The Second Moment of omega under Twin-Centre Conditioning on the 6N Skeleton

Closes a question the programme had left open since Volume I: the second moment (Erdos-Kac variance) of omega>3(N) -- the number of distinct prime factors >3 of a centre N -- and its response to conditioning on N being a twin centre (6N+-1 both prime). Sieving all centres up to 6N=6e8 (2,166,300 twin centres), the distribution of omega>3 is measured over all centres and over twin centres at six cutoffs. Findings: (i) twin-conditioning shifts the mean by a SATURATING, small-prime-dominated amount (+0.171 to +0.215 as 6N runs 1.5e6 to 6e8, decelerating toward a finite limit ~0.21-0.22; it does NOT track lnln); (ii) it inflates the variance by a stable factor 1.104 +- 0.005; (iii) the dispersion ratio Var/mean is CONSERVED -- its twin-to-all ratio is 1.011 +- 0.004 across two and a half orders of magnitude, not a small-data fluctuation. Interpretation: writing omega as a sum of near-independent Bernoulli indicators 1[q|N], the twin weight prodq|N q/(q-2) nudges each inclusion probability up by ~2/(q(q-2)), a convergent small-prime-dominated series; this shifts the mean to a finite limit and rescales mean and variance nearly in step -- a NEAR-POISSON TILT under which the Erdos-Kac dispersion survives. SCOPE: new to the programme, not new mathematics; it confirms (does not supersede) classical Erdos-Kac and claims no infinitude. A clarification ties off a separate thread: the annihilation/tiling criterion of Part XXI equals the classical Hardy-Littlewood admissibility condition (inadmissible iff the tuple covers Z/p for some prime p), connecting to Erdos covering systems.