Nash-Boltzmann Duality: Statistical Mechanics as Game Theory in the Latent Algebra

We establish an exact duality between strategic games and statistical mechanical systems through the Latent algebra's grade decomposition. The Nash-Boltzmann map identifies the Hamiltonian \(H(\sigma)\) of a physical system with the negative payoff \(-u(s)\) of a game, the inverse temperature \(\beta\) with the rationality parameter, and the Boltzmann distribution \(\pi(\sigma) \propto e^{-\beta H(\sigma)}\) with the logit quantal response equilibrium. Under this identification, three fundamental correspondences emerge: (i) the cluster expansion of statistical mechanics is the ANOVA decomposition of the payoff function — both decompose a multivariate function by interaction order; (ii) the Kadanoff-Wilson renormalization group is Latent grade truncation — both coarse-grain by eliminating high-order interactions; (iii) critical phenomena (\(\rho = 1\) in the Latent, \(T = T_c\) in physics) mark the same structural transition — the point where all interaction grades contribute equally and no finite truncation suffices. We prove that the Latent Number \(\rho\) of a physical system equals \(e^{J/k_BT}\) for nearest-neighbor Ising models (where \(J\) is the coupling strength), providing a closed-form complexity measure for spin systems. We extend the duality to quantum systems, where the grade decomposition corresponds to entanglement order and \(\rho\) measures the decay rate of connected correlation functions. We demonstrate the duality on three systems: the 2D Ising model (exact solution via Onsager maps to a grade-2 congestion game), the Lennard-Jones fluid (grade-2 dominant with grade-3 corrections from three-body forces), and the Hubbard model of strongly correlated electrons (quantum game with on-site + nearest-neighbor interactions).