SAT ∈ NP — the headline theorem

This file ties together the two halves of the SAT ∈ NP proof:

• SAT.VerifierTM.verifyPairTM_decidesInTime (in VerifierTM) — the deterministic three-tape verifier decides pairLang Witness within the quadratic budget verifyPairTMTime, so pairLang Witness ∈ P.
• SAT.language_mem_NP_of_verifierP_direct (in GuessVerify) — the SAT-specialized guess-and-verify NTM turns pairLang Witness ∈ P into language ∈ NP.

Combining them yields the unconditional theorem SAT.language_mem_NP : language ∈ NP.

theorem Complexity.SAT.pairLang_witness_mem_P : pairLang Witness ∈ P

The paired SAT verifier language is decided in polynomial (in fact quadratic) time by verifyPairTM, hence pairLang Witness ∈ P.

theorem Complexity.SAT.language_mem_NP : language ∈ NP

SAT ∈ NP. The Boolean satisfiability language is in NP, witnessed by the SAT-specialized guess-and-verify NTM running over the polynomial-time deterministic pair verifier.