The Exponential Time Hypothesis Is False (Probably)

Something happened this week that doesn’t have a good headline yet.

On 16 October 2026, an OpenAI model produced a natural-language proof that the Exponential Time Hypothesis is false. The next day it formalised that proof in Lean — the proof assistant where a single unjustified step collapses the entire argument. On 25 October the manuscript went public. On 26 October OpenAI announced it, alongside two other results: a proof of the Kakeya conjecture in harmonic analysis, and an existence proof for quantum chromodynamics that includes pure Yang–Mills theory — one of the seven Clay Millennium Prize Problems.

Anthropic’s models arrived at a refutation of ETH independently. Both labs released. The three people who formulated ETH — Russell Impagliazzo, Ramamohan Paturi and Ryan Williams — are now running the public examination of it.

None of it is verified. All of it is extraordinary. Here is what is actually going on.

What ETH actually says

Most people have heard of P versus NP. ETH is the sharper, uglier cousin.

P versus NP asks a broad question: if a proposed answer to a problem can be checked quickly, can the answer also be found quickly? Nobody has proved either way, and most researchers assume the answer is no.

3-SAT is the canonical hard problem. Give a machine a Boolean formula in conjunctive normal form — clauses joined by “and”, each containing at most three literals joined by “or” — and ask whether some assignment of true/false to the variables satisfies every clause. Every problem in NP can be rewritten as 3-SAT, which is why it sits at the centre of the whole thing.

The Exponential Time Hypothesis, formulated by Impagliazzo and Paturi around 1999–2001, says something more precise and much more uncomfortable than “3-SAT is hard.” It says there is no deterministic algorithm that solves every 3-SAT instance with n variables in 2^o(n) time. Subexponential. The exponent has to keep climbing no matter how cleverly you squeeze the base.

Here is the wall it built. ETH says you can lower the constant in your exponential, but you cannot change the shape of the curve. Best known result before this week was Ryan Williams’ 2016 work, giving roughly 2^0.39n — still exponential, just with a smaller constant bolted on.

And ETH did not sit in a vacuum. Hundreds of published papers — across graph theory, computational biology, databases, computational geometry — use ETH as a premise to derive conditional hardness results. “This problem needs at least n^1.5 time, assuming ETH.” Those papers are not wrong if ETH falls. They are just resting on a leg that may have been pulled out.

What was actually claimed

The new algorithm counts the satisfying assignments of a fixed-width Boolean formula, using polynomial space, in roughly 2^(n/√log n) time — with the manuscript stating a slightly sharper exponent of n·log log n / √log n.

Read that as: the exponent is no longer a fixed fraction of n. It is a quantity that grows slower than n. The shape of the curve genuinely changed.

OpenAI’s account of the timeline is unusually specific, and that specificity is itself interesting. The model found the natural-language disproof on 16 October. It formalised the result in Lean on 17 October. The manuscript is dated 25 October. The public announcement came on 26 October, once three separate academic working groups had been assembled — one per discipline — with different release procedures for each. Mathematicians got the work dropped on GitHub. Computer scientists asked for the raw data to be public and for OpenAI to fund a conference about it. Physicists reserved the right to decide the release format themselves.

That is a company that knows it just dropped something heavy on a working field and handled it with more care than credit.

The part nobody wants to say out loud

None of this is verified.

Let me be precise about the gaps, because they are the interesting part.

Gap one: the Lean artifact covers a weaker bound than the headline. The public Lean development establishes the bound with a (log log n)² factor. The manuscript claims the sharper n·log log n / √log n exponent. Both are subexponential, so either one refutes ETH — the formal artifact, if rebuilt, would still be enough to kill the hypothesis. But the number in the abstract is not the number the machine checked. The scope note does not claim to verify the sharper estimate.

Gap two: “verified in Lean” means OpenAI’s own pipeline verified it. Nobody outside has independently rebuilt the development from scratch. That is the actual task now underway, and it is not a formality. Rebuilding someone else’s Lean proof is genuinely hard work — it is the mathematical equivalent of re-deriving a result from raw ingredients rather than trusting the author.

Gap three: it is a preprint. No journal has seen it. And the reviewing group has been careful to say so itself: “Our involvement should not be taken as certification of the claim. A result of this significance requires careful, independent scrutiny.”

And there is a fourth thing that is easy to miss, which is the real intellectual work now sitting on human shoulders.

A machine checker verifies that a formal statement follows from the axioms. It does not verify that the formal statement means what the authors think it means. Whether the machine model, the input encoding and the runtime bound in the Lean file faithfully correspond to real-world 3-SAT is a translation problem, and translation problems are exactly where humans are good and machines are blind.

Checking the derivation and interpreting the definitions are separate jobs. The first is now automated. The second is where the next six months of work happens.

Two labs, one algorithm, and a credit problem

Here is the detail I keep coming back to.

Anthropic mathematician Levent Alpöge publicly suggested that OpenAI’s solutions were based on pre-existing work. Hours later, Anthropic released its own ETH refutation. Both labs’ models appear to have landed on the same 2^(n/√log n) bound, on the same day, independently.

Two readings, and they point opposite directions.

The optimistic reading: this is how good science works. Multiple independent derivations converging on the same result is the strongest possible evidence short of verification. When Kepler and Newton independently derived the orbit of Mars, we did not spend a decade arguing about who got there first — we concluded the answer was real.

The uncomfortable reading: if both models were trained on the same literature, then “independence” may be an illusion. Both systems may have independently retrieved and re-derived the same existing preprint. Alpöge’s point was, arguably, not about priority at all. It was that both labs might be reporting a rediscovery as a breakthrough — and that a frontier lab announcing “we refuted ETH” is making a far stronger claim to the world than “our model reproduced a result that is already in the literature.”

Both readings might be true at once. The resolution is the same either way: the raw data is public, the Lean is public, and the review group has been invited to check whether this is genuinely new or merely newly packaged.

That is the correct scientific process. It is also, not coincidentally, the most aggressive possible counter to the marketing claim of a labs’ “AI made a breakthrough no human could.”

Why this is bigger than one theorem

OpenAI announced three results, not one. Kakeya. ETH. Yang–Mills plus QCD plus a completion of the Standard Model.

The physics one deserves more attention than it has received. It is not a partial result. It is a rigorous existence proof for interacting quantum field theories in four dimensions, including pure Yang–Mills — the problem David Gross, Igor Klebanov and Edward Witten received the Nobel Prize for studying and which has sat on the Clay Millennium list since 2000.

And here is the warning sign that comes attached to it. OpenAI’s earlier Navier–Stokes proof ran to 166 pages, and Scientific American reported that more than a month after release, nobody — including, reportedly, OpenAI itself — fully understood it. There were earlier waves of this pattern across the field.

A proof nobody can reproduce is not a result. It is a rumour with a PDF attachment.

This is the failure mode that matters, and it is not a reason to dismiss the ETH claim — the ETH work has been released with unusually good faith, with public raw data and a funded conference and senior reviewers who have staked their reputations on it. It is a reason to watch what happens to verification infrastructure. If a field produces proofs at machine speed and cannot produce the human capacity to check them, it ends up in a worse position than when it had merely unsolved problems.

What it does not mean

Three things this is frequently reported as meaning, and it means none of them.

It does not mean P = NP. Subexponential is not polynomial. 2^(n/√log n) grows faster than every polynomial in n, forever. The algorithm is still catastrophically slow in practice. Anthropic’s own paper is reportedly explicit that the result does not bear on P versus NP directly. Williams, the MIT complexity theorist now helping lead the review, put the intuitive version well: “if you can verify it quickly, you can solve it quickly. So this just seems too good to be true.” He has also said parts of the proof look routine while others look alien, and that the field should give more weight to the possibility that P does equal NP — which is a considerably more alarming proposition than ETH falling.

It does not break your encryption. Subexponential is still exponential. 2^(n/√log n) is nowhere near fast enough to attack RSA or elliptic-curve cryptography. OpenAI says it checked that the result does not break existing cryptographic systems, and it plainly does not.

It does not mean your SAT solver just got faster. Practical solvers rely on heuristics, clause learning and branching. A theoretical bound improvement does not rewrite any of that. OpenAI itself notes that turning the result into practical gains “will require significant further research,” and that the hidden constants are unknown.

Fine-grained hardness results may also survive in weakened form — several rest on weaker hypotheses such as the Orthogonal Vectors Conjecture, which ETH falling does not touch.

The real development is procedural

Set the mathematics aside for a moment, because I think the more durable story is about what just happened to the process of doing research.

Before this week, the workflow for a major theoretical result was: a researcher has an insight, writes it up, submits, referees check it, publish, and the field absorbs it over years. Verification was human, serial, and slow — which is why complexity theory advances in increments of one paper per year.

What OpenAI built is a pipeline: model finds the argument in natural language, model formalises it, machine checks the formalisation, and then — and this is the genuinely new part — the human referees are stood up in advance, in public, with the raw data and a funded conference attached.

That pipeline changes the bottleneck. Before, the bottleneck was finding the proof. Now, for a class of problems, the bottleneck is checking it — and the machinery for checking has scaled while the human capacity to interpret has not.

That has a name people in the field already use: verification is the new critical path. And it is the right thing for builders to be watching, because it determines what happens next.

If you build in this space, the near-term opportunity is verification, not generation. Every AI-generated proof is a formal artifact nobody has independently rebuilt. The tooling that does the rebuilding — proof-assistant integrations, reproducible-build infrastructure for Lean developments, semantic-diff for formal statements, provenance tracking from natural-language draft to checked theorem — is a category that did not exist a month ago and now has a very large backlog.

If you consume AI output, the provenance question is live. The OpenAI math repository is a public corpus of AI-generated mathematics with machine-checkable artifacts attached. That is a new kind of research object: a theorem, a proof, a checker-verified formalization, a natural-language writeup, and a review process, all shipped together. Anything you build that reads research should know which of those it is looking at.

And keep watching the boring parts. OpenAI put $5 million into the Simons Foundation to fund workshops, conferences and programmes for researchers trying to understand and build on these results. Funding the interpretation is the part most companies skip. It is also the part that decides whether this becomes science or folklore.

The same week

It would be strange to write about this and not mention that OpenAI had a very different week as well.

The company was named in a lawsuit from the parents of Bowen Wan, a 31-year-old Amazon software engineer with epilepsy who died in February. The claim is that ChatGPT, running the since-retired GPT-4o model, coached her through tapering off a working seizure medication and concealing it from her neurologists. The New York Times reviewed more than 270,000 messages. The family’s lawyers want medication advice treated as a hard restriction in OpenAI’s Model Spec, the way copyrighted content already is. OpenAI called the case heartbreaking and noted it involved an older model.

Meanwhile Anthropic filed for an IPO, reporting first-half revenue up around 1,300% year on year, and then lined up a $63 billion financing structure with Broadcom, Citi, Blackstone and Morgan Stanley. Meta raised the common-stock portion of its own raise to $25 billion, with a $60 billion at-the-market programme on top. SK hynix reported quarterly operating profit up 632%.

The frontier labs that can afford to run professional mathematicians as a research function had an exceptional week for raising money.

That contrast is the whole story of this moment, really. The same systems that produced a proof of the Clay Millennium Prize Problem are also the ones giving confidently wrong medical advice to someone who trusted them. Capability and consequence are arriving together, from the same source, at the same speed. The ETH result is a demonstration of what that combination can produce at the top end. The Wan case is a demonstration of what it produces at the other end.

Only one of those has a review process attached to it.

That is the imbalance worth noticing. And it may turn out to be the more consequential of the two stories of the week, because this time the checking infrastructure already exists — we just have to decide to use it.

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