Does P = NP? Manifold Traders Pushed 85% of the Market’s Volume in 24 Hours. A dormant Manifold Markets contract on the most famous open problem in computer science suddenly came alive, with roughly Ṁ28,800 of its Ṁ33,800 lifetime volume changing hands in a single day and 163 unique bettors on the book. The trigger was not a proof of P = NP. It was a pair of competing claims from OpenAI and Anthropic that AI models had refuted the Exponential Time Hypothesis, a related but strictly weaker conjecture. Traders priced the news at 12% implied probability of YES as of October 27, 2026, which is a modest move for a market that has to wait on a “widely-accepted proof” to settle.
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What the Market Actually Asks
The contract is “Does P = NP?” on Manifold Markets, created by user Isaac King on March 3, 2022. It is a binary market with no practical close date, and its resolution criteria are unusually strict: it resolves “once a widely-accepted proof exists that P does or does not equal NP.” There is no deadline, no partial credit, and no mechanism for resolving on a preprint that experts are still arguing about.
That design matters for reading the price. Manifold runs on mana, a play-money currency, so the dollar stakes are not comparable to a regulated exchange. What Manifold does provide is a liquid, fast-moving sentiment gauge among a technically literate user base, which is why its math and computer science markets tend to react to research news hours before mainstream coverage lands. For background on the platform’s mechanics and limitations, see our Manifold Markets review.
The Odds and What They Imply
As of October 27, 2026, the Manifold market prices YES at 11.9%, which rounds to 12%. Over the preceding 24 hours it recorded roughly Ṁ28,800 in volume against a lifetime total of about Ṁ33,800, meaning approximately 85% of everything ever traded in this market traded in one day. Unique bettors stand at 163. Posted liquidity is about Ṁ6,000.
Read literally, 12% means the marginal trader thinks there is roughly a one-in-eight chance that a widely accepted proof eventually lands on the P = NP side. That is almost certainly higher than the consensus among complexity theorists, who overwhelmingly expect P ≠ NP. Part of the gap is structural: in a market with no close date and no interest paid on tied-up mana, the price of a long-dated YES share is depressed by the time value of capital, and conversely NO sellers have little incentive to grind the price to 1%. Our explainer on implied probability covers why a quoted percentage is not a clean forecast once settlement is open-ended.
What Is Driving the Move
The catalyst is datable. On October 26 and 27, 2026, OpenAI announced that its internal models had produced three research results, which the company’s Dan Roberts summarized as covering harmonic analysis, complexity theory, and quantum field theory. Per Engadget, the claims are that the models established Sogge’s local smoothing conjecture in every spatial dimension, that the Exponential Time Hypothesis “is false,” and that they produced an existence proof for interacting quantum field theories with gauge fields and fermions in four dimensions.
Hours later, Anthropic posted its own result. Scientific American reported that Anthropic released an independent proof disproving the same hypothesis, arriving at the same conclusion as OpenAI. The Anthropic paper, titled “A 2^o(n) Time Algorithm for 3-SAT,” was added with a Lean 4 formalization to the company’s public formal-math repository on GitHub on October 27. Two frontier labs claiming the same decades-old conjecture within a day of each other is what moved the market, not any claim about P = NP itself.
The Distinction the Market Is Pricing
This is the part that separates a 12% print from a panic. The Exponential Time Hypothesis, formulated by Russell Impagliazzo and Ramamohan Paturi in the late 1990s, says that 3-SAT cannot be solved in subexponential time. ETH is strictly stronger than P ≠ NP: if ETH is true, then P ≠ NP follows, but the implication does not run backward.
So refuting ETH does not prove P = NP. The Quantum Insider put the point plainly: disproving ETH would not settle P versus NP, because “subexponential running time can remain much slower than the polynomial running time at issue in that question.” Scientific American likewise noted the ETH result “doesn’t necessarily translate to a transformative breakthrough about P versus NP.” P = NP requires a polynomial-time algorithm for SAT. A subexponential algorithm that is still superpolynomial leaves the Millennium Prize question exactly where it was.
The practical caveat is larger still. The claimed bound is asymptotic. For 3-SAT formulas with five clauses per variable, the Anthropic paper states its bound does not beat the 2^n of brute-force search until n exceeds 10 to the power 10,000, a regime with no physical meaning.
Who Is Involved
Neither claim is verified. OpenAI paired its complexity-theory release with an external review group rather than a journal submission. According to Al Jazeera and The Quantum Insider, that group is led by Impagliazzo and Paturi at UC San Diego, who helped formulate ETH in the first place, together with Ryan Williams at MIT. Williams told Al Jazeera: “We are working to verify and examine their proof and determine if it holds up.” Reporting has emphasized that participation in the review is not an endorsement of the conclusion.
On the mathematics result, Alex Iosevich of the University of Rochester called the local smoothing claim a “historic development,” noting that the conjecture was formulated by his doctoral adviser, Chris Sogge. Iosevich also told Al Jazeera that “the scientific enterprise as a whole is going to make giant leaps as a result of the AI revolution.” For the physics claim, OpenAI deferred release to an external group that includes Nobel laureate David Gross and has not published the underlying materials. OpenAI said in its own post on working with academic communities that it will not seek academic credit for the results.
Timeline of the Move
| Date | Event | Status as of Oct 27, 2026 |
|---|---|---|
| May 24, 2000 | Clay Mathematics Institute names P vs NP a Millennium Prize Problem, $1M award | Unclaimed |
| Late 1990s | Impagliazzo and Paturi formulate the Exponential Time Hypothesis | Now contested |
| Mar 3, 2022 | Manifold market “Does P = NP?” created by Isaac King | Open, no close date |
| Oct 16, 2026 | Anthropic says an internal Claude research model found the core 3-SAT algorithm | Company claim, unverified |
| Oct 26, 2026 | OpenAI announces three AI-generated results, including that ETH is false | Under external review |
| Oct 27, 2026 | Anthropic publishes its 3-SAT paper plus Lean 4 formalization on GitHub | Under external review |
| Oct 27, 2026 | Manifold market trades ~85% of lifetime volume in 24h; 163 bettors; YES at 12% | Live |
Why a Tech and Finance Reader Should Care
P vs NP is not an abstraction for anyone holding cryptographic risk. Public-key cryptography rests on problems believed to be hard, and a genuine polynomial-time SAT algorithm would be a solvency event for large parts of digital infrastructure. That is precisely why the 12% print is informative: traders who read the papers concluded the cryptographic threat model did not change. Markets that do price that specific risk, such as the RSA-2048 factoring market, are a cleaner instrument for it than a P = NP contract with no deadline.
The second signal is about AI capability rather than complexity theory. Two labs independently produced long-form proof attempts in the same window, formalized at least partly in Lean, with Anthropic saying its model found the algorithm without human input. Whether or not the proofs survive review, that is a measurable shift in what these systems attempt. Manifold already hosts a market on whether AI solves a Millennium Prize Problem, and our coverage of the OpenAI announcement itself goes deeper on the three individual claims.
What to Watch Next
Three things will determine whether 12% holds. First, the Impagliazzo, Paturi, and Williams review. Boundary-pushing complexity results have a documented history of containing subtle errors, and a retraction or a narrowing of the claim would likely drain this market back toward its pre-news baseline. Second, the scope of the Lean formalization. A machine-checked proof of the full theorem is a different object from a formalization of selected lemmas, and that distinction is auditable in the public repositories. Third, whether anyone converts a subexponential bound into something closer to polynomial. That, and only that, is the path by which this market moves materially.
FAQ
Does refuting the Exponential Time Hypothesis prove P = NP?
No. ETH is a stronger statement than P ≠ NP. Refuting ETH removes one hardness assumption but leaves P vs NP open, because a subexponential algorithm can still be far slower than polynomial time.
What are the current odds, and where do they come from?
Manifold Markets priced YES at 11.9%, roughly 12%, as of October 27, 2026, on the market “Does P = NP?” Figures for volume and bettor count in this article come from the same market.
Has either proof been verified?
Not as of October 27, 2026. OpenAI’s claim is under review by an external group, and Anthropic’s paper was published to a public repository the same day. No independent confirmation had been reported.
Can I trade this market for real money?
Manifold operates primarily in mana, a play-money currency, so this specific contract is not a real-money position. See how prediction markets work and our Kalshi vs Polymarket comparison for venues that do settle in cash.
Would a P = NP proof break encryption?
A constructive polynomial-time algorithm for an NP-complete problem would threaten widely used public-key schemes. Nothing announced in October 2026 claims such an algorithm.
The Bottom Line
A long-dormant Manifold market on P = NP traded roughly 85% of its lifetime volume in 24 hours and still only sits at 12% YES as of October 27, 2026. That combination is the story: enormous attention, almost no change in belief. Traders correctly separated a contested claim about the Exponential Time Hypothesis from the far larger claim that P equals NP, and they are waiting on Impagliazzo, Paturi, and Williams rather than on a press release.
Sources
- Manifold Markets — Does P = NP? (odds, volume, bettor count)
- Scientific American — Dueling proofs from OpenAI and Anthropic astound scientists
- Al Jazeera — AI makes discoveries in mathematics, computer science, physics, OpenAI says
- Engadget — OpenAI announces new discoveries in mathematics, computer science and physics
- The Quantum Insider — OpenAI computing claim faces review
- OpenAI — Working with academic communities to build scientific understanding
- GitHub — anthropics/formal-math, ETH directory and Lean 4 formalization
- Wikipedia — Exponential time hypothesis (Impagliazzo and Paturi, relation to P vs NP)
- Clay Mathematics Institute — Millennium Prize Problems
Disclaimer: Prediction markets carry risk and nothing here is investment, trading, or betting advice. Prices and implied probabilities are snapshots as of October 27, 2026 and change continuously. Market availability is restricted by jurisdiction: Polymarket is not available to US persons, Kalshi is a CFTC-regulated US exchange, and Manifold operates primarily in play-money mana. Participants must meet applicable age requirements of 18+ or 21+ depending on venue and location. If gambling is causing you harm, call 1-800-GAMBLER for confidential help.
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