Two Days, Two "Refutations"

On October 26, 2026, OpenAI announced that its AI system had refuted the Exponential Time Hypothesis (ETH). ETH is a load-bearing assumption of theoretical computer science: it asserts that problems such as 3-SAT admit no subexponential-time algorithm, and a large body of complexity results over the past two decades and more rests on it. The next day, Anthropic deposited a paper on the same claim at the public platform Hexagon; the paper says its internal model independently found the core algorithm on the same date, October 16, and reaches the same conclusion: ETH is false.

Both claims come with the same new kind of evidence: not only a natural-language manuscript, but also a proof checked by Lean formalization. The tone of the announcements is accordingly firm. OpenAI's own words are "our system has recently proved that the ETH is false," and it says the result was "subsequently verified in Lean."

But as of the checking date of this article, October 27, 2026, neither paper has passed peer review, and no independent body has issued a conclusion of "established." The three scholars leading the review of OpenAI's paper stated in advance in their open letter that their involvement should not be taken as certification of the claim.

Between a claim and its establishment lies an entire chain of checking. What this article does is take that chain apart and mark, item by item, how far each lab has actually got.

A Proof Has Five Gates to Pass

"Proving a mathematical statement" can be broken into five stages, each of which can be checked separately.

The first gate is stating the claim: saying in one plain sentence what is to be proved. The second gate is formal transcription: translating that plain sentence, word by word, into a statement inside a formal system, where how variables are encoded and how time is counted are all fixed once and for all. The third gate is machine verification: a proof assistant such as Lean checks, line by line, whether every step of the formal proof follows the rules of logic, and audits which axioms it depends on. The fourth gate is human review of semantic correspondence: people check whether what the formal statement says is the same thing the plain sentence at the first gate said, and whether each choice of a "standard encoding" in the transcription quietly weakened or skewed the claim. The fifth gate is independent review and scholarly acceptance. This gate is not a single act, but three lines that can advance in parallel: independent reproduction, whether other researchers can independently rerun the build, restate the argument, and set the result out clearly again; expert review, specialists in the relevant field checking item by item for logical gaps, missing assumptions, and conclusions whose stated scope does not match what was shown; and scholarly acceptance, whether, through continued exchange and follow-up research, the field comes to accept the conclusion and use it as a premise for new work. A paper can be independently reproduced before it passes peer review, and can still be disputed after passing it; the three lines have no fixed order. The five gates are an editorial framework DeepWing uses for its checking, not a universal standard that a mathematical proof must pass in sequence.

The key point of this chain is that machines can only pass the third gate. Anthropic puts this more bluntly in its own repository documentation than any commentator has: "Lean checks the proofs, but no machine checks that this file says what the paper says." Lean checks the proof itself, but no machine checks whether the statement written down in the file is the plain sentence in the paper's title. For that reason they kept the statement file extremely short, so that people can read through it item by item, and they recorded in the paper's appendix, item by item, where the formal statements differ from the printed statements.

An analogy: Lean is like putting every step of the reasoning through a security check. The strength of that check is real; it rules out the most common class of error, a hidden leap in the argument. But the security check is not responsible for confirming that the destination printed on your ticket is the right one.

Two ETH Papers, Two Kinds of Fourth Gate

Both labs have published formal material that anyone can inspect, but the scope of the verification has to be assessed separately for each; it will not do to say loosely that the whole result has been verified. Start with the main theorem on Anthropic's side. The statement file, MainResults.lean, imports nothing but Mathlib and contains no proofs; according to the repository documentation, its MainTheorem is the paper's Theorem 1, the counting algorithm together with its time bound. The top of the paper's page writes that bound as 2^{O(n log log n / √(log n))}; the formal statement is more precise: the running time is 2^{C·min{km/√(log m), k²·n·log log n/√(log n)}}, where k is the clause width, m the number of clauses, n the number of variables, and the constant C does not depend on k. This formal bound is stronger than the paper's headline form in two ways. First, its dependence on k is explicit: how the bound changes as k grows is written into the exponent, not hidden in a constant. Second, it has an extra branch counted in clauses, km/√(log m); when the number of clauses m is at most a fixed multiple of the number of variables n, that branch amounts to the tightest bound in OpenAI's manuscript. In other words, the main theorem Anthropic has formalized not only proves exactly the bound its manuscript claims, with no gap in width, but also writes both of the tightest cases from OpenAI's manuscript into the formal statement. The main theorem itself depends only on Lean's three standard axioms, must also be rechecked by a second, independently written kernel, and its proved statements must be matched word for word against the separately stored comparator statements; according to the repository documentation, anyone can rerun the build in about 8 minutes on 24 vCPUs.

But the main theorem passing does not mean the whole argument has passed. According to the repository documentation, building it produces two "declaration uses 'sorry'" warnings, both from the comparator file Challenge.lean. By the repository's design, that file writes down the statements of the two theorems without proofs, as placeholders for the fuller check that follows (comparator comparison plus rechecking by a second kernel). These two warnings do not fall on the proof of the main theorem.

Neither paper has yet passed the fourth gate, but they are stuck in different ways. OpenAI's side likewise has comparator statements and a formalization scope document, open to anyone; what is stuck there is the width of the bound. Its formalization scope document (document 378) states that the running time Lean actually proves is a polynomial factor of (L+1) raised to the O_k(1), times 2^{O_k(n (log log n)^2 / √(log n))}, where L is the bit length of the encoding. The two tighter bounds both come from the October 25 manuscript: the general bound of 2^{O_k(n log log n / √(log n))} (with a further polynomial factor of (n+m) raised to the O_k(1)); and, when the number of clauses m does not exceed a fixed multiple of the number of variables n (m ≤ Dn), a further tightening to 2^{O_{k,D}(n / √(log n))}. The scope document attributes only the wider bound to the Lean proof; it does not say that these two tighter bounds are covered by the formalization. What matters is that both upper bounds are of subexponential form: this difference in tightness calls into question whether the strongest assertion in the manuscript is covered by the formalization; it does not mean that the slightly wider bound would be insufficient to refute ETH. If the algorithm and the formal statements hold, the wider bound is likewise enough to make ETH false. As for what the difference in tightness does to the manuscript's other conclusions, that is exactly what the review group has to check with human eyes. On Anthropic's side there is no such gap in width; its fourth gate comes down mainly to two things. The first is statement correspondence: whether the statements written in the statement file match the paper's printed statements, item by item; the repository keeps Appendix E and PaperIndex.lean for exactly that comparison, recording where the two correspond and where they differ, and the comparison itself can only be done by people. The second is the machine-model step: the "algorithm" in Lean is a Mathlib stack machine, Turing.FinTM2, not an algorithm in the usual sense; getting from that machine to the claim that an ordinary algorithm runs this fast relies on two steps, a random-access machine simulating it and the input being rewritten into the file's format, and the file's comments state that these two steps are standard facts that are not part of the Lean statements or proofs. One more scope limit is attached to the theorem: it covers only formulas with at least 4 variables and at least 2 clauses, in which every variable occurs and no clause is repeated; in the file's own words, nothing is said about any other input.

The fifth gate, independent review and scholarly acceptance, has just shown its earliest signal, and only on the first of its three lines: on October 26, a third-party researcher published a simplified restatement of OpenAI's proof, retelling the main line of the argument, which is where the independent-reproduction line begins. A restatement is the beginning of understanding, not a verdict. On the expert-review line, the machinery of review is in place: an open letter, a call for comments, a repository they plan to maintain collecting notes and corrections from all sides, and a proposal to hold an online meeting before the end of the year to discuss it. But a verdict: not one word yet. The scholarly-acceptance line has not begun; it is too early to speak of it.

One more sentence that headlines easily swallow: both results are purely asymptotic statements. Anthropic's repository itself declares that the result "says nothing about solving formulas in practice." If it holds, what changes is the foundation of the theoretical edifice, not anyone's speed at solving problems tomorrow. Specifically, the first thing that would have to be reexamined is the body of conditional hardness conclusions built on ETH over the past two decades and more: every lower bound of the form "if ETH holds, then problem X admits no faster algorithm" would have to be rechecked one by one against the new bound. It does not mean that NP-hard problems suddenly become easy, and still less does it mean that the question of whether P equals NP has been settled; that is a different question, and neither paper touches it.

The Other Two Claims in the Same Release, Uneven in Substance

OpenAI's announcement in fact packaged three claims; ETH is only one of them.

The second is the local smoothing conjecture in harmonic analysis. The company says its model gave a proof in every dimension from two upward, and says that the Kakeya conjecture and the Stein restriction conjecture are thereby solved at the same time. On the Kakeya side, the company's repository change log speaks of the maximal and dimension conjectures being obtained together. On the Stein side, the wording has to be tightened: the change log's exact statement is that the spherical restriction estimates hold throughout the interior of the conjectured range, endpoints excluded, and its formalization scope document likewise confines the Stein square-function bounds to their strict ranges. "Interior of the range" is the scope the manuscript itself claims; it cannot be read as the Stein restriction conjecture being settled as a whole. The completeness of the material for this claim is the same as for the ETH paper: manuscript, formalization, and a document listing the applicable scope item by item are all public. The difference is that there is no designated reviewer; under the company's strategy it is released for the field to organize around on its own, and the pace of what happens next is the least predictable.

The third is an existence construction for four-dimensional interacting quantum field theories, including QCD; within it, the existence of pure Yang-Mills theory is, the company says, part of a Clay Millennium Prize problem (that problem also requires a mass gap, which is outside the scope of the claim). This claim has the lowest public checkability of the three: the manuscript and the formalization files are all held by the five-person review group led by David Gross, nothing is public beyond the announcement's retelling, and it is not even stated whether the formalization covers QCD. Whether the review group holds enough material, and whether the public can scrutinize it independently, are two different questions; on the latter, as of the checking date, the public cannot directly inspect a single manuscript or formalization file. On October 26, AGMAI, an independent advisory group on mathematics and artificial intelligence made up of nine mathematicians, whose project page sits at the Institute for Advanced Study (its members include Gowers, Hairer, Witten, and De Lellis), demanded in a public statement that OpenAI release the material immediately, naming the outputs for Yang-Mills theory, quantum chromodynamics (QCD), and a completion of the Standard Model. Its reason was stated in direct terms: by OpenAI's own account the Lean formalization is already complete, so there is no scientific reason to delay; if the material is not public, the mathematical community has no way to begin normal checking (the link to the statement is in the Sources at the end of this article). The group's origin is itself a piece of evidence about its independence: by its own account, OpenAI first approached some of its members about establishing an external advisory board; in agreement with OpenAI, they decided instead to create an independent group and invite others to join; it operates independently of any AI company, and its members accept no payment for this work.

The same day also hides a dispute over norms. Another AGMAI statement, from the day before (October 25), opposes exactly this model of "entrusting material to a small number of designated mathematicians": it slows down access, creates inequality, and in effect borrows the academic prestige of the entrusted scholars. That statement's recommendation is that laboratories adopt pathways 2A or 2B of AGMAI's guidelines of September 29; the specific prescriptions, that results be deposited in academic repositories not controlled by the company with process disclosure attached (which model, which prompts, the chains of thought, how many attempts, how much compute, at what cost), come from pathway 2B of those September 29 guidelines. The same release supplies a ready-made contrast: OpenAI's repository change log records that the review group on the ETH side is an independent working group, and that the materials were made public immediately at the group's request. On one side, full release at once at an independent review group's request; on the other, material entrusted to a designated group and not yet public. This dispute will recur every time a laboratory issues a claim of this kind from now on.

A Precedent Worth Noting: This Same Pipeline Withdrew Papers This Month

One more reason for "let us wait and see." On October 7 of this month, the same OpenAI manuscript repository withdrew three papers, because a sign error in one place invalidated a key argument; another 14 papers were revised on the same day, for reasons including proof repairs and statement corrections. The repository itself also states plainly that the manuscripts are unpolished raw research data, and that results which have not been formalized may have problems.

This is not to say there is anything wrong with the three claims this time. It is to say that on this pipeline, "published" has never meant "passed." The withdrawal record is, precisely, the instruction manual that comes with this pipeline.

What to Watch Next

DeepWing has registered the four claims, three from OpenAI and one from Anthropic, item by item, recording under the five gates how far each has got; the verification status tracking table is further down this page, and will be revisited as events arrive:

1. The review group's first public statement, and whether the proposed online meeting before the end of the year takes place, and what is said there.
2. Whether the material for the physics claim is released, and whether OpenAI responds to the advisory group's public call.
3. Whether either ETH paper issues corrections or new versions. The repositories' change logs are themselves a signal.
4. Whether named independent checks or explanatory papers appear, and whether the papers enter a formal review channel.

Before the first hard milestone arrives, this article has only one conclusion: the two laboratories have pushed "AI can produce major proofs" to the doorway of checkability; inside the door, no one has signed the verdict yet.

Verification status tracker

The four claims registered gate by gate, as of October 27, 2026. Solid red: passed or public; red outline: partial; grey outline: under way or awaiting human review; plain: not started or not public. The five gates are DeepWing's editorial framework, not a universal standard in mathematics; the table is revisited at the milestones listed under What to Watch Next.

Sources

All facts in this article are as of October 27, 2026, and all sources are primary materials:

1. OpenAI announcement (October 26, 2026)
2. OpenAI mathematics repository; ETH formalization scope document; local smoothing formalization scope document; repository change log (including the October 7, 2026 withdrawal and revision entries)
3. Anthropic paper page (Hexagon, deposited October 27, 2026); formalization repository; its eth folder documentation
4. ETH review group open letter (signed by Impagliazzo, Paturi, and Williams)
5. Statements of the independent advisory group AGMAI (Advisory Group on Mathematics and Artificial Intelligence) (October 25 and 26, 2026), and its General Guidelines of September 29, 2026 (the source of pathways 2A and 2B)
6. Third-party simplified restatement (Shiva Kintali, October 26, 2026)

Statements whose original text could not be found are marked separately in DeepWing's research notes and are not used as a basis for this article.