Thermodynamics, Statistical Mechanics, and Information as Structural Representations

Version 3: Final structural revision and full formal consolidation. This version substantially revises and strengthens Version 2. The paper has been reorganized around a fully explicit lifted antisymmetric-response framework. The basic object is now clearly distinguished from an ordinary scalar Hessian: it is a full second-order structural response form whose antisymmetric sector defines an alternating response form ω_x, an induced operator A_x, and the reversible kernel ker A_x. After choosing a compatible lift J_x on the nondegenerate sector, the entropic projection is defined byΔS_x(u)=ω_x(u,J_xu),and the structural second law is obtained in the exact formΔS_x(u)≥0, with equality if and only if u∈ker A_x. Compared with Version 2, this version further clarifies that ΔS_x is a nonnegative quadratic separation on the quotient V_x/ker A_x and does not, by itself, determine a temporal direction. A thermodynamic or process orientation is obtained only after an admissible process cone, an oriented class of admissible variations, or a selected dynamics has been fixed. The treatment of heat, work, temperature, and internal energy has also been revised. These quantities are now formulated as calibrated thermodynamic readouts rather than primitive structural objects. The paper separates structural entropy production from thermal readout, energy readout, temperature calibration, and first-law balance. The roles of statistical mechanics and information theory have been sharpened. Statistical mechanics is presented as a reduced structural representation, not as the foundation of irreversibility. Information-theoretic quantities, including relative entropy, mutual information, channel monotonicity, and data-processing inequalities, are treated as reduced distinguishability statements under admissible reduction. Appendices A–D have been added or fully rewritten to provide the formal support for the main text. Appendix A gives the linear algebra of lifted antisymmetric response. Appendix B proves the exact additivity, interaction, and quotient identities of the entropic projection. Appendix C develops the readout calculus for heat, work, temperature, and calibration. Appendix D supplies explicit finite-dimensional models, including a model showing that heat-form integrability is an additional readout-level condition and not a consequence of the lifted antisymmetric structure alone. Version 3 therefore consolidates the paper into a self-contained structural account of thermodynamics, statistical mechanics, information theory, and macroscopic universality as coordinate expressions of one admissible second-order structure.