Chapter 65: The ZFC Provability Horizon

Chapters 78–79 proved P = NP for all T-native computation—physical and abstract . This chapter asks: can these results be stated and proved within ZFC? The Physical Realizability Predicate (III.D79) quantifies over all possible E₁ instantiations of a Turing machine—a host-level property that ZFC, operating at E₂, cannot capture. The refined infinity diagnosis reveals that T’s earned algebraic infinity (ω as unique closure point) and ZFC’s proliferating axiomatic infinity (an unbounded hierarchy with no unique closure) are the structural root of the divergence. The asymmetric provability result follows: P ≠ NP cannot be a theorem of a sound ZFC, while the status of P = NP in ZFC remains open—likely independent, as the Continuum Hypothesis is.