A Bridge Theorem for the 6N Twin-Gap Distribution: Gap Preference as Normalised Right-Centre Survival, and the Resolution of the 42 Residual

Part VI of the 6N twin-prime project. Part V gave a measured two-centre shield law but left open the bridge from the right-centre conditional survival back to the omega-stratified gap preference r(d|omega) of Parts II-IV, and with it the fate of the high-omega rise of the gap 6dN=42 (observed 1.55 at omega=6). Here we close that bridge. Measuring, on the 23,988,173 twin centres of S10, both the gap preference r(d|omega) and the right-centre survival P(N+d twin | N twin, omega), we find their ratio is constant in omega: r(d|omega) = P(N+d twin | N twin, omega) / C(d), C(d) independent of omega. For 42 the constant is C=41.17 with a coefficient of variation of 0.49% across omega=1..6; for 210, C=27.68 with CV 1.37% (a single omega=6 point, where both quantities are smallest and noise-sensitive, accounts for most of the spread, which we report rather than smooth). The gap preference is therefore nothing but the right-centre survival rescaled by an omega-independent constant. This resolves the 42 residual that ran through Parts II-V: its rise to 1.55 is exactly the rise of the right-centre survival from 0.018 to 0.038, rescaled by the constant, and that rise is in turn the cumulative effect of the Part V two-centre shield (factor-rich centres carry more small primes, hence more modular-shift protection of the right centre; for 42 dominated by q=5, since N+7 = N+2 mod 5 lands the right centre on a safe residue when 5|N). No separate mechanism is needed. The six-part chain closes end to end: two-centre shield (V) implies P(N+d twin|omega), which divided by C(d) (VI) gives r(d|omega). One narrower problem remains open: a closed-form combination of the single-prime shields into the absolute survival P(N+d twin|omega), since a naive independent product overstates it (the Part V cross-prime hedge is non-multiplicative at the full-survival level). Also open is whether C(d) has an arithmetic expression in d, and whether the 1.37% spread on 210 is a finite-size effect or a genuine breakdown at the sparsest stratum. No claim is made about the infinitude of twin primes or any prime k-tuple conjecture. This is a measured, factor-resolved account of the conditional gap structure of the 6N twin skeleton.