Will P vs NP be resolved before 2030?

- [2026-10-28] No source I found shows P vs NP resolved. MathWorld (last updated 2026-10-26) lists P vs NP as "unsolved" on the Clay Mathematics Institute (CMI) status table. https://mathworld.wolfram.com/MillenniumPrizeProblems.html - [2026-10-26/27] OpenAI announced that an internal model disproved the Exponential Time Hypothesis (ETH), with a Lean-verified proof. ETH is a stronger assumption than P≠NP, and OpenAI calls it closely related to P vs NP. A reported Anthropic result is also in the news. Reports say Ryan Williams, Impagliazzo and Paturi are reviewing the ETH claim. MindStudio states it is "not yet" confirmed, and says P=NP is not implied. OpenAI president Greg Brockman is quoted saying P probably still does not equal NP. https://www.mindstudio.ai/blog/exponential-time-hypothesis-disproven - [2026-10-27] The same announcement reportedly also covered the Kakeya conjecture and Yang-Mills existence. Sources describing it are secondary, such as winzheng.com. They say the three results were submitted for independent verification. https://www.winzheng.com/en/article/openai-models-solve-three-math-problems-verification-debate - [2026-10-27] A Manifold "Does P = NP?" market showed about 12% YES after the ETH news. This is a different venue and a differently worded question. https://tech-insider.org/does-p-np-manifold-market-odds-volume-spike-2026/ - [2026-09-08] OpenAI claimed an AI-generated counterexample for Navier–Stokes, with a Lean formalization. It said it would not claim the prize. The claim is disputed and faces a priority dispute. https://openai.com/index/navier-stokes-solution/ - [2026-09-11] CMI called Navier–Stokes "apparently" settled but awarded no prize. It still labels the problem "Active". P vs NP remains "unsolved". https://mathworld.wolfram.com/MillenniumPrizeProblems.html - [2026-09-11] CMI rules, as relayed by MathWorld, require publication in a qualifying outlet, general acceptance, and at least two years since publication before CMI considers a solution. A CMI prize for P vs NP before 2030 therefore requires a publication by about early 2028. The Kalshi rules also allow other routes: a peer-reviewed article, recognition by an authoritative body, or two Source Agencies reporting a resolution with no unaddressed error. https://mathworld.wolfram.com/MillenniumPrizeProblems.html - [2026-09-11] OpenAI told the New York Times it has made "substantial progress on another Millennium Prize problem". It did not name the problem. Yang-Mills now appears to be that problem, per the October announcement. https://explainx.ai/blog/millennium-prize-problems-ai-solved-fact-check-2026 - [2026-09 snapshot] A fact-check article reported no AI progress on the core P vs NP question as of September 2026. https://explainx.ai/blog/millennium-prize-problems-ai-solved-fact-check-2026 - [2026-06-10] Computational Complexity blog (Fortnow/Gasarch) commenters agree a P≠NP proof is not around the corner. Some argue it should not be hard to formalize complexity theory. https://blog.computationalcomplexity.org/2026/06/respect-p-v-np-problem.html - [2026-06] A June 2026 arXiv preprint claims a Lean-verified proof of P=NP. ProofAtlas says it is under review, that its repository admits gaps, and that Clay still labels the problem unsolved. Fringe claims are common (Zenodo, SSRN) and none is accepted. https://www.proofatlas.ai/collaboration/p-vs-np/ - [2026-07-19] A Zenodo survey of P vs NP barriers, checked against the literature through 2026-07-19, claims no proof. https://zenodo.org/records/21440860 BOTTOM LINE: As of 2026-10-28 P vs NP is unresolved. AI labs are now producing fast, Lean-verified claims on related and adjacent problems (ETH refuted, Navier–Stokes counterexample). That raises the tail risk of a credible claim before 2030, but no credible P vs NP claim exists yet. The CMI two-year rule makes a CMI prize unlikely before 2030. The Kalshi resolution paths (peer-reviewed publication, recognition by an authoritative body, or two Source Agencies reporting it) do not require the two-year wait. NOT FOUND: I could not load claymath.org directly and relied on MathWorld's relay of its status. I found no scheduled event before 2030 tied to P vs NP resolution. I found no peer-reviewed or authoritative-body confirmation of any P vs NP claim. I could not fetch the OpenAI or Anthropic primary posts on ETH. The June arXiv Lean-verified P=NP preprint is known only through ProofAtlas.

- [2026-09-11] Clay Mathematics Institute (CMI) still lists P vs NP as unsolved. Only Poincaré is solved, and Navier–Stokes is labelled "Active" after an OpenAI counterexample claim in September 2026. CMI's own rules say a proposed solution must be published in a Qualifying Outlet, must have been published for at least two years, and must have received general acceptance. https://mathworld.wolfram.com/MillenniumPrizeProblems.html - [2026 (page current)] I fetched CMI's rules page, which confirms those three conditions. The Kalshi rules here are looser (an authoritative-body statement, or two source agencies reporting). The CMI process is much slower than the Kalshi route. Any claim made now could not reach CMI prize eligibility before 2029 at the earliest. https://www.claymath.org/millennium-problems/rules/ - [2010-03-18] Base rate: of 7 Millennium problems announced in 2000, 1 has been resolved. Perelman posted his proofs in 2002–03, and CMI awarded the prize on 18 March 2010, about 7 years after the preprints. That is roughly 1 in 26 years across the set, and P vs NP specifically has stayed open for 55 years (posed 1971). https://www.claymath.org/events/millennium-prize-problems-lecture-series/ - [2026-06-10] Lance Fortnow, a leading complexity theorist, says an AI-generated P≠NP proof is not around the corner. He doesn't expect to see a proof "in my lifetime", by human or machine, and says there is currently no viable approach to settling it. I fetched and read this post. https://blog.computationalcomplexity.org/2026/06/respect-p-v-np-problem.html - [2018 poll, per Wikipedia] Gasarch's polls found 61% (2001), 83% (2011) and 88% (2018) of respondents believed P≠NP. In 2018, 99% of the experts did. The polls say nothing about timing. https://en.wikipedia.org/wiki/P_versus_NP_problem - [2012] Timing opinions in Gasarch's 2012 poll varied widely. Manindra Agrawal predicted resolution by 2030, so that forecast is due now. Eric Allender said within 25 years, with low confidence. Daniel Apon said "2000 years". Agrawal's 2030 prediction is the only one I saw that targets this window, and it has not come true so far. https://www.cs.umd.edu/~gasarch/papers/poll2012.pdf - [2026-08-07] ProofAtlas says there is a recent proof claim under review. A June 2026 arXiv preprint (Arthanari) claims a Lean-verified proof of P=NP, but it relies on axioms. A July package claims P≠NP with two axioms. Neither has been independently audited or published in a qualifying outlet. Clay still lists the problem as unsolved. Crank and unreviewed claims are routine (a Zenodo "complete proof" from February 2026, SSRN papers, and others). https://www.proofatlas.ai/collaboration/p-vs-np/ - [2026-10-27] Manifold has a "Does P = NP?" market at about 11.9% YES. The source article says it is a play-money market with thin history and a spike tied to an AI "ETH disproved" announcement. That announcement does not resolve P vs NP, and the article says as much. I did not verify the exact resolution wording of that market. https://tech-insider.org/does-p-np-manifold-market-odds-volume-spike-2026/ - [2026-10-28] The AI-math context is the main upside risk. AI claims include the Navier–Stokes counterexample and a Lean-checked disproof of the Exponential Time Hypothesis (ETH). OpenAI's Greg Brockman said P probably still ≠ NP. An explainer says no AI progress exists on the core P vs NP question. https://www.mindstudio.ai/blog/exponential-time-hypothesis-disproven - [2026-09-11] Related Kalshi contracts: "Clay awards a Millennium Prize before 2029" was quoted at 27¢/32¢ on 2026-09-11, a figure that is most likely mostly Navier–Stokes. An OpenAI "another Millennium solution" contract at that date priced around 34¢ and later spiked on a Hodge rumour. There is also a Kalshi "solved next before 2045" series (KXMILLENNIUMNEXT-45). These are thin markets driven by AI hype, not calibrated forecasts. https://www.oddsshopper.com/articles/prediction-markets/openai-millennium-prize-odds - [2026-10-17] A Kalshi Yang–Mills "next before 2045" contract jumped about 20 points on volume equal to its lifetime turnover with no underlying breakthrough. This shows these Millennium contracts react to sentiment. https://tech-insider.org/yang-mills-solved-next-before-2045-kalshi-odds/ - [2026-10-22] Secondary sources reviewed in late October 2026 still show P vs NP as open, with no accepted proof or counterexample. https://geniuses.club/blog/millennium-prize-problems BOTTOM LINE: The outside view is strongly against resolution. P vs NP has been open for 55 years, only 1 of 7 Millennium problems has fallen in 26 years, leading experts see no viable approach, and CMI's two-year rule alone would delay any prize. The main tail risk is a Kalshi trigger that needs only two news outlets or one authoritative-body statement, which an AI-driven or crank claim could in principle meet. I found no sign of an accepted claim. NOT FOUND: I found no formal expert survey with a 2030-specific probability since 2018. I did not find the Gasarch 2018 poll's timing answers. I did not find a Metaculus or Polymarket P vs NP market. I did not find a Kalshi P vs NP price, and I did not look for one on purpose.

YES means: P vs NP is resolved before 1 Jan 2030 under the Kalshi rules. That needs a peer-reviewed proof or disproof, an authoritative-body statement, or two Source Agencies reporting a resolution tied to a public written work, with no unaddressed error. - [2026-10-07] OpenAI released 722 manuscripts claiming to solve 372 open problems. The TCS items include a unique games proof, a Lean-formalized matrix multiplication bound of n^2.25, and L=BPL. Fortnow wrote that, if they hold up, they are more TCS progress in a day than in the prior three decades. He added that AI solved under 10% of the problems it was given. He also said none of it gets us closer to P vs NP. He had Claude build a page indexing these results. A commenter posted an unverifiable rumor that OpenAI had resolved P vs NP. Gasarch replied that if it were resolved he would bet on P=NP. He said he cannot imagine a P≠NP proof with 2026 math, but he has been surprised by AI before. https://blog.computationalcomplexity.org/2026/10/open-no-more.html - [2026-10-26] The ETH refutation is a deterministic 2^o(n) algorithm for 3-SAT and #3-SAT. OpenAI's paper is 67 pages, and OpenAI says it was Lean-verified. It is the first widely reported subexponential-time SAT algorithm. That moves the question from "no approach" toward "an algorithm exists", which is the direction a P=NP route would need. The paper itself disclaims any bearing on P vs NP. https://www.kucoin.com/news/flash/openai-claims-ai-disproves-exponential-time-hypothesis - [2026-10-27] Ryan Williams, who is organizing the field's review, told TechCrunch that computer scientists should put a little more weight on P=NP. He said the result calls into question intuitions about how tractable hard NP problems are. He also said OpenAI's algorithm is probably not the best possible. This is a shift in tone from a leading complexity theorist. https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ - [2026-10-27] Two labs reached the same ETH result independently on 16 Oct. Anthropic's paper is credited to "Claude, Anthropic" and used about 14M output tokens with no human input in discovery. Both are Lean-checked. A rapid replicated pace on core complexity problems supports the idea that a next step could be a polynomial-time algorithm. That step would also be a quickly checkable artifact. https://officechai.com/ai/anthropic-says-claude-too-independently-disproved-exponential-time-hypothesis-on-the-same-day-as-openai/ - [2026-10-26] OpenAI's pipeline now pairs each result with an expert panel: Gross for Yang–Mills, and Impagliazzo, Paturi and Williams for ETH. OpenAI funds a conference and publishes the raw data. That gives an "authoritative written communication" route. It also gives fast two-outlet coverage (Washington Times, TechCrunch, Qz, TechTimes all covered ETH within a day). A P vs NP claim run through the same process could meet the Kalshi "two Source Agencies" test within days. https://www.unite.ai/openai-says-its-models-produced-three-proofs-in-math-cs-and-physics/ - [2026-10-26] The Yang–Mills claim shows the path: lab claim, Lean proof, named expert panel, wide media coverage. It also shows a problem for the Yes case. A claim in this form has not yet produced a CMI prize or peer-reviewed publication. https://www.techtimes.com/articles/329596/20261027/openai-claims-yang-mills-proof-nobel-laureate-whose-discovery-enables-it-now-reviews-it - [2026-08 arXiv] A paper on verification and adjudication notes that in May 2026 OpenAI's Erdős unit distance disproof was matched the same day by a human-verified writeup from Alon, Bloom, Gowers, Litt and Sawin. Expert validation of AI proofs can therefore come within hours when the result is tractable. This is the mechanism that would let the "no unaddressed error" clause hold quickly. https://arxiv.org/pdf/2608.28997 - [2026-10] The most likely Yes path: a lab model finds a polynomial-time SAT algorithm (P=NP) or a lower-bound proof, with a Lean proof of the formal statement, as the ETH work was done. A named panel then confirms it, and two outlets report it citing the public paper. This would settle under the third Kalshi route before the CMI two-year wait. Prerequisites: (a) the Lean statement must match the CMI formulation, which the ProofAtlas-covered June P=NP preprint relies on axioms to avoid; (b) no unaddressed error; (c) both the Kalshi market and Fortnow-style experts treat the formal result as the CMI statement. - [2026-10-27] Counter-weight: reviews are still pending even for ETH. None of the sources I read say any lab has a P vs NP result. Reporting uniformly says ETH does not imply P=NP. https://www.popularai.org/p/openai-exponential-time-hypothesis-proof BOTTOM LINE: The strongest Yes case rests on the pace of AI progress. Within a month, AI labs have produced Lean-checked results on adjacent problems, such as ETH, and Williams now says P=NP deserves slightly more weight. Kalshi's looser two-outlet route needs no two-year wait. No source shows a credible P vs NP claim yet. NOT FOUND: Any credible lab or expert claim of a P vs NP resolution, and any peer-reviewed or authoritative-body statement on one. I could not read the OpenAI or Anthropic primary posts. I did not look for this contract's own price or the Kalshi list of Source Agencies.

YES means: P vs NP, in the Clay Mathematics Institute (CMI) formulation, is resolved before 1 Jan 2030. Resolution can come through a peer-reviewed publication, a statement from a recognised authoritative body, or two Source Agencies reporting a resolution with no unaddressed error. The resolution must not have been retracted. - [2026-10-27] The new ETH results cut against the "AI is closing in" story. The OpenAI paper itself says it proves no superpolynomial lower bound for NP. The algorithm's gain is tiny: running time of about 2^(n/√log n). Anthropic's paper puts the crossover with brute force at roughly 10^10,000 variables. A subexponential SAT algorithm is far from a P vs NP resolution. https://pasqualepillitteri.it/en/news/27422/openai-eth-disproved-local-smoothing-kakeya ; https://www.techtimes.com/articles/329569/20261027/two-ai-systems-disproved-same-complexity-conjecture-independently-same-day-800-papers-risk.htm - [2026-10-27] ETH is a strengthening of P≠NP. A refutation therefore removes a conditional hardness assumption. It does not move toward settling P≠NP. Techtimes says so directly, and so does the OpenAI president's remark that P probably still ≠ NP. The one reviewer-side comment I saw that touches P=NP is from Ryan Williams. He reportedly said theorists should assign somewhat more plausibility to P=NP, and that individual views differ. This is a reported comment, not a proof. https://www.kucoin.com/news/flash/openai-and-anthropic-claim-to-refute-exponential-time-hypothesis - [2026-10-27] Even the ETH claim is unverified. The Impagliazzo–Paturi–Williams open letter says their involvement is not certification. Anthropic's own footnote says its Lean code was reviewed only by other AI instances, and no person is known to have checked it by hand. Lean/natural-language mismatches are a documented failure mode. If the best-credentialed AI result in complexity theory is still pending expert review, a P vs NP claim would need a longer vetting path before two outlets could report it as "resolved". https://www.techtimes.com/articles/329569/20261027/two-ai-systems-disproved-same-complexity-conjecture-independently-same-day-800-papers-risk.htm - [2026-10-27] Labs have shown restraint on formal claims. OpenAI said it would not claim a prize for the Yang–Mills material, and the Gross physics panel is withholding the AI-produced Yang–Mills materials for now. This suggests labs will not rush to declare a P vs NP "resolution". Labs also now route claims through named reviewers who say they are not certifying, which slows any "authoritative" confirmation. https://www.technology.org/2026/10/28/openai-exponential-time-hypothesis-disproof-lean-kakeya/ - [2026-10-04] The explainx fact-check of Millennium problems, dated October 4, says no AI system has made progress on the core P vs NP question. It gives a structural reason: no partial bound to nudge and no "almost there" proof to complete. It also judges the Navier–Stokes "solved" claim as not having held up. That shows how fast headline claims erode. https://explainx.ai/blog/millennium-prize-problems-ai-solved-fact-check-2026 - [2026-10-01] Crank-claim noise is high. A researchgate preprint dated 1 Oct 2026 asserts P=NP from the PCP/"holographic proof" theorem. That is another unreviewed claim, adding to the June/July Lean-axiom packages. Such claims are routine and none is peer reviewed. They are the main way a "two Source Agencies" trigger could be gamed, but the clause's no-unaddressed-error condition defeats any claim with an identified error. https://www.researchgate.net/publication/415111780_PNP_by_proving_that_verifying_in_polylogarithmic_time_allows_you_to_prove_in_polynomial_time - [2026-10-28] Fetched the CMI rules page (last modified 2024-06-17). Its three conditions are unchanged: a Qualifying Outlet, two years since publication, and general acceptance. Kalshi's text says "official Clay Mathematics Institute formulation" and the "Payout Criterion". A reader could therefore require CMI-style acceptance. Kalshi's wording is looser, but a literal reading leaves ambiguity that favors NO when a claim is merely contested. https://www.claymath.org/millennium-problems/rules/ - [2026-10-28] Deadline arithmetic. About 14 months remain (to 2030-01-07; the rules say 1 Jan 2030). A journal route needs a submission, refereeing and publication. For a proof of this length and novelty, refereeing is typically many months to years. Perelman's preprints (2002) took about 4 years to reach confirmatory exposition papers. A new P vs NP paper would have to appear and clear peer-reviewed publication within roughly one year. https://www.sciencenews.org/article/million-dollar-math-prize-awarded-not-necessarily-accepted - [2026-10-04] The Kalshi "Source Agency" wording is vague. Kalshi's own rulebook defines a Source Agency as the agency that publishes the underlying value. Here no named agency exists, which adds adjudication friction. Ambiguous or disputed claims are likely to be settled conservatively or delayed. Kalshi's help page says markets are not re-settled because an event appears to have occurred. https://help.kalshi.com/en/articles/13823821-market-faqs - [2026-10-27] The overall rate of AI-claimed results is high, but they cluster on problems with incremental routes (ETH, Kakeya, Navier–Stokes). The ETH disproof also strengthens the case that P vs NP is hard. It shows that progress on "adjacent" assumptions does not carry over. The Fortnow post of 2026-06-10 (already cited by colleagues) predates all of this and has not been contradicted by any named expert. https://www.winzheng.com/en/article/openai-models-solve-three-math-problems-verification-debate BOTTOM LINE: A YES needs a credible, accepted P vs NP resolution to appear and be confirmed within about 14 months. The newest AI results (ETH, Oct 26–27) are narrow, still unreviewed, and explicitly leave P vs NP untouched. The status quo of no accepted claim is the default outcome. NOT FOUND: I found no peer-reviewed P vs NP paper, no authoritative-body statement, and no credible expert endorsing any P vs NP claim. I could not find a named Source Agency for this Kalshi contract. I did not retrieve the Kalshi contract page, by design. I did not find any 2026 expert survey with a timing estimate.