Spectral Closure Systems: Local Access, Entropic Defect, and Reconstruction

This preprint develops the theory of spectral closure systems: an operator-algebraic framework for local access to a global compatibility object through typed, lossy closures, organized as a calculus of admissible sections of access maps. Truth is treated as extendability along an inverse system, with finite-stage obstruction certificates by compactness. Recovery of local charts is canonical exactly on the sufficiency sector (Petz recovery, conditioning); off it, every totalization is impotent or pays an exact entropy price. Time enters as the Radon–Nikodym cocycle of a nonsingular dynamics in its crossed-product envelope; a nontrivial clock forces a noncommutative envelope, with Murray–von Neumann–Connes type read off the ratio set and the flow of weights as envelope invariant. In the quantum perimeter, point-valued global truth is empty (Kochen–Specker), state-valued compatibility survives (Gleason), Bell violations measure the classical gluing defect, and stable classical recordability is characterized as the commutative, modular-invariant, asymptotically sufficient chart sector; on the reflection-positive sector the record process reconstructs a positive-generator dynamics. Further results include an independence theorem for the underlying skeleton, a two-tier classification theorem with provable Foreman–Rudolph–Weiss wildness inside its fibers, a sector-split Orlicz envelope for relative entropy beyond bounded densities, and a complete logarithmic Sobolev inequality for the temporal module, uniform in the modular cocycle, by transference from the circle. Every external input is a named theorem-import with matched hypotheses; the interpretive vocabulary is proved to be a conservative definitional extension of the formal core.