Structural Admissibility Regimes (SAR): Unified Algebra I — Foundations and Internal Consistency

This paper develops the algebraic foundation of the Structural Admissibility Regimes (SAR) framework by formulating a unified operator-algebraic system defined on a σ-finite semifinite von Neumann algebra equipped with a faithful normal trace and a single primitive constraint: a finite admissibility capacity element . The admissible domain is given by the operator inequality , from which all structural and dynamical properties are derived without introducing probabilistic, thermodynamic, geometric, or physical assumptions. Within this setting, a canonical capacity-bounded entropy functional generates an intrinsic admissibility geometry through its Hessian, producing an entropy–Hessian metric and associated curvature operator on the interior of the admissible domain. The framework further yields a nonlinear, positivity-preserving, capacity-bounded strongly continuous semigroup governing admissibility evolution. Global well-posedness, forward invariance of the admissible set, and convergence to a unique stationary state proportional to the capacity element are established under suitable structural hypotheses. Entropy, curvature, generator, and semigroup together form a structurally closed system determined entirely by the finite capacity constraint. The resulting theory provides a mathematically self-contained operator-algebraic foundation for the Structural Admissibility Regimes program. Part of the Structural Admissibility Regimes (SAR) research series, following the foundational SAR trilogy and preceding Unified Algebra II, which develops the associated operator-algebraic transport law and spectral phase structure.