Part III of the 6N twin-prime project. Part II reported that the twin-prime gap distribution on the 6N±1 skeleton depends on the factor count ω>₃ (N) of the centre — the preference for the gap 6ΔN=42 rises with ω while that for 210 falls — and left the closed-form conditional gap singular series as an open problem. Here we give its explicit first-order form. Conditioning the unconditional Hardy–Littlewood correlation factor C2 (d) on the centre's factor set via the congruence-lockdown rule of Part II (q | N forbids the gap residues d ≡ ±6^-1 mod q, equivalently q | (6d±1) ), we obtain S₂, ⏨ (d) ≈ C2 (d) · L_ω (d), where L_ω (d) is the lockdown-allowance of the gap d in the stratum. The admissibility of a gap is governed by the factorisation of 6d±1: 6·35∓1 = 209 = 11×19, so the long gap 210 is locked by the common small primes 11 and 19; whereas 6·7∓1 = 41, 43 are both prime, so the short gap 42 is locked only by the rare primes 41 and 43. Tested against the 23, 988, 173 twin centres of S10 (where 6N ∈ [10⁹, 10¹0) ), the first-order series reproduces the ω-trends of the gaps 30, 42, 60 (residuals ≤ 0. 13) and the direction of the collapse of 210. A systematic residual remains at high ω on the gap 210: the observed preference falls to 0. 41 where the first-order series predicts 0. 62 (residual ≈ −0. 2 at ω = 5, 6, stable across S9 and S10). We attribute it to the second-order correlation between the factorisations of the two centres N and N+d, which the first-order (per-prime independent) series omits, and pose the two-centre conditional singular series as the next open problem. The series reduces to the unconditional Hardy–Littlewood series in the mean. No claim is made about the infinitude of twin primes or about any prime k-tuple conjecture. This advances the Part II open problem from fully open to first-order-solved with a quantified second-order remainder.
Ruqing Chen (2026) studied this question.