The Semiprime Neighbour on the 6N Skeleton: A Slower omega-Enrichment and Its Mechanism

Part XVI of the 6N project. Part XII established that twin centres enrich with the factor count omega_{>3}(N) through the single-prime law P(q | N | twin) = 1/(q-2). Here we relax the right member from prime to prime-or-semiprime: we count centres with 6N-1 prime and 6N+1 either prime or a product of exactly two primes (Omega(6N+1) in {1,2}, where Omega counts prime factors with multiplicity).

This neighbour count also enriches with omega_{>3}(N), but systematically MORE SLOWLY than the twin. On S9 (3.5x10^6 centres with 6N-1 prime at omega=1), the twin rate rises to 1.81x its omega=1 value at omega=5, while the exactly-semiprime part rises only to 1.37x. The enrichment ratio (semiprime-only over twin) falls monotonically from 1.00 at omega=1 to 0.76 at omega=5, so the semiprime follows a genuinely flatter omega-law, not a scaling of the twin enrichment.

MECHANISM. The twin enrichment of Part XII is a small-prime avoidance effect: a factor-rich N forces, through the dead residues dead(q) = {+-6^-1 mod q}, that 6N+-1 avoid divisibility by the small primes q | N. For the right wing to be PRIME this avoidance is all-or-nothing and maximally helpful. For it to be SEMIPRIME it is less helpful: a semiprime may carry one small prime factor, so suppressing small factors removes some of the very semiprimes being counted. The enrichment is therefore weaker.

This predicts a direct signature: as omega grows, the surviving semiprimes 6N+1 should increasingly be products of two LARGE primes rather than small x large. We confirm it on the same shell (S9): among semiprime 6N+1 with 6N-1 prime, the fraction having a prime factor <=97 falls monotonically from 0.56 at omega=1 to 0.41 at omega=5. Factor-rich N pushes its semiprime neighbour toward the large x large regime, exactly as the mechanism requires.

ATTRIBUTION. The single-prime and semiprime local densities are classical (Hardy-Littlewood for the prime wing; Landau's semiprime density for the relaxed wing). This paper introduces NO new density formula. The contribution is the measured fact that the semiprime neighbour enriches more slowly than the twin, and the factor-structure mechanism for it, confirmed by the falling small-factor fraction.

OPEN: the exact double-factor closed form. The twin enrichment reduces to the clean single-prime law 1/(q-2). The semiprime condition Omega(6N+1)=2 couples TWO prime factors, and the omega-dependence does not separate into a single per-prime factor; a faithful model would be a double-factor convolution that does not reduce to a clean product at the percent level. We report the enrichment and its mechanism empirically and leave the exact closed form open, rather than force an approximate product.

SCOPE. No claim is made about the infinitude of twins, semiprime neighbours, or any pattern. This is a measured, factor-resolved account of a conditional rate.