Spectral Isotropy and the Exact Temperature of the Prime Gas

We prove that the forward distance matrix D on the admissible residues (Z/mZ) * satisfies D + DT = m (J + I) for every modulus m. This algebraic identity forces the Boltzmann prime transition operator to be spectrally isotropic: all oscillatory modes decay at identical rate, and the limiting eigenvalue angles are equally spaced at π/φ (m) intervals. The result holds for all moduli, not only primorials. The thermodynamic temperature of the prime gas is T = N/π (N) (the mean prime gap), not the asymptotic approximation T = ln (N). This zero-parameter correction improves R² from 0. 966 to 0. 970 (mod 30) and from 0. 980 to 0. 986 (mod 210) at N = 10⁹, verified across 50, 847, 534 primes with monotonic improvement at every checkpoint. Additional results: (1) every consecutive prime pair with p > 2 is automatically a Goldbach pair, so the Boltzmann model governs consecutive-prime Goldbach representations at r = 0. 986; (2) the Boltzmann transition weight and the Hardy–Littlewood singular series are orthogonal predictors — the Boltzmann weight breaks degeneracies within tiers of equal singular series values at r = 0. 93, a prediction no finite extension of the singular series can make; (3) the prime eigenphase spectrum is Poisson-distributed (integrable), not Wigner–Dyson (chaotic) — the spectral gap tends to 1, giving O (1) mixing time. Companion paper: "The Prime Column Transition Matrix Is a Boltzmann Distribution at Temperature ln (N), " Matos (2026), DOI: 10. 5281/zenodo. 19076680.