Phase V of the AAIS Framework: Axiomatic Closure and Identity Kernel Stabilization"

This repository contains Phase V of a mathematical study in operator algebras, addressing structural invariants in infinite-dimensional von Neumann systems. The work develops a framework based on direct integrals of Type III₁ factors, central projections, and superselection sector decomposition. A distinguished central projection is used to define invariant sectors, leading to sectorial irreducibility and operational isolation. Stability is established through spectral–topological methods, including Weyl-class invariance, Fredholm index persistence, and K-theoretic rigidity. Operational observables are treated via the Calkin algebra, providing essential isolation of the central structure from compact perturbations. Modular dynamics and categorical techniques are employed to demonstrate non-reconstructibility of the core algebraic structure from operational data. The absence of left adjoints and essential saturation yield a topological obstruction to inverse recovery. All constructions are formulated constructively within standard operator-algebraic foundations, without reliance on choice-dependent arguments. The appendix labeled “Zero-Point” refers solely to an operator-theoretic baseline and carries no physical interpretation. This submission is intended as a mathematical contribution to operator algebras and mathematical physics, focusing on structural invariance, superselection mechanisms, and categorical stability in Type III systems.