Tractability Proof System: Tractability Certificates, Normal-Form Escape, and Clocked Enumeration

An important correction: deciding, for arbitrary Turing-machine code, whether it runs in polynomial time is indeed undecidable in general — but this doesn't prevent building an extensionally complete (extensionally complete) normal-form language for P. Cobham's bounded recursion and Bellantoni-Cook's safe recursion have long since given exactly this kind of classic characterization, where membership in the syntax itself guarantees polynomial time. Distinguishes three kinds of completeness: machine-index completeness (too strong — undecidable), proof completeness, and extensional normal-form completeness (already a mature theory). The Equals Team therefore no longer needs to prove polynomiality solver by solver for arbitrary SAT solvers; it can instead construct SAT directly within a normal-form language where “membership in this syntax guarantees polynomial time” — if this succeeds, P=NP follows immediately. The Not-Equals Team's symmetric task is to find a semantic invariant that is preserved by the initial functions, by composition, and by safe recursion alike, but violated by SAT's characteristic function (a Grammar Invariant Program). Also establishes the clocked Turing machine as a second effective normal form, paving the way for next round's diagonalization.