The Geometry of Recovery: Spectral Admissibility as a Structural Principle

We introduce the Geometry of Recovery, a framework in which physical admissibility is characterized by the bounded invertibility of an admissibility defect operator rather than spatial curvature or energetic bounds. We formalize a pseudo-metric space where distance is measured strictly by systemic recoverability, constrained to a trace-class-controlled spectral geometry. By elevating uniform recoverability to a structural principle, we rigorously link physical breakdown to the vanishing of the regularized spectral determinant (det₂) via a Birman-Schwinger reduction. Applying this framework across distinct physical regimes, we provide a unified operator-theoretic criterion for stability. The manuscript demonstrates how fluid dynamic blow-up (Navier-Stokes), quantum decoherence (Lindblad evolution), and control system failure can all be identified directly with the collapse of inverse admissibility structure. In this framework, physics is redefined as the study of systems that dynamically maintain their presence within the recovery manifold, and singularities are precisely the points at which uniform recoverability becomes mathematically impossible. The analytic theorems are supported by finite-dimensional numerical validations demonstrating the concurrent collapse of the recoverability radius, resolvent blow-up, and determinant degeneration.