjecture. This is a measured, closed-form, factor-resolved account of the conditional gap structure of the 6N twin skeleton.