Thermodynamics, Statistical Mechanics, and Information as Structural Representations

Version 2: Mathematical strengthening of structural temperature definition.Correction of pointwise temperature normalization.Appendices A–D added for full formal rigor. This paper unifies thermodynamics, statistical mechanics, and information theory under a single structural principle derived from Axiomatic Second-Variation Geometry (ASVG), where the second variation operator on a kinematic state manifold decomposes into symmetric (reversible) and antisymmetric (irreversible) sectors without statistical assumptions. Irreversibility emerges geometrically as progression along curvature-defined order in the antisymmetric sector, enforcing monotonic free-energy decrease and structural entropy increase. Classical thermodynamic relations, including fluctuation-dissipation theorems, follow exactly from this decomposition. Statistical mechanics arises secondarily as a reduced representation of structural states, not as foundational; information measures like relative entropy capture degeneracy in these reductions, aligning precisely with free-energy differences. Macroscopic universality stems from hierarchical coarse-graining in the reversible sector. Thus, these domains are coordinate projections of one intrinsic order, with the second law as an inevitable geometric identity—ΔS ≥ 0, equality holding solely in the antisymmetric kernel—admitting no reversal. The framework resolves foundational puzzles like the Loschmidt paradox structurally, independent of probability or dynamics.