Fine Structure of the Wing-Transition Memory on the 6N Skeleton: the Lag-Two Echo and the 1/log Decay

Part XXXIV established that the prime wing string — right (6N+1) versus left (6N−1) — is not a first-order Markov chain but carries an effective second-order memory, the switch sequence having a lag-one autocorrelation of −0.033. Two fine questions were left open, and this note settles them with the full prime census to 10⁹ (50,847,531 steps). First: is the small lag-two echo a mere shadow of the lag-one term, or independent? For a first-order (AR(1)) switch process the lag-two autocorrelation would be exactly ρ₁² = +0.00098; we measure ρ₂ = +0.00417, 4.3 times larger, a partial autocorrelation PACF(2) = +0.00319 at 23σ, with ρ₃ and beyond vanishing. The switch process therefore has memory length exactly two — a negative nearest-neighbour term and a positive lag-two echo about a tenth its size, then nothing. (Picturesquely, an overcorrection echo following the strong nearest-neighbour rebound; we offer the image only as a mnemonic for the measured signs, not as a mechanism.) Second: how fast does the bias fade? Across thirteen quarter-decade windows the first-order deficit δ₁ = ½ − P(same) falls from 0.074 to 0.053 and obeys a leading 1/log N law (δ₁·log N ≈ 1.05, nearly flat; a single power fit gives an effective exponent 0.85, the shortfall a slowly-varying correction the range cannot resolve). The second-order signal |ρ₁| fades in lock-step: the ratio δ₁/|ρ₁| holds near 1.7 (drifting only from 1.6 to 1.8) across five decades, so the first- and second-order biases are controlled by a single fading amplitude rather than decaying independently. None of this is new as theory. The structure itself is contained in the Hardy–Littlewood r-tuple heuristic, and the vanishing of the bias with scale is the Oliver–Soundararajan prediction; we prove no theorem and propose no mechanism. What this note adds are the specific measured numbers — the size and sign of the lag-two partial autocorrelation, the exact-two memory length, the leading 1/log N decay exponent, and the locked first-to-second-order ratio. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXV of "Arithmetic Geodynamics on the 6N Skeleton," a fine-structure refinement of Part XXXIV. Code and measured data: https://github.com/Ruqing1963/6N-wing-finestructure