A Second-Order Conditional Singular Series for Twin-Prime Gaps on the 6N Skeleton: the Spatial-Compression Penalty and a Residual Shield-Gain Term

Part IV of the 6N twin-prime project. Part III gave a first-order conditional singular series S_{2,ω}(d) ≈ C2(d)·L_ω(d) for the twin-prime gap distribution on the 6N±1 skeleton, reproducing the ω-trends of the gaps 30, 42, 60 and the direction of the collapse of 210, but leaving a stable high-ω residual on 210 (observed 0.41 where the first-order series predicts 0.62 at ω=6). Here we identify the second-order term responsible.

The first-order series treats the primes q ∤ N by their unconditional Hardy–Littlewood factor; but conditioning on q ∤ N is itself a constraint — it removes the residue N ≡ 0 and crowds N into the remaining q−1 classes, where the same fatal residues now occupy a larger fraction. This spatial-compression penalty, pen_q(d) = [(q−1−k_q)/(q−1)] / [(q−k_q)/q] < 1 (for k_q = 4 this is (q²−5q)/(q²−5q+4)), lowers the survival of factor-rich centres on gaps locked by small primes.

Evaluated on the 23,988,173 twin centres of S10 with each centre's real factor set, the second-order series closes the 210 residual essentially completely (at ω=6, observed 0.413, second order 0.426, against first order 0.357), across all strata. It also corrects the direction on 42 but only partially: about half of the high-ω rise of 42 remains unexplained.

We are explicit about the limitation: the second-order series is by construction a single-centre condition, conditioning on N while treating N+d as independent. The two centres are in fact correlated modulo each prime. The 210 residual closes because its lock primes 11,19 rarely divide both centres; the 42 residual remains because the gap's factor 7 shields both centres when 7|N — a two-centre, factor-coincidence shield gain that the single-centre series omits. This shield-gain mechanism is at present only a qualitative hypothesis, not a verified closed form. The two-centre conditional singular series is posed as the open problem.

No claim is made about the infinitude of twin primes or about any prime k-tuple conjecture. This is a conditional, factor-resolved refinement of the Hardy–Littlewood gap heuristic, demonstrated empirically and offered with its open remainder.