Part IV of the 6N twin-prime project. Part III gave a first-order conditional singular series S₂, ⏨ (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, penq (d) = (q−1−kq) / (q−1) / (q−kq) /q < 1 (for kq = 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.
Ruqing Chen (Tue,) studied this question.