Equivalence of Thermodynamic Potential Equations and Special Relativity: A Gravitational-Thermodynamization Perspective

Thermodynamic potentials provide equivalent descriptions of equilibrium systems through Legendre transforms and associated Maxwell relations. Motivated by recent developments in the "thermodynamization" of gravitational dynamics and by the RVB-residue method viewpoint (attributed to Wen-Xiang Chen), we propose a unified framework comparing two notions of equivalence: (i) the algebraic equivalence classes of thermodynamic potential equations, and (ii) the covariance constraints of special relativity. We formulate a map between potential-based contact geometry identities in thermodynamics and invariant structures in relativity via a variational "residue" extraction on constrained manifolds. Conceptually, we show that the equivalence of different thermodynamic potentials can be viewed as a gauge-like redundancy, while Lorentz covariance in relativity acts as a compatibility condition selecting physically admissible gauges. The framework is illustrated on relativistic ideal fluids and a simple horizon thermodynamics toy model.