Geometric Completion of Thermodynamic Response: From the (T)-(V)-(S)-(P) Compass to Recursive $λ$-Jacobian Geometry

Abstract

We develop a geometric framework for thermodynamic response in a simple compressible system using the cyclic state-variable structure T-V-S-P. Standard thermodynamic potentials are placed on an oriented compass, and constraint-dependent response variables $λ_p$, $λ_v$, $λ_s$, and $λ_t$ are introduced for isobaric, isochoric, isentropic, and isothermal directions. These variables extend the familiar caloric responses $C_p$ and $C_v$ by completing the corresponding mechanical response sectors and by satisfying conjugate product relations derived from the fundamental thermodynamic differentials. The equilibrium structure is formulated through a Jacobian representation intended to encode Maxwell-type relations and generate a recursive hierarchy of higher-order response derivatives. The ideal gas is used as a consistency example. For nonequilibrium processes, response one-forms are introduced whose failure of exactness is represented by a curvature two-form. By Stokes' theorem, the integrated curvature provides a geometric candidate for cyclic return residue, memory, or hysteresis, subject to explicit classification, tolerance, dimensional calibration, and comparison with standard Onsager entropy production. Onsager reciprocity is retained in the conventional flux-force frame, while asymmetry in entropy-scaled variables is treated as frame-dependent. The framework unifies thermodynamic potentials, constrained response functions, recursive derivative structure, and calibrated cyclic effects. The curvature and information-sector extensions are presented as testable constitutive hypotheses requiring further analytical and experimental validation.