Structural Admissibility Regimes (SAR): Unified Algebra II — Operator Transport and Spectral Phase Structure

This paper develops the second part of the Unified Structural Admissibility System within the Structural Admissibility Regimes (SAR) program. Building on the operator-algebraic foundations established in Unified Algebra I, we construct a single structural transport equation for admissible operators on a finite von Neumann algebra subject to a fixed capacity constraint . The transport equation combines three components: an entropy–Hessian gradient sector generating contractive entropy dynamics, a compatibility-preserving skew sector generating derivation-type transport, and a bounded admissibility perturbation representing structurally permitted non-gradient activity within the finite capacity budget. The associated entropy functional induces a dual-barrier Hessian metric and curvature operator on the admissible interior. A capacity-weighted commutator operator is introduced to quantify incompatibility structure. The spectral gap of the corresponding positive operator defines a classification of admissible regimes into strict-gap, gapless, and fully commuting phases. Conditional expectations onto invariant subalgebras yield consistent subsystem projections of the structural transport equation while preserving the global capacity constraint. Representation results show that familiar dynamical forms—unitary operator evolution, dissipative entropy flows, geometric Hessian dynamics, symmetry transport, and stabilized subsystem behavior—arise as distinct sectors of the same structural transport law. Finite capacity imposes bounded incompatibility load, bounded entropy dissipation, and non-reconstructibility of projected subsystem dynamics. An explicit realization on the matrix algebra is provided, where the spectral gap reduces to squared eigenvalue separation and the entropy–Hessian metric coincides with the classical Fisher information metric on the eigenvalue simplex. Part of the Structural Admissibility Regimes (SAR) Unified Algebra series, following Unified Algebra I and preceding the applied SAR regime analyses.