Conditional Distributions of Linear Prime Patterns on the 6N Skeleton: A Factor-Resolved Survey of Parts I-XIV

This survey consolidates fourteen computational studies (Parts I-XIV) of how the small-prime factor structure of an integer conditions the local density of linear prime patterns on the residue skeleton 6N +- 1. The unifying object is the count omega>3(N) of distinct prime factors greater than 3, treated as a stratifying variable. Across the prime difference (twins, cousins, sexy pairs, and the general Polignac gap) and the prime sum (Goldbach), the conditional counts are organised, to within a fraction of a percent at the 10^7-10^8 scale, by the classical Hardy-Littlewood singular series multiplied by closed-form Chinese-Remainder factors. The entire conditional twin-gap distribution reduces to a single scale, the twin-centre density rho, with no fitted parameters: r(d | omega) = S(d) * rho * P(N+d twin | omega), P = rho * tail_S(d) * B * prod_POOL f_q(d,N). The survey states this synthesis and its components; the two-centre CRT mechanism reproducing the cousin/sexy omega-distortions (to 2.3%); the Polignac universal collapse, in which every even gap divided by its singular series yields one shared constant (CV 0.068% over 500 gaps at X=10^8); the Goldbach comet collapse (pointwise CV 0.22%); and the resulting sum-difference symmetry, prod(q-1)/(q-2) read on the two linear constraints p1-p2=d and p1+p2=2N. Provenance is stated explicitly: the singular series S, S_G, tail_S and the twin constant Pi_2 are classical Hardy-Littlewood; rho is a known density. The contribution is the omega-stratified viewpoint, the closed-form two-centre CRT mechanism, the reduction of all constants to the single scale rho, and the large-sample verification of all of the above. A dedicated section records the limits of the programme as an integral part of it: the open problem of the Erdos-Kac limit law for omega on twin centres (the accessible scale lnln(6N) ~ 3 has not reached a Gaussian, and no closed form for the variance is claimed); the analytic trap of the divergent naive variance sum 1/(q-2)(1-1/(q-2)); and the null result that the Goldbach comet carries no post-main-term residual structure beyond Hardy-Littlewood (the residual's dependence on q|N is unity to <0.001%). No claim is made about the infinitude of twin, cousin, sexy, or Polignac prime pairs, or about the Goldbach conjecture.