ONE AXIOM : The Entropy-Coherence Isomorphism

Title: The Entropy-Coherence Isomorphism: Local Thermodynamic Projections from Finite Group Structure Description: Abstract: This paper, designated as Document 1B of the ONE AXIOM framework, presents a formal derivation of the five laws of thermodynamics from the structural invariants of the finite group G = S₄ Z₂³. Using a novel dual-track methodology—distinguishing between Ontological Coherence Resolution (OCR) and Ontological Epistemic Resolution (OER) —we resolve the long-standing problem of energy circularity by defining temperature and entropy as emergent projections of group-theoretic constraints. Key Technical Pillars: Entropy-Coherence Isomorphism (Theorem T8): We establish the central identity = 1 - Sₑ₄₋/ (2k), linking the coherence metric () directly to relational entropy (Sₑ₄₋). Capacity Invariant (IF): We define the structural capacity S₂₀₏ and prove it is constrained by the invariant S₂₀₏ = IF = 192. This ensures that the total structural information of the domain remains constant. Triadic Layer Architecture: The derivation maps the transition of physical laws across three ontological layers: ₅ (Arché/Potential), ₅. ₅ (Logos/Transition), and ₆ (Physis/Super-diffusive reality). Unified Foundation: The framework provides a unified basis for both equilibrium and non-equilibrium thermodynamics through group-theoretic coherence constraints, framing the Second Law as a local ₆ projection of the Ontological Expansion Principle (OEP). Experimental Predictions: We propose an experimental protocol targeting "coherence excess" in the Dynamic Casimir Effect (DCE), predicting deviations of the order of 10^-4. BBN Compatibility: The model shows a 0. 07% alignment with Big Bang Nucleosynthesis (BBN) data for the q=3/2 super-diffusive state. Methodology: The paper utilizes 13 dual-track theorems to bridge the gap between categorical symmetry and measurable physical phenomena. By treating the domain as an open, flow-based system, we demonstrate how structural stability is maintained across the ₅ ₆ gradient.