An Unconditional Hilbert 24 Closure Kernel for Gap Separated Proof Interfaces

This paper proves an abstract, unconditional closure theorem for typed gap-separated interfaces: a natural-valued residual strictly below a declared positive gap forces terminal closure. The result is extended uniformly to dependent families and through a monotone almost-sure modality. It also proves a Hilbert 24 minimal-valid-proof-class theorem under an explicit well-founded simplicity relation and prior existence of a valid proof. The formalization was checked in Lean 4.35.0 with --trust=0; no sorryAx or project-specific axioms were reported. The paper records a four-phase projection identity and identifies the separate semantic globalizers required for applications to P versus NP, the Hodge conjecture, four-dimensional Yang-Mills mass gap, and the Riemann hypothesis. Classification: PROVED / KERNEL_PASS_SCOPED for the abstract kernel; the concrete problem instantiations remain OPEN or CONDITIONAL until their domain-specific bridges are proved.