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 * prodq>3 f_q(d, N), where f_q 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 f_q = 1 if d mod q is q-safe and 0 otherwise; when q does not divide N, f_q 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 <= 5 and <= 2.5% at the sparsest stratum omega = 6, for both 6dN = 42 and 210. The two tail constants are nearly equal (K = 0.1033 for 42, 0.1052 for 210), indicating K is gap-independent. No correction term is needed for the Part V cross-prime hedge: it is captured automatically by selecting the q|N or q-not-N branch of f_q per prime according to the centre's factorisation. Composed with the Part VI bridge constant, this gives an end-to-end closed form r(d|omega) = (K/C(d)) * <prod_q f_q>_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 f_q 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.