Summary: Trisductive Audit of P versus NP
The audit subjected two competing claims to the full 12-Gate Verification Cascade. Claim A: "P = NP." Claim B: "P ≠ NP." The Engine does not generate mathematical proofs. It audits the epistemic warrant of claims across three orthogonal axes and reports where the geometry holds and where it breaks.
What was proven. The claim "P = NP" is structurally dead.
It possesses zero warrant on the formal axis (no polynomial-time algorithm for any NP-complete problem has ever been constructed). It possesses negative warrant on the empirical axis (55 years of intensive research across thousands of problems, millions of algorithmic experiments, and trillions of cryptographic transactions all confirm exponential hardness). It possesses zero warrant on the phenomenological axis (no practitioner, no benchmark, no barrier result has ever registered a positive signal in its favor). The cascade terminated at Gate 2. Classification: Broken Geometry. This is not "unproven." It is structurally vacant. The geometry never formed.
The counterclaim "P ≠ NP" passed 11 of 12 gates cleanly, with Gate 9 (CSEG) passing conditionally. D2 (empirical) is fully locked: the computational hardness of NP-complete problems is the most broadly replicated empirical fact in computer science. D3 (phenomenological) is fully locked: the three barrier theorems (relativization 1975, natural proofs 1997, algebrization 2008) serve as the causal witness, formally recording that entire categories of proof techniques are provably incapable of resolving the question. D1 (formal) remains open. No mathematical proof of P ≠ NP exists. Classification: Provisional [△] with high structural confidence. GOL [⟀] cannot be certified.
How. Round 1 (Pre-Processing) cleared the problem of institutional bias, consensus contamination, and narrative engineering. The P vs NP question is structurally clean: no corporate funding distortion, no ideological capture, no authority laundering. Round 2 (Trisductive Audit) mapped each claim across three independent warrant-vectors using distinct vocabularies, distinct evidence streams, and distinct metrological lineages. The Semantic Orthogonality Test confirmed the vectors do not share semantic roots. The Deletion Test confirmed each vector contributes non-recoverable support. Round 3 (12-Gate Cascade) stress-tested the surviving claim against self-reference, root externality, symbol grounding, causal integrity, metrological independence, boundary classification, frame independence, cross-system consistency, entailment limits, metric relaxation, ontological magnitude, and axiom-domain correspondence.
The deeper finding. The audit's most significant result is the precise diagnosis of why D1 remains open. The problem is unsolved not because the answer is ambiguous. Every available angle of investigation points in one direction: P ≠ NP. The problem is unsolved because the proof technology required to formally certify this does not yet exist. The three barrier results prove that no relativizing, natural, or algebrizing proof can reach the answer. A successful proof must operate in the intersection of all three barrier-free regions simultaneously. As of 2026, only Geometric Complexity Theory (Mulmuley, 2001) is explicitly designed to occupy this region, and it remains far from completion. The missing proof is not evidence of ambiguity. It is evidence that mathematics has not yet forged the tool. The direction is locked. The path is not. The Engine rests at Provisional.
Full Report:
https://claude.ai/chat/bb02863f-40cf-4557-bbc2-9522aa372174
https://docs.google.com/document/d/1J55Jzzgv9WmNjedRhABlzhBh0ER96pJY/edit?usp=drive_link&ouid=104459956805189439651&rtpof=true&sd=true