Non-Markovian Residue Kernels of Consecutive Primes

Let P⁽ᵏ⁾ₘ(x) be the empirical transition kernel from pₙ mod m to pₙ₊ₖ mod m. The earlier Prime-Symmetry Instrument treated Pₘ(x)ᵏ, the ordinary matrix power of the one-step kernel, as if it were P⁽ᵏ⁾ₘ(x). That identification is not valid: Pₘᵏ is the prediction obtained by artificially closing the one-step law as a Markov chain, while P⁽ᵏ⁾ₘ is measured directly from the prime sequence. We define the Markov defect Δ⁽ᵏ⁾ₘ(x) = P⁽ᵏ⁾ₘ(x) − Pₘ(x)ᵏ. [V] An independent sieve to x = 10⁸ reproduces every headline figure of the original paper to six decimal places: for m = 10 the actual lag-2 diagonal mean is 0.232754, the Markov square predicts 0.254523, and the Frobenius defect is 0.058700; the one-step diagonal suppression falls from 40.21% at 5×10⁵ to 31.27% at 10⁸, confirming it is not a fixed invariant. [D] This edition then resolves, numerically, three problems the original left open. (i) Defect spectrum: the symmetrised lag-2 defect is negative semidefinite for m = 4, 6, 10 and indefinite for m = 30. (ii) Character content: the mod-10 defect concentrates in the quartic-character cross-modes χ⊗χ̄, not the quadratic sector. (iii) Cross-modulus structure: the mod-30 joint law is CRT-consistent by construction but does not factor over its prime components — a genuine ≈13% cross-prime gap coupling remains. [C] Finally we flag a correction: the paper's leading conditional asymptotic uses a 1/((k−1)log x) scaling, whereas the established Lemke Oliver–Soundararajan bias for consecutive primes carries a log log x / log x factor. Tested against data to 10⁸, neither simple form gives a stable ratio, so the leading coefficient should be treated as heuristic, not data-confirmed. No claim is made about the Riemann Hypothesis.