The First-Principles Origin of the Tail Constant K on the 6N Skeleton: K = rho * S-tail(d) * B and a Single Scale

Part XI of the 6N twin-prime project. The closed-form right-centre survival of Part VII, P(N+d twin | omega) = K * prod_{q in POOL} f_q(d,N) with POOL = {5,...,47}, carried one empirical constant K ~ 0.105, absorbing the primes q>47 and the relative-to-absolute normalisation. We give its first-principles origin.

The constant factors as

    K(d) = rho * tail_S(d) * B,

where rho is the twin-centre line density (Part VIII), tail_S(d) = S(d)/S_POOL(d) is the part of the Hardy-Littlewood singular series S(d) carried by primes q>47, and B = prod_{q in POOL} S_q(d)/<f_q> is a pure Chinese-Remainder basis-conversion constant, predicted analytically as B ~ 6.535 and d-independent. The derivation equates the Part VIII merged identity P_merged(d) = rho*S(d) with the omega-merged Part VII form P_merged(d) = K*<prod_POOL f_q>, giving K = rho*S(d)/<prod_POOL f_q> = rho*tail_S(d)*B.

Measured across d on S9 and S10, the ratio K/(rho*tail_S) is d-independent to a coefficient of variation of 0.31% (S9) and 0.16% (S10), and equals the predicted B to within 0.4%. K is therefore not a free parameter.

The decisive evidence that the only scale is rho: from S9 to S10 the density drops from 0.0199 to 0.0160 (factor 0.80), and K drops by the same factor (0.128 to 0.103) while K/(rho*tail) stays at 6.55. The shell dependence of K is entirely that of rho; tail_S(d) and B are shell-independent arithmetic.

One scale for the whole theory: Part VIII gave the bridge constant C0 = 1/rho, and this paper gives K = rho*tail_S(d)*B, so both constants of the series reduce to the single density rho. The assembled conditional gap preference r(d|omega) = S(d)*rho*P(N+d twin|omega), with P = rho*tail_S(d)*B*prod_POOL f_q, contains no fitted constant: S and tail_S are the Hardy-Littlewood singular series, f_q and B are closed-form CRT, and rho is the twin-centre density (given, to leading order, by the Hardy-Littlewood twin constant and the shell's mean 1/ln^2). The single empirical input is rho, itself a known density rather than a constant of this construction.

The 0.2-0.4% excess of the measured ratio over B is the uniform-residue approximation in the q-not-N branch (Part VII), the same few-tenths-of-a-percent effect seen throughout. Results are for d tested 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, closed-form, factor-resolved account of the conditional gap structure of the 6N twin skeleton.