Structural Admissibility Regimes (SAR): Finite-Capacity Constraints on Algebraic Decompositions

This work establishes the algebraic foundation of the Structural Admissibility Regimes (SAR) program. It introduces structural admissibility as a cut-relative constraint on algebraic decompositions within finite-dimensional operator algebras. Rather than assuming dynamical laws or probabilistic postulates, the framework begins from a finite structural capacity governing which collections of commuting subalgebras remain mutually admissible. The relative commutant of a cut is identified as the central object controlling admissibility. Its dimension provides a measure of remaining structural capacity available to support additional commuting refinements. Structural exhaustion occurs when the relative commutant becomes trivial, yielding a sharp algebraic criterion for incompatibility. Large admissible families of commuting refinements organize into tensor product structures of the relative commutant, while generic refinements rapidly exhaust structural capacity and prevent persistence of extended commuting structure. The admissible state space supported on the relative commutant is formalized, and finite capacity is shown to bound admissible informational degrees of freedom. Positivity of admissible states yields convex state structure without introducing independent probabilistic assumptions. Standard operator-algebraic representation results provide closure at the state level. All results are purely algebraic and kinematic within the finite-dimensional ∗-algebraic setting. No dynamical, thermodynamic, or geometric assumptions are introduced. This paper provides the structural foundation upon which subsequent SAR developments are built. This paper is the first installment of the Structural Admissibility Regimes (SAR) foundational series.