The Arithmetic Black Hole: Softmax Thermodynamics and the Four Eigenvalue Laws of the Prime Gas

We present a unified physical model of prime number statistics as a thermalized gas on the coprime residue lattice. The central observation is an exact operator identity: the Boltzmann transition matrix governing consecutive prime residue classes is algebraically identical to the softmax attention mechanism in Transformer neural networks, with temperature T = N/pi(N) supplied by the Prime Number Theorem. The model achieves R-squared = 0.970 against empirical prime transition data at N = 10^9 with one discrete structural choice and zero continuous free parameters.

The 3.0% residual is not noise: it is the Lemke Oliver-Soundararajan diagonal suppression, and its scaled trace converges to -ln(pi) within 0.05%. Dense verification resolves the residual into a damped complex spiral governed by four closed-form eigenvalue laws depending on exactly two integers at each primorial level.

The phase law generalizes into the Hyper-Radix Tower: at any primorial m, each odd prime p dividing m contributes a distinct eigenvalue mode whose phase advances at rate 1/(p-1) per log_p T. This is verified at m = 210 for three CRT fiber modes (p = 3 at 1.6% error, p = 5 at 0.1% error, p = 7 at 1.3% error). The tower is holographic: the 2x2 boundary matrix at m_0 = 6 faithfully encodes the CRT projection of all higher-floor modes.

An eleven-module simulator (740 invariant bounds, all passing) extends the model to spectral statistics, entanglement entropy, and holographic scrambling. The coprime boundary entanglement saturates at half the Page limit, exhibiting an arithmetic Hawking-Page transition that becomes discontinuous in the thermodynamic limit. Holographic duality is topological: replacing von Mangoldt weights with binary coupling preserves all scrambling signatures (52/52 verdict match).

Companion papers: "Spectral Isotropy and the Exact Temperature of the Prime Gas" (DOI: 10.5281/zenodo.19156532), "The Prime Column Transition Matrix Is a Boltzmann Distribution" (DOI: 10.5281/zenodo.19076680), "Active Transport on the Prime Gas" (DOI: 10.5281/zenodo.19243258).