Topological Strain Memory on the 6N Skeleton: the Effective Second-Order Markov Dynamics of Prime Wing Transitions

Every prime > 3 sits on a wing of the 6N lattice — right (6N+1) or left (6N−1) — so the prime sequence is a binary string of wing symbols. Lemke Oliver and Soundararajan showed that consecutive primes avoid repeating their residue class; here we ask a sharper question of the wing string: how deep is its memory? Using all 26,355,865 primes 3 < p ≤ 5×10⁸, we test the Markov order of the chain against three targets: the zeroth (Cramér fair coin), the first (a switch bias confined to the current step), and a genuinely higher-order chain. It is decisively not first-order: the probability of staying on a wing depends on the previous two primes, not one (P(R|RR) = 0.4252 versus P(R|LR) = 0.4580, a gap of 0.033 at 168σ). The cleanest probe is the switch (step) sequence, which for a symmetric first-order chain would be independent noise: instead it carries an autocorrelation of −0.0328 at lag one (a switch is followed by a switch less often than chance — an anti-persistence we name a topological strain memory), a weak +0.0046 echo at lag two, and nothing measurable beyond. The incremental memory contributed by each further prime halves per order (0.033, 0.016, 0.009, …), so the chain is effectively — not exactly — second-order: a dominant nearest-neighbour memory with a geometrically decaying tail, not an abrupt cutoff. We then show the wing memory and the well-known anti-correlation of consecutive prime gaps are the same effect: the switch indicator is identically the event "gap not divisible by 6," and the step lag-one autocorrelation (−0.0328) equals the normalised-gap lag-one correlation (−0.0332). All signals fade slowly with scale (P(same) rises from 0.404 below 10⁵ to 0.446 near 5×10⁸), as the Hardy–Littlewood framework requires. None of this is new as theory — the anti-repetition bias is Lemke Oliver and Soundararajan's, and the heuristic that predicts every consecutive-pattern frequency (hence all the structure measured here) is Hardy and Littlewood's. We prove no theorem and propose no mechanism. What this paper adds is the explicit memory-depth characterisation of the wing chain — its order, the sign and magnitude of each term, and the geometric decay — together with the identification of that memory with the consecutive-gap anti-correlation, in the 6N geometric language, measured against the full prime census to 5×10⁸. The words "strain" and "anti-clustering" name measured correlations, not a physical force. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXIV of "Arithmetic Geodynamics on the 6N Skeleton." Code and measured data: https://github.com/Ruqing1963/6N-wing-transitions