P = NP by Exclusion of the SAT Separator Endpoint

Overview This record contains the manuscript and project materials for (P=NP) by Exclusion of the SAT Separator Endpoint: An AASC Endpoint-Rigidity Proof for Arbitrary Finite CNF-SAT. The manuscript presents an AASC endpoint-rigidity proof of the positive mathematical endpoint for the P versus NP problem. The official carrier used is unrestricted finite CNF-SAT. The positive raw branch is deterministic polynomial-time decidability of arbitrary finite CNF-SAT, and the bare negative branch is its raw complement on the same carrier. Central Result The central result is that the SAT separator endpoint is unavailable. The proof establishes the chain: (\SepBare \Rightarrow \SepImg \Rightarrow \SepDisc(model) \Rightarrow \bot). Therefore the bare negative branch is excluded. By raw complementarity, the positive branch follows. Since the positive branch is (\CnfSATP), arbitrary finite CNF-SAT is decidable in deterministic polynomial time. The standard Cook–Levin/Karp endpoint correspondence then yields (P=NP). Method The proof is kernel-first and AASC-owned. AASC is not treated as an external classifier, optional foundational system, or source of theorem standing. The dependency order is the reverse: theorem-bearing endpoint use is governed construction; governed construction is possible only under the kernel of admissibility, standing, reference, and irreversibility; and the downstream AASC machinery records the fixed-domain consequence layer forced by that kernel. Ordinary theoremhood, positive or negative, is therefore kernel-internal when it is genuine theoremhood rather than raw trace. SAT Endpoint Asymmetry The positive branch directly occupies the constructive endpoint carrier. The negative SAT branch is different in role, not because negative theoremhood is defective, but because the SAT image-separator branch performs same-domain endpoint exclusion by separator classification. On the fixed CNF-SAT carrier, that separator route induces an independent same-domain endpoint-status discriminator. Kernel-forced structural rigidity excludes independent same-domain endpoint-status discriminators. This excludes the SAT separator route and forces the positive endpoint. Relation to Cook–Levin/Karp Classical complexity theory is used only at the endpoint-correspondence layer. The manuscript does not alter the standard definitions of (P), (NP), SAT, CNF-SAT, polynomial time, or NP-completeness. AASC supplies the proof of separator-endpoint exclusion. Cook–Levin/Karp supplies the standard correspondence from polynomial-time arbitrary finite CNF-SAT to (P=NP). Lean4 Support The record includes Lean4 appendix support for the SAT-operator bridge and negative-branch closure. The Lean material formalizes the local proof spine connecting the bare negative branch, the SAT image-separator branch, the independent separator discriminator, and the contradiction under no-independent-discriminator closure. Lean is included as reproducibility and audit support; the manuscript proof remains primary. Scope The result is a mathematical endpoint claim for the official P versus NP problem. It is not a conventional diagonalization, circuit lower-bound, relativization, natural-proofs, proof-complexity, or algorithm-search argument. It is an AASC endpoint-structure proof on the unrestricted finite CNF-SAT carrier. Record Contents This record contains the manuscript PDF, LaTeX project source, verification notes, theorem and notation ledgers, Lean4 appendix-support material, and the associated SAT-operator Lean audit package.