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 $G = S_4 \times \mathbb{Z}_2^3$. 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 $\heart = 1 - S_{rel}/(2k)$, linking the coherence metric ($\heart$) directly to relational entropy ($S_{rel}$). Capacity Invariant ($I_F$): We define the structural capacity $S_{cap}$ and prove it is constrained by the invariant $\heart \times S_{cap} = I_F = 192$. 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: $\pi_5$ (Arché/Potential), $\pi_{5.5}$ (Logos/Transition), and $\pi_6$ (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 $\pi_6$ 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 $\pi_5 \to \pi_6$ gradient.