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We prove that the forward distance matrix D on the admissible residues (Z/mZ)* satisfies D + D^T = 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 R^2 from 0.966 to 0.970 (mod 30) and from 0.980 to 0.986 (mod 210) at N = 10^9, 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.