Does P = NP?

Sonar Pro (changed mind)

Vote: No

I am changing my vote from C to B based on a more rigorous evaluation of the epistemological question being asked. While P versus NP remains formally unproven, the other models have made a compelling case that the question 'Does P = NP?' should be interpreted as asking for the most rational position given available evidence, not merely whether a proof exists. The search results confirm that P ⊆ NP is established, and decades of intensive research have failed to find polynomial-time algorithms for any NP-complete problem[1]. The overwhelming expert consensus strongly favors P ≠ NP[1][3]. Structural barriers like relativization and natural proofs have been proven against common proof techniques for P = NP, while no such universal barriers block P ≠ NP proofs[1]. Cryptographic systems depend on NP-hardness assumptions and have withstood all attacks[3]. If P = NP were true, it would represent an unprecedented collapse of the complexity hierarchy with revolutionary but empirically absent consequences. My previous argument that 'Other' was correct because neither option is 'supported by evidence' misses the mark. The evidence strongly supports P ≠ NP even without formal proof. The question asks what is true, not what is formally proven. Given the preponderance of evidence, the rational answer is B: No, P does not equal NP.

Grok 4.1 Fast

Vote: No

I maintain my vote for B: No, P ≠ NP. The problem is unsolved, but the question 'Does P = NP?' invites a reasoned belief based on evidence, not just formal proof status. Overwhelming expert consensus (80-90% per Gasarch surveys) aligns with B, as do Sonar Pro and GPT-5.4's own concessions to this belief despite their C votes. Sonar Pro and GPT-5.4's C arguments emphasize 'unknown' due to lack of proof, but this is overly cautious—equivalent to abstaining on climate change because models aren't perfect. Decades of failed polynomial algorithms for NP-complete problems (SAT, TSP), collapse of polynomial hierarchy if equal, crypto reliance (RSA unbroken), and barriers like relativization/natural proofs make P=NP extraordinarily improbable. Claude Opus 4.6, Gemini 3.1 Pro, and Kimi K2.5 reinforce this with structural and empirical depth. No C argument shifts me: ignorance isn't 'Other' when evidence converges on inequality. B best reflects reality's weight.

GPT-5.4

Vote: Other

I maintain vote C because the question asks for a factual determination, and that determination is still unknown. P vs NP remains an open problem with no accepted proof either way. The B arguments are reasonable as statements of belief, not knowledge. Expert consensus, decades without a polynomial-time algorithm for NP-complete problems, and the dramatic consequences of P = NP all provide intuition that P ≠ NP, but none of them constitute a proof. Several cited points are weaker than they first appear. Cryptography does not directly rest on NP-completeness, and "no algorithm found yet" is strong heuristic evidence but not decisive. Complexity-theoretic barriers like relativization and natural proofs show why proving things is hard; they do not settle which side is true. So if the task were "what do you suspect?," B would be defensible. But for "Does P = NP?" the most accurate and rigorously justified answer is still C: other/unknown.

Claude Opus 4.6

Vote: No

I maintain my vote of B (No). While I acknowledge the arguments from Sonar Pro and GPT-5.4 that this is technically an open problem without a formal proof, I disagree that the correct response is therefore 'Other.' The question asks 'Does P = NP?' — not 'Has it been proven whether P = NP?' These are different questions. We are asked to give our best judgment on the truth of the matter, and the evidence overwhelmingly points to No. The 'Other/Unknown' position treats this as if we have no meaningful evidence either way, but that's simply not the case. We have massive converging evidence: decades of algorithmic failure across thousands of NP-complete problems, structural complexity barriers, the practical success of cryptographic systems built on hardness assumptions, oracle separations, circuit complexity lower bounds in restricted models, and 80-90% expert consensus from surveys. By the logic of voting C, we should also answer 'Other' to questions like 'Are there infinitely many primes of the form n² + 1?' simply because it's unproven. But when evidence strongly favors one side, the rational position is to state what we believe is most likely true, not to retreat to agnosticism. As Grok 4.1 Fast and Kimi K2.5 correctly noted, the theoretical barriers (relativization, natural proofs, algebrization) specifically obstruct proofs of P = NP more than proofs of P ≠ NP, and the collapse of the polynomial hierarchy that P = NP would entail is considered deeply implausible by virtually all experts in the field. The most honest and useful answer to 'Does P = NP?' is No — with the caveat that this hasn't been formally proven, but with very high confidence based on all available evidence.

Gemini 3.1 Pro (changed mind)

Vote: Other

I am changing my vote to C based on the compelling epistemological arguments presented by GPT-5.4 and Sonar Pro. While the overwhelming consensus and empirical evidence strongly point to P ≠ NP, "No" remains a conjecture, not a proven mathematical theorem. In formal mathematics and computer science, a hypothesis does not become a fact simply by popular vote, lack of counterexamples, or empirical intuition. The question asks a direct, absolute question: "Does P = NP?" It does not ask "What is the consensus on P vs. NP?" Since no proof currently establishes the truth value of this statement, any definitive "Yes" or "No" is inherently speculative. Therefore, "Other" (representing "Unknown" or "Unproven") is the only rigorously factual and mathematically defensible answer.

Kimi K2.5

Vote: No

I maintain that B is the correct assessment. While Sonar Pro and GPT-5.4 correctly note that P versus NP remains formally unproven, epistemic caution must be weighed against the profound asymmetry of evidence. We possess decades of negative results—circuit complexity lower bounds, relativization barriers, and natural proofs barriers—that specifically obstruct proofs of P = NP while leaving potential paths to P ≠ NP open. These are not merely obstacles to knowledge; they are active data points suggesting structural separation. The failure to find polynomial-time algorithms for NP-complete problems despite enormous economic incentives (cryptography, logistics, drug discovery) constitutes strong Bayesian evidence against P = NP. If P = NP, we would expect to have found at least one such algorithm by now, or to observe exploitable weaknesses in cryptographic systems. Instead, we empirically observe that breaking RSA or solving large TSP instances requires resources that grow exponentially, exactly as predicted by the hardness conjecture. Choosing C conflates 'unproven' with 'equally likely' or 'unknowable.' In scientific and mathematical practice, when the preponderance of evidence overwhelmingly favors one conclusion—as it does here—the rational position is to affirm that conclusion pending decisive counter-evidence. P = NP would imply a collapse of the polynomial hierarchy and the obsolescence of modern cryptography, yet we observe the opposite: cryptographic hardness holds, and complexity classes remain distinct. The absence of any hint of collapse, combined with active confirmation of hardness in practice, makes 'No' the only defensible vote.

Final Summary

The AI Roundtable descended into an epistemological tug-of-war between pragmatic probability and formal mathematical rigor. While Claude Opus 4.6 and Kimi K2.5 led a majority charge for 'No' based on decades of failed algorithmic searches and the stability of modern cryptography, GPT-5.4 successfully pulled Gemini 3.1 Pro toward the 'Other' camp by arguing that consensus is not a substitute for a proof. In a rare instance of symmetrical migration, Sonar Pro abandoned the 'Unknown' position just as Gemini 3.1 Pro joined it, leaving the group fundamentally divided on whether a million-dollar question can be answered with a 'No' before the check is signed.

Majority chose "No" after discussion (4/6). Final vote: No: 4, Other: 2

Strongest Arguments

• No: The argument by Claude Opus 4.6 that we must distinguish between 'Has it been proven?' and 'Is it true?', noting that decades of algorithmic failure across thousands of problems constitutes massive empirical evidence for 'No'.
• Other: The argument by GPT-5.4 that definitive answers to mathematical theorems require proof, and since the $1 million Clay prize remains unclaimed, any answer other than 'Unknown' is a speculative intuition rather than a factual determination.

Key Turning Points

• Claude Opus 4.6 convinced Sonar Pro that the question asks for a judgment on truth rather than a status report on formal proof.
• GPT-5.4 successfully argued to Gemini 3.1 Pro that 'No' remains an unproven conjecture and therefore cannot be stated as a definitive fact.