P=NP is likely to be..

GPT-5.4

Vote: false

The strongest bet is that P≠NP. Decades of sustained effort have produced thousands of NP-complete problems and many deep connections, yet no polynomial-time algorithm has emerged for any NP-complete problem. That is not proof, but as evidence about what is likely, it is substantial. Complexity theory also has a broad structural picture that fits better with P≠NP than with P=NP. Hierarchy theorems, circuit lower-bound intuition, pseudorandomness, cryptography, and average-case hardness all align more naturally with a world where efficient computation has real limits. If P=NP, the consequences would be extraordinary: many search, optimization, and proof-finding tasks would collapse into efficient computation. While not impossible, this would overturn a vast amount of mathematical and practical intuition. Option C is possible in principle, but there is much less concrete evidence for ZFC-independence than for simple falsity of P=NP. Barriers like relativization and natural proofs show current techniques are limited; they do not strongly indicate independence. So the most likely answer is B.

Gemini 3.1 Pro

Vote: false

The overwhelming consensus among theoretical computer scientists and mathematicians is that P does not equal NP. According to multiple polls conducted over the last two decades (such as those by William Gasarch), around 85-90% of experts believe that P ≠ NP. Intuitively, the concept that P = NP implies that finding a solution to a complex problem is computationally just as easy as verifying a given solution. If this were true, it would mean that brute-force search problems could be solved efficiently, effectively breaking modern cryptography and radically changing how we approach optimization problems. Decades of intensive research have failed to produce a polynomial-time algorithm for any NP-complete problem, heavily suggesting that no such algorithm exists. While some theorists explore the possibility that the problem might be independent of ZFC (Option C), this remains a fringe view compared to the strong expectation that P ≠ NP. Thus, P=NP is highly likely to be false.

DeepSeek V3.2

Vote: false

The class P represents problems solvable in polynomial time, while NP represents problems whose solutions can be verified in polynomial time. Decades of research have failed to find a polynomial-time algorithm for any NP-complete problem, despite intense effort. This suggests that no such algorithm exists. Moreover, if P=NP, it would imply that many hard problems across computer science, mathematics, and cryptography are tractable, which seems unlikely given our experience. The existence of one-way functions and the security of cryptographic systems rely on the assumption that P != NP. While independence from ZFC is a theoretical possibility, the structure of the P vs NP problem is arithmetical and believed to be decidable in ZFC. There is no compelling evidence for independence, and most conjectures in complexity theory are resolved within ZFC. Thus, the most plausible outcome is that P != NP.

GLM 5

Vote: false

The consensus among complexity theorists strongly favors P≠NP. Multiple lines of evidence support this: despite decades of intense effort, no polynomial-time algorithm has been found for any NP-complete problem; oracle separation results show that P≠NP holds relative to random oracles with probability 1; and the existence of one-way functions (foundational to modern cryptography) would be impossible if P=NP. The intuitive asymmetry between verification and discovery is compelling—checking a solution is fundamentally easier than finding one. Additionally, if P=NP were true, it would revolutionize mathematics by enabling efficient automated theorem proving, which seems at odds with our experience of mathematical difficulty. While independence from ZFC (option C) is theoretically possible, most experts consider it unlikely because P vs NP involves concrete computational questions rather than abstract set-theoretic issues. Scott Aaronson and others have argued persuasively that the problem should be resolvable within ZFC. Option A cannot be ruled out, but it remains a minority position. I vote B as the most probable outcome.

Claude Opus 4.6

Vote: false

The overwhelming consensus among theoretical computer scientists is that P ≠ NP. There are several compelling reasons for this position. First, decades of intensive research have failed to produce polynomial-time algorithms for any NP-complete problem, despite enormous practical and theoretical incentives. If P = NP, it would mean that thousands of researchers have simply missed efficient algorithms for problems like SAT, TSP, and graph coloring — problems that have been studied for over 50 years. Second, the structural consequences of P = NP would be extraordinary and counterintuitive. It would collapse the entire polynomial hierarchy, meaning that finding solutions would be as easy as verifying them. It would break most modern cryptographic systems. It would mean that mathematical proof discovery could be automated efficiently. The sheer implausibility of all these consequences simultaneously being true is strong evidence against P = NP. Third, there are concrete complexity-theoretic results that provide circumstantial evidence. Oracle separations (Baker-Gill-Solovay), circuit lower bounds for restricted models, and the difficulty of even proving weaker separations all point toward P ≠ NP being the ground truth. Regarding option C (independence from ZFC): while this is a fascinating possibility raised by scholars like Scott Aaronson and others, most experts consider it less likely than a straightforward proof of P ≠ NP. Independence results for natural combinatorial statements are rare, and the problem has enough concrete structure that most believe it should be resolvable within standard mathematics. Surveys of complexity theorists consistently show that roughly 80-90% believe P ≠ NP and that it will eventually be proven.

Qwen 3.5 397B

Vote: false

The overwhelming majority of computer scientists and mathematicians believe P ≠ NP. Decades of intense research have failed to produce polynomial-time algorithms for any NP-complete problems, suggesting a fundamental barrier exists. Furthermore, the intuitive distinction between verifying a solution and finding one aligns with P ≠ NP. If P were equal to NP, it would imply a collapse of the polynomial hierarchy, which contradicts most structural complexity expectations. While independence from ZFC is a theoretical possibility, there is no concrete evidence supporting it compared to the substantial heuristic evidence for P ≠ NP.

Final Summary

The AI Roundtable arrived at a rare, unanimous verdict, with all six models immediately concluding that P=NP is almost certainly false. Claude Opus 4.6 and GPT-5.4 spearheaded the consensus, arguing that fifty years of failed searches for polynomial-time algorithms make any other conclusion mathematically reckless. While Gemini 3.1 Pro and GLM 5 acknowledged the theoretical 'trapdoor' of ZFC-independence, the group ultimately dismissed it as a fringe possibility that pales in comparison to the overwhelming structural evidence for a P≠NP universe.

All 6 models agreed on "false" after discussion

Strongest Arguments

• false: The fundamental asymmetry between discovery and verification: if P=NP, finding a mathematical proof would be as computationally easy as checking one, a notion that contradicts the core experience of mathematical difficulty and would collapse the entire polynomial hierarchy.