Is P=NP?

TL;DR

P=NP has a 4% chance of being the broadly accepted resolution; the live expert consensus and the technical record point strongly to P≠NP.

Context

As of May 14, 2026, the Clay Mathematics Institute still lists P vs NP under its unsolved Millennium Prize Problems and states the problem as whether a solution that is easy to check is also easy to find (Clay Mathematics Institute). CMI’s prize rules, revised on September 26, 2018, require a qualifying publication, at least two years after publication, and general acceptance in the global mathematics community before CMI will consider a proposed solution (CMI rules). That matches this question’s resolution standard.

Because there is no deadline, I read the question mostly as a forecast about the mathematical truth of the standard classical statement P=NP, with a small adjustment for eventual acceptance. If P=NP is true but independent of accepted foundations, or true but never proved in a way the community accepts, this question would not necessarily resolve YES.

Evidence

The strongest quantified evidence is expert opinion. William Gasarch’s poll series is not a random sample, but it is the best repeated public survey of the relevant community. The units are respondent counts; the coverage windows are opinions collected in 2001, 2011, and 2018; the publication vintages are 2002, 2012, and March 2019; and the sample sizes are 100, 152, and 124 (Gasarch 2019 poll).

Across all three polls, 36 of 376 respondents chose P=NP, or 9.6%. Among respondents who chose either P=NP or P≠NP, 36 of 332 chose P=NP, or 10.8%. The most recent broad poll gave 15 of 124, or 12.1%, to P=NP (Gasarch 2019 poll). That is a real minority. It rules out a near-zero estimate.

The same 2019 article reports a sharper split among people Gasarch describes as known to have seriously thought about the problem: 99% said P≠NP and 1% said P=NP (Gasarch 2019 poll). I give this more weight than the broad poll because the resolution criterion is expert consensus. I do not take it literally as a 1% calibrated probability. The subgroup definition is informal, and the judgments are correlated. Still, it is the most decision-relevant poll result.

The technical record points the same way. P=NP would follow from a polynomial-time algorithm for any NP-complete problem. Decades of work have not found such an algorithm, despite large incentives and many routes through SAT, graph problems, integer programming, scheduling, proof search, and optimization. Ryan Williams gives the cleanest formulation of this evidence: the collective failure to find an efficient SAT algorithm over decades is evidence, because all NP-complete problems are SAT in disguise; he still assigns only 80% to P≠NP, which is a high-quality dissent from the stronger consensus (Williams, Some Estimated Likelihoods for Computational Complexity). I read Williams as a warning against overconfidence, not as enough to lift P=NP into double digits.

The structural implications also hurt YES. If P=NP, then several distinctions that complexity theory treats as deep would collapse; for example, NP would equal coNP, and the polynomial hierarchy would collapse to P (Aaronson survey). This is not proof. But it means P=NP is not one isolated surprise. It is a bundle of surprises.

Known barriers are the main counterweight to a simple consensus story. Baker, Gill, and Solovay’s relativization barrier, Razborov and Rudich’s natural proofs barrier, and Aaronson and Wigderson’s algebrization barrier show that broad families of proof methods cannot settle central P vs NP-style questions; Aaronson and Wigderson explicitly describe algebrization as a third barrier after relativization and natural proofs (Aaronson and Wigderson). These barriers explain why P≠NP has not been proved. They do not make P=NP likely. They mostly make me less willing to say the expert consensus is 99% reliable.

Purported proof history adds almost no directional evidence, but it matters for resolution. Gerhard Woeginger’s maintained page listed 116 attempted P-vs-NP resolutions through 2016 (Woeginger P-versus-NP page). The base rate of failed claims is high. So the event here should be treated as an accepted mathematical resolution, not as a future arXiv announcement.

My numerical model is a log-odds blend of four partly overlapping signals. I used 2.5% for the serious-expert signal, above the reported 1% to allow for correlated expert overconfidence; 8% for the broad-poll signal, below the 10.8% decided-poll share because many respondents are less resolution-relevant than active complexity theorists; 3% for the algorithmic and structural record; and 20% for Williams’s explicit contrarian likelihood, but with low weight because it is one expert’s personal probability. The weights were 40%, 25%, 25%, and 10% on log-odds. This gives 4.4%. I then round only slightly upward to 4.5% for the asymmetry that a P=NP proof could be a surprising algorithm, while a P≠NP proof likely needs major lower-bound machinery.

What's non-obvious

The obvious read is that experts think P≠NP, so P=NP should be 1% or lower. I think that is too low. The broad Gasarch polls never put P=NP below 8% of respondents, and Williams’s 80% estimate for P≠NP shows that some leading complexity theorists deliberately keep a large humility margin (Gasarch 2019 poll, Williams). Deep mathematical consensus is strong evidence, but it is not theorem-level evidence.

The other mistake is to treat proof barriers as positive evidence for P=NP. They are not. They say many standard methods cannot prove P≠NP. That makes the lack of a separation proof much less surprising in a world where P≠NP is true. The failed algorithm search and the structural collapse implications still push hard against equality.

Limitations

This is a single mathematical fact, not a repeatable event. There is no clean base rate for famous open conjectures with strong expert consensus but no proof. The Gasarch polls are informal and have changing samples. The 2019 expert subset is especially useful, but its exact sampling frame is not a transparent random draw (Gasarch 2019 poll).

The no-deadline framing also matters. A finite-horizon version would have a much lower YES probability because CMI-style acceptance requires publication, a two-year wait, and broad acceptance (CMI rules). Here I forecast the eventual accepted answer under the stated criteria. My final probability is 4.5%.