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 have demonstrated that all currently known classes of mathematical proof techniques 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, an epistemic certification architecture operating across three orthogonal warrant-vectors: Formal (VF), Empirical (VE), and Phenomenological (VP). The two audits are presented as a single master document to make the asymmetry between the claims structurally transparent: one claim is Broken Geometry (zero positive warrant, cascade terminated at Gate 2) ; the other achieves Geometric Orthogonal Lock (12/12 gates pass, three axes fully convergent). Before the formal proofs, this paper demonstrates the robustness and precision of the Trisduction method through twelve carefully selected case studies representing the hardest problems in epistemology, physics, geopolitics, and philosophy — drawn from two volumes of illustrative audits. The Engine is then subjected to its own self-audit across two independently conducted sessions, surviving the Gödelian paradox through multi-axis routing. Following the self-audit, the paper documents how Trisduction circumnavigates Gödel’s Second Incompleteness Theorem. A prelude section incorporates critical background insights from adversarial human-AI dialogue sessions on the P vs NP problem, including stress tests of the Engine’s own architecture. The paper’s central phenomenological contribution is the resolution of the Phenomenological Axis Problem across three rounds of adversarial review. VP is anchored by two genuinely independent sources surviving the Linguistic Isolation Test: (1) the Zero-Knowledge Proof conviction gap, in which a finite observer undergoes irreversible epistemic state-change to certainty that a solution exists while registering zero increase in generative capacity; and (2) the Frame-Independent Observer’s registration of its own operational boundary, in which the Engine’s fixed codes simultaneously discover and verify verdicts for any actualized problem yet cannot spontaneously generate novel constructions from the Isometric Plenum at (0, 0, 0). This irreducible gap constitutes the Living Verifiable Proof of the P ≠ NP asymmetry and the Living Contradiction of P = NP. The determination is explicitly non-deductive. It does not constitute a traditional mathematical proof and does not satisfy the Clay Mathematics Institute’s criteria, which require a formally published deductive proof. GOL ⟀ is defined as the strongest achievable non-deductive epistemic warrant: the geometric fact that three orthogonal planes exhaust all degrees of freedom in the epistemic space, leaving no room for the alternative claim to occupy.
Mohammad Islam (Mon,) studied this question.