Tropical-Spectral Hybrid Algebra

We introduce a tropical-spectral hybrid algebra that enriches tropical numbers with a spectral entropy component, unifying optimization (tropical) and equilibrium (spectral) descriptions of the same phenomena. Tropical algebra uses (min, +) arithmetic where addition is minimum and multiplication is plus, naturally encoding shortest paths, optimal allocation, and LP relaxation. The hybrid extends this by attaching an entropy measure to each tropical element, with free energy F = cost − entropy providing the bridge between the two viewpoints. We establish fundamental properties including cost additivity under tropical multiplication, entropy subadditivity, and free energy superadditivity. The framework is formalized in the Platonic proof language with 10 verified theorems and 8 structural hypotheses, all exportable to Lean 4. Applications include supply chain optimization with dynamic pricing, shortest paths in networks with time-varying costs, and game-theoretic Nash equilibria via tropical eigenvalues.