Structural Admissibility Regimes (SAR): Quantum and Thermodynamic Regimes from Finite Admissibility

This paper develops the quantum and thermodynamic regime structure of the Structural Admissibility Regimes (SAR) framework under finite admissibility capacity. Building on the algebraic foundations of the SAR series and the Unified Structural Admissibility System, we analyze how compatibility structure and refinement algebra generate quantum-type and thermodynamic-type behavior as admissible finite-capacity regimes. Within this framework, complex amplitude geometry arises from compatibility relations among admissible refinements, and admissible participation acquires a quadratic structure determined by compatibility organization. Cross-cut incompatibility under finite capacity produces structural selection, yielding an intrinsic mechanism for structural measurement and quantum entanglement without introducing collapse postulates or external probabilistic assumptions. Structural entropy is defined through accessible compatibility organization within the refinement algebra. Its monotonic behavior under non-invertible admissible evolution establishes an intrinsic structural arrow of time. Redistribution of admissible accessibility generates transport processes, fluctuations, and nonequilibrium thermodynamic regimes. Amplitude-level refinement dynamics and entropy-ordered accessibility evolution therefore emerge as complementary manifestations of a single admissibility-driven refinement algebra. The resulting analysis identifies structural constraints that any finite-capacity quantum or thermodynamic regime must satisfy, providing a pre-dynamical algebraic infrastructure within which domain-specific quantum and thermodynamic theories may be consistently formulated. Part of the Structural Admissibility Regimes (SAR) Applied Series, following the Unified Structural Admissibility System and preceding applications to gravitational and cosmological regimes.