Renormalization as Physical Elimination in the Spectral Theory of Subspaces: Operator-Induced Flows, Closure Domains, and an RG-Thermodynamic Coordinate System

This preprint proposes an operational reading of renormalization in the Spectral Theory of Subspaces (STS). Instead of starting from fields and fluctuations, the approach treats renormalization as a property of an elimination and coarse-graining protocol that induces a scale-indexed family of effective operators. The paper translates this principle into testable constraints: approximate semigroup consistency, controlled backreaction via Schur-complement diagnostics, and chart-local closure realism via low-dimensional closure fits. Phases are defined as certified scale windows where these constraints hold simultaneously, while universality and ordering are treated as empirical properties that can be checked within those windows. A Tier-0 mini-battery on synthetic toy instances demonstrates the diagnostic suite, the separation from decoy controls, and a certified parameter-window workflow that can be carried to higher-tier studies.