Prime Gaps on the 6N Skeleton: An ω-Dependent Shift in the Twin-Prime Gap Distribution, and a Congruence-Lockdown Mechanism
Year: 2026
Type: preprint
Abstract: Part II of the 6N twin-prime project. In Part I we showed that the conditional density of twin primes on the 6N±1 skeleton depends on the factorisation of the centre N through the local enrichment factor E(N)=∏_{q|N,q>3}(q−1)/(q−3). Here we ask the complementary question: not how many twin centres survive, but how they are spaced. Using the exact set of 23,988,173 twin centres in the decade shell S10 (where 6N ∈ [10^9,10^10)), and the corresponding sets in S8 and S9, we study the distribution of the gap ΔN between consecutive twin centres, stratified by ω_{>3}(N), the number of distinct prime factors >3 of the left centre. The aggregate gap distribution is the known Hardy–Littlewood shape, peaking at the highly divisible values 6ΔN = 30, 42, 210. The new finding is that the shape shifts systematically with ω_{>3}: the relative preference for the short gap 6ΔN=42 rises monotonically with ω (from 0.73 to 1.55 across ω=1..6 in S10) while that for 6ΔN=210 falls (from 1.15 to 0.41); the trend reproduces across S8, S9, S10 over a ~64× range of sample size. This contradicts the unconditional Hardy–Littlewood prediction, in which gap preference depends only on the gap and not on the centre. We identify a first-order mechanism — congruence lockdown: when a prime q divides the centre N, the right centre N+d can host a twin only if d ≢ ±6^{-1} (mod q), equivalently q ∤ (6d±1). Each prime factor of the centre forbids two gap residues. Because 6·35∓1 = 209,211 with 209 = 11×19, 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. This reproduces the rise of 42 and 30 and the collapse of the absolute survival of 210, but a residual at high ω remains. We make no claim about the infinitude of twin primes. The closed-form conditional gap singular series 𝔖_ω(ΔN), together with the high-ω residual, is posed as an open problem. The aggregate gap shape is the known jumping-champion effect (Odlyzko–Rubinstein–Wolf 1999; Goldston–Ledoan 2011); empirical twin-gap modelling is due to Kelly–Pilling (2001); almost-prime gap bounds to Goldston–Graham–Pintz–Yıldırım and Sono. The contribution here is the conditional, ω-resolved layer and the congruence-lockdown mechanism. All data and verification scripts are openly released.
Author: Chen, Ruqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162933428
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