Closure Calculus, Composition Rules, and Proof-Carrying Operators

Converges Papers 00–06 into the first executable calculus. Every closure operator is given an explicit type signature, preconditions, transformation, postconditions, certificate, debt, and version, packaged as a “Proof-Carrying Closure Operator” (PCO). Defines a first-version, 18-member operator family (Infer, Block, Refute, Prove, Condition, Bridge, Project, Transfer, Quotient, Split, Reopen, Discharge, Cut, Cover, Exhaust, Promote, Replay, Compile). Key claim: legality does not compose freely — two individually legal operators still require an explicit “composition certificate” before their composite operator is legal, covering type/scope/certificate/debt composability, non-inflation of authority, version consistency, bridge consistency, and reopening invalidation. Introduces a closure normal-form pipeline (Normalize → Validate → Infer → Propagate → Resolve → Rebuild → Project), and a minimal NS compiler interface requiring legacy document labels to first be parsed into candidate-layer objects before they can enter the calculus.