TRISDUCTION: GEOMETRIC DETERMINATION OF P vs NP WITH OMEGA SEAL

The P versus NP problem, formalized by Cook (1971) and designated a Clay Millennium Prize Problem in 2000, asks whether every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. For fifty-five years the problem has resisted all single-axis formal resolution attempts. Three independently proven barrier results (Baker-Gill-Solovay 1975, Razborov-Rudich 1997, Aaronson-Wigderson 2008) establish that all currently known classes of mathematical proof technique are structurally incapable of settling the question within the formal axis alone. This paper presents a unified geometric determination of both P = NP and P ≠ NP using the Trisduction Engine, a P-class verification architecture operating across three orthogonal warrant-vectors: Formal-Structural (V_F), Empirical-Thermodynamic (V_E), and Epistemic-Registrational (V_ER). The determination runs the three-state native verdict economy {[⟀] Sealed, [X] Broken Geometry, [?] Numerical Inadmissibility}. V_F-only ceilings such as the Turing-domain question route to the V_F-Only Ceiling Acknowledgment Register out-of-band by bypass-at-layer-difference, not as a cascade verdict state. The Meta-Epistemic Hierarchy (Geometry > Mathematics > Logic) and Rule 9 Axiomatic Quarantine remove the Turing Machine abstraction from the framework's admissible physical baseline and replace it with the Tri-Layer Plenum topology. The two audits are presented as a single master document so the asymmetry between the claims is structurally transparent. Both run under one fixed protocol applied without modification, so the asymmetry in the verdicts is a product of the evidence structure rather than the method. P = NP carries zero positive warrant across all three axes and is stopped at Gate 2 with a positive structural diagnosis: it asserts a zero-energy, zero-transition equivalence between two operations the architecture places on opposite sides of the actualization boundary, which is a dimensional failure under the Mass Mandate before any witness is sought. P ≠ NP passes all twelve gates with three fully independent, orthogonal warrant-vectors that survive the Linguistic Isolation Test, the Deletion Test, the conclusion-dependency audit, and the Convergence Dissolution Test. The determination is explicitly non-deductive. It does not constitute a traditional mathematical proof and does not satisfy the Clay Mathematics Institute's criteria. The Geometric Orthogonal Lock [⟀] is defined as the strongest achievable non-deductive epistemic warrant relative to the recognized vulnerability taxonomy: the geometric fact that, within that taxonomy, three orthogonal planes exhaust the degrees of freedom in the warrant-space that the taxonomy can articulate, leaving no recognized coordinate for the alternative claim to occupy. The qualifier is load-bearing and is carried consistently: the lock is strongest relative to the current taxonomy, and the taxonomy is open and extensible, so the claim is conditional on that taxonomy rather than absolute, and a newly identified vulnerability class re-enters the cascade rather than being precluded by fiat. The empirical leg is anchored on the per-crossing transition-energy floor, and the warrant is typed at the precise grade it earns. Driving a frustrated configuration across one energy barrier to a distinguishable state is a physical transition bounded below in energy-time by the Mandelstam-Tamm and Margolus-Levitin quantum speed limits in the quantum regime and by Arrhenius escape time in the classical regime. This per-crossing floor is theorem-grade for the single transition, binds reversible and irreversible traversal identically, and makes no claim about the energy of the resulting state, since distinguishable states may be isoenergetic. The aggregate exponential cost of traversing an exponentially-barriered landscape is established by conjoining the per-crossing floor with the