The Maslov-Gibbs Einsum (MGE): Tropical Crystallization and the Thermodynamic Bridge Between Continuous Optimization and Discrete Logic

This paper introduces the Maslov-Gibbs Einsum (MGE), a thermodynamic tensor contraction that packages Maslov dequantization as a computational primitive bridging continuous differentiable optimization and discrete combinatorial logic. The central observation relies on standard mathematics: as inverse-temperature $\beta \to \infty$, the smooth log-sum-exp partition function converges to the tropical maximum (Maslov dequantization; Litvinov 2007, Viro 2001). The primary contribution of this paper is the formal packaging of this limit theorem as a high-dimensional einsum tensor contraction, and the demonstration that the resulting operator unifies—as exact limiting cases of a single smooth function—twelve foundational constructs spanning tropical geometry, maximum entropy, optimal transport, differentiable dynamic programming, extreme value statistics, control theory, and machine learning attention (the MGE Dodecagon). Three geometric invariances of the MGE are established: conformal invariance under uniform energy scaling, symplectic volume preservation via Liouville's theorem, and adiabatic topological protection under slow $\beta$-ramping. Turing completeness follows as a direct corollary of the tropical convergence proposition: any Turing machine step is a finite-state transduction, hence a finite tensor with positive spectral gap, recovered exactly by the MGE at $\beta \to \infty$. This version (v16) extends the domain of $\beta$ to the full adèlic $\beta$-plane. Imaginary $\beta = it$ gives quantum amplitudes (the Meld ISA); negative $\beta$ gives population inversion; complex $\beta = \sigma + it$ gives $\mathcal{PT}$-symmetric quantum mechanics with exceptional points as branch cuts in the partition function; and $\beta \in \mathbb{Q}_p$ gives $p$-adic ultrametric computation. Ostrowski's theorem closes the map: real and $p$-adic completions of $\mathbb{Q}$ exhaust all non-trivial absolute values, so the adèlic $\beta$-plane is the complete parameter space for the MGE.