The Arithmetic Structure of the Bridge Constant on the 6N Skeleton: C(d) = C0/S(d) and C0 as the Inverse Twin Density

Part VIII of the 6N twin-prime project. Part VI introduced the bridge constant C(d) = r(d|omega) / P(N+d twin|omega), an omega-independent number relating the conditional gap preference to the right-centre survival, and left its dependence on the separation d unexplained. We determine it.

Measuring C(d) for d = 1..30 on the 23,988,173 twin centres of S10 and comparing to the Hardy-Littlewood admissibility S(d) of the separation (the standard singular-series factor for a twin pair at centre-step d), we find

    C(d) = C0 / S(d),   C0 ~ 62.75,

with the product C(d)*S(d) constant to a coefficient of variation of 0.41% across all thirty separations, while the raw C(d) ranges over a factor of four (35 to 158). The constant C0 is stable under the prime cutoff of S(d) and is gap-independent.

We then identify C0 itself. At the omega-merged level the definitions give P_merged(d) = rho * S(d), the standard two-point correlation form (density times admissibility), holding to CV 0.39%. Hence

    C0 = 1/rho,

where rho is the twin-centre line density. This is verified across shells: 1/rho = 50.3 against fitted C0 = 50.1 on S9, and 1/rho = 62.5 against 62.8 on S10, each to 0.4%; the growth of C0 as the shell deepens is exactly the thinning of rho. Since rho is itself the twin-centre density (given to leading order by the Hardy-Littlewood twin constant and the shell's mean 1/ln^2), no undetermined constant remains.

Substituting into the Part VI bridge gives the fully resolved conditional gap preference

    r(d|omega) = S(d) * rho * P(N+d twin|omega),

with the separation dependence carried by the classical admissibility S(d), the factor-count dependence by the closed-form right-centre survival P of Part VII (K * prod_q f_q), and the two meeting through the twin density rho. The bridge constant of Part VI is thus the density factor of the standard two-point correlation, and the conditional 6N theory shares its d-dependence with the classical singular series.

Results are for d = 1..30 on shells S9 and S10. 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.