The Prime Column Transition Matrix Is a Boltzmann Distribution at Temperature lnN

We show that the transition matrix governing consecutive primes between residue classes modulo m is predicted by a Boltzmann distribution on the forward cyclic distances of the admissible residue group (Z/mZ)∗, with temperature equal to the mean prime gap lnN. The model has zero free parameters. At modulus 30, it achieves R2 ≈ 0.970 against empirical matrices measured across six orders of magnitude (103 to 109), with the fitted decay rate converging to within 1.1% of the PNT prediction λ = 1/ lnN at the largest measured scale. At modulus 210 (48 columns), R2 = 0.988 at the largest scale, improving monotonically with N. The diagonal suppression discovered by Lemke Oliver and Soundararajan (2016) follows as a one-line corollary: self-transitions cost energy m (a full modular cycle), making them the least probable transition at any finite temperature. The 3% residual is structured: it decomposes into a circulant component scaling as O(1/ lnN) and a non-circulant component scaling as O(1/ ln1.6 N). The Hardy-Littlewood singular series does not appear at any tested order (§5). The Boltzmann framing was first proposed by an AI worker (Gemini HELICASE) during aconstrained swarm convergence run, then confirmed through a 22-wave adversarial falsification protocol (Appendix B).