This paper proves a conditional separation between P and NP inside Six Birds Theory, a typed framework for closure formation in mathematical structures. For SAT, the framework supplies a saturated SAT layer T!SAT — a typed carrier of polynomial-time SAT computations together with their audit data — observed through a lawful polynomial-time instrument Ipoly that declares which observations on the layer count as current. Two translation theorems connect this setup to standard SAT semantics. Translation A identifies SAT ∈ P with decision currentization: the SAT decision bit is itself a current observation on T!SAT. Translation B identifies decision currentization with currentization of the canonical witness producer of the autonomous lift ΩSAT, using SAT self-reducibility and closure of current observables under post-processing, finite products, and composition. The structural hypothesis ΓCSL-SAT-hidden of Tsiokos (2026, Hiddenness as a Structural Law of Emergence) says that this canonical witness producer is not a lawful current observable on T!SAT. Under that hypothesis, the four-step contradiction chain gives SAT ∉ P; combined with SAT ∈ NP from the standard verifier package, the framework accepts PSB ≠ NPSB at recognition grade. Outside the framework, the result is the conditional theorem: formed-SAT closure carrying ΓCSL-SAT-hidden under Ipoly implies P ≠ NP. The same closure records also tag T!SAT, in the audit-closure register of Tsiokos (2026, Audited Operational Realisability), as a carrier-attached cascade-stable AOR package under ΓCSL-SAT-hidden. This is packaging of the same conditional theorem. The hypothesis further localizes to one residual: the canonical lexicographic SAT-branching readout is not a lawful current observable on the standard fixed-presentation polynomial current carrier. A scoped Class-B Irremovability theorem rules out one removal route: no basis-generated Class-B carrier-feature method can remove that residual, except by leaving the basis or importing a target-equivalent lower bound. The paper does not claim an unconditional standard-ZFC separation of P and NP, derivability of ΓCSL-SAT-hidden from framework primitives, circumvention of the classical barriers in their native senses, absolute irremovability of ΓCSL-SAT-hidden, or any AOR claim outside the stated formed-SAT, standard-basis scope.
Ioannis Tsiokos (Wed,) studied this question.