Cousin and Sexy Prime Pairs on the 6N Skeleton: Three omega-Distortions Set by Pairing Geometry

Part IX of the 6N twin-prime project. Parts I-VIII built a conditional theory of twin pairs (6N-1, 6N+1) in which the pair rate rises sharply with the factor count omega_{>3}(N) of the single shared centre. Here we ask whether the same omega-dependence carries over to cousin pairs (difference 4) and sexy pairs (difference 6). It does not carry over identically; it distorts into three distinct, geometry-determined shapes.

On the 6N skeleton a cousin pair is (6N+1, 6(N+1)-1) -- the alternative 6N+3 is always divisible by 3 -- and a sexy pair is either (6N-1, 6(N+1)-1) ("sexy-A") or (6N+1, 6(N+1)+1) ("sexy-B"). Unlike the twin, all of these straddle two consecutive centres N and N+1. We bin by the left centre's omega, the analogue of the twin attribution.

On the 1.5x10^9 centres of S10 (over 5x10^8 pairs), normalised to omega=1:
 - the twin rate rises monotonically (to 4.8x at omega=7);
 - the cousin and sexy-A rates fall monotonically (to 0.13-0.15x) and are numerically identical to three decimals at every omega;
 - the sexy-B rate is non-monotone, peaking near omega=4 before falling.

The cousin/sexy-A coincidence has a clean cause: the two share the right member 6(N+1)-1, so after normalisation the left member's own omega-response divides out, leaving the shared right member's response on the adjacent centre N+1 -- common to both, hence the curves coincide. The fall (rather than the twin's rise) reflects that the partner lives on N+1, not N: a factor-rich left centre does not protect a prime on the neighbouring centre. Sexy-B's non-monotonicity suggests two competing effects, an early enrichment-like gain and a later straddle suppression, not separated here.

The single-centre twin enrichment thus reverses or becomes non-monotone once the pair straddles two centres, with the sign and shape of the distortion fixed by which centres and which wings the pair occupies -- not by the difference alone (difference 4 and difference 6 can coincide, while two difference-6 forms differ). We report the phenomenon and its geometric account; a quantitative two-centre mechanism over (N, N+1), via the modular-shift/lockdown mechanism of Part V, is posed as the open problem.

No claim is made about the infinitude of twin, cousin, or sexy primes, or any prime k-tuple conjecture. This is a measured, geometry-resolved account of conditional pair rates on the 6N skeleton.