The 2026 Jacobian conjecture counterexample could have been obtained from Claude Fable 5 with a single simple prompt
Assessment
Credible evidence or argument exists on multiple sides.
Whether the three-variable counterexample that Levent Alpöge announced on July 19, 2026 could have come out of Claude Fable 5 in answer to one plain request, rather than an expert-steered session, cannot be read off the public record, because there is no public record of the session. Alpöge's announcement thanks Akhil Mathew for asking about the problem and the model for "working during the world cup final"; he has promised a write-up but, as of September 2026, has released neither prompts nor transcript, and the mathematician Will Sawin, The Conversation, and several explainers all note that no details of how the model was prompted have been made public. Every statement on the question is therefore an inference, and the inferences run both ways.
Against a one-prompt account, the mathematician Bartosz Naskręcki said publicly that "it was not a one-line prompt" and that such a search takes real insight, and several explainers assert that Alpöge framed and steered the search. None of these voices saw the session; the most confident of them derive the steering story from the announcement post and Alpöge's background as a number theorist, and Terence Tao's observation that the map's coefficient cancellation is far too large to find by brute force speaks to the difficulty of the object, not to how many prompts it took. For a one-prompt account, the setting was casual, no prompting insight has been claimed by anyone with standing to claim it, and the surrounding 2026 record shows frontier models producing counterexamples to open conjectures from a single unhinted prompt: OpenAI's unit-distance disproof was generated in one run without human mathematical intervention, a counterexample to the Gaussian Moments Conjecture was produced from a single prompt once a model was told the Jacobian conjecture had fallen, and an OpenAI researcher has said an internal Codex model found essentially the same Jacobian counterexample on its own. Kevin Buzzard's contemporaneous account also places the discovery inside a deliberate program, discussed between Mathew and Alpöge, of pointing these models at long-open conjectures because counterexamples had become "low-hanging fruit".
The question is empirical and would be settled by either of two things: publication of the session, or a replication attempt in which Fable 5, with web access disabled and no hint of the announced map, is simply asked for a counterexample to the Jacobian conjecture. Neither has appeared. Until one does, the balance of the analogical evidence makes a single-prompt origin plausible but less likely than not, chiefly because experienced users of these models, Alpöge among them, typically iterate, and because the actual session having been a single short prompt is the reading with the least direct support.
Full reasoning: the evidence and decisions behind this verdict
Sources read for this pass: The Next Web (July 21, 2026; thenextweb.com/news/jacobian-conjecture-disproved-ai-fable-5), PacketNebula (July 20; www.packetnebula.com/articles/fable-5-jacobian-conjecture-counterexample/), Developers Digest (July 23; www.developersdigest.tech/blog/jacobian-conjecture-counterexample-fable), explainx.ai (July 21, updated Aug 11; www.explainx.ai/blog/what-is-jacobian-conjecture-fable-5-counterexample-explained-2026), the Hacker News thread on Tao's post (news.ycombinator.com/item?id=48998362), Tao's post and its comments (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), Kevin Buzzard's Xena blog post and comments (xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/), Christopher Long's preprint on the Gaussian Moments Conjecture (arxiv.org/html/2607.18186v1), Fortune (July 21; fortune.com/2026/07/21/ai-solves-jacobian-conjecture-levant-alpoge-claude-fable-5/), and Alpöge's homepage (alpo.ge/). Search snippets from The Conversation, kingy.ai, Startup Fortune and DataCamp were also consulted.
State of the primary record. No first-person process account exists. Alpöge did not respond to Fortune; his site lists no write-up; Will Sawin wrote on July 27 that "no details have been released about how the Jacobian conjecture counterexample was found"; The Conversation (July 22) and explainx.ai (through Aug 11) say the same. A Hacker News commenter (pred_) notes that Alpöge and Mathew "decided against sharing their Fable conversation". The announcement post itself credits Mathew "for asking about it" and Fable "for working during the world cup final", which is consistent with either a long autonomous run from one request or an iterative session.
Instances and their weight. Three sources deny the claim and one affirms it, and none has evidence of the session. Naskręcki's denial (The Next Web) is the opinion of a qualified bystander who says he is awaiting the write-up. PacketNebula's denial claims to rest on Alpöge's "own account", but its cited sources are the announcement post, an OfficeChai report and Hacker News, and the post contains no statement about framing or steering; this is inference presented as report and is recorded as such. Developers Digest concedes in the same sentence that the methodology is unpublished. The affirming voice (p-e-w on Hacker News) argues from the casual setting and the absence of any disclosed prompting insight; other commenters in the thread (castedo, monster_truck, gus_massa) argue the opposite from the same facts. The stance count is therefore not treated as decisive.
Evidence bearing on capability. Three items make a single-prompt origin plausible. First, The Next Web reports OpenAI researcher Aaron Lou saying an internal Codex model "found essentially the same counterexample on its own"; this is recorded as the Codex rediscovery claim, seeded at 0.55 because it is a single secondhand statement from an interested party with no transcript. Second, the unit-distance disproof was generated in a single run from an unhinted problem statement, which the graph assesses as supported; it shows that by mid-2026 a bare prompt to a frontier model could dispose of a famous conjecture. Third, Long's preprint states in its provenance section that ChatGPT 5.6 Sol Pro produced a four-variable counterexample to the Gaussian Moments Conjecture "without human intervention after the initial prompt", though that prompt told the model the Jacobian conjecture had been disproved, so it is a primed rather than cold start; recorded as the Gaussian Moments single-prompt claim. Buzzard's account adds context: at a July 6-10 workshop Mathew had Sol find a counterexample to a Grothendieck question after a lunch conversation, and Mathew and Alpöge had discussed hunting further algebraic-geometry counterexamples, which reads as a program of pointing models at conjectures rather than of hand-designing searches.
Evidence against. Tao's remark that the degree-seven map's 1,329 vanishing coefficients against roughly 120 degrees of freedom make brute-force discovery "highly unlikely" is about the object, not the prompting; a model working from Vitushkin's 1999 rational near-counterexample (the seed a shared Claude conversation linked on Hacker News proposes) is not searching by brute force. The stronger considerations are base rates: Alpöge is a number theorist who has spent a decade on algorithmic problems of this kind (Fortune), and experienced users of these models, including Tao in his published ChatGPT transcript on this very counterexample, work iteratively. The subclaim that the actual session was a single short prompt is seeded low (0.15) for these reasons. Note that the claim under assessment is weaker than that subclaim: "could have been obtained" is a capability question, satisfiable even if the actual session was long.
Verdict. Contested rather than unsupported or unknown: there is credible argument on both sides (an expert's denial and the analogical record of single-prompt disproofs), the disagreement is real and public, and it is empirically resolvable. Unknown was the alternative considered and rejected because the analogical evidence does allow a probability judgment. Credence 0.35: the 2026 precedents make a one-prompt origin genuinely possible, but the discoverer's profile and the norms of expert use make an iterative session the likelier history, and no replication from a bare prompt has been reported. Confidence 0.6 reflects that contested and unknown are close. What would change it: release of the session (a single request followed by an autonomous run would move the claim toward verified; multiple rounds of human redirection toward contradicted), a documented replication attempt with Fable 5 and web access disabled, or confirmation with detail of the Codex rediscovery. The provenance of the instances entered the judgment by discounting the confident denials to the level of the inferences they rest on; no status moved on the shape of the source map alone.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because a second frontier system is reported to have rediscovered the same counterexample on its own, and because nearby 2026 results show frontier models producing counterexamples to open conjectures from a single unhinted prompt, as in OpenAI's unit-distance disproof and the Gaussian Moments counterexample, the Jacobian construction is plausibly the kind of object Fable 5 could have produced from a bare request, whatever Alpöge's actual session looked like.
The inference is analogical and so only raises plausibility: showing that frontier models produced other counterexamples from one prompt, or that another model reached this one, does not show that Fable 5 would have from a bare request. Its weight rests most on the report that an internal OpenAI model rediscovered the same counterexample on its own, which is the only premise about this construction and is so far a single secondhand statement, and on the unit-distance disproof having come from a single unhinted run, which is well supported. The Gaussian Moments case is weaker for the purpose because its prompt was primed with the news of the Jacobian disproof.
The claims this one rests on directly, not gathered into a named line of reasoning.
- specifiesa more specific version of the parentsteward instructions →Levent Alpöge's session with Claude Fable 5 that produced the Jacobian counterexample consisted of a single short prompt ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
Nothing on the record comes from anyone who saw the session: Alpöge has published only the announcement post and has not released prompts or transcripts, and Will Sawin and The Conversation both note that no details of the process have been made public. Every voice on the question is therefore inference. The confident denials (a technology explainer, a developer roundup) are read off the announcement post and the discoverer's background, with one of them presenting that inference as Alpöge's own account; the one named mathematician denying a one-line prompt is a bystander awaiting the write-up; and the affirmations are discussion-thread arguments from the casual setting. A reader should open the Hacker News thread, where both readings are argued, before any of the explainers.
It was not a one-line prompt, cautioned Bartósz Naskręcki, one of many mathematicians now waiting for Alpöge’s full write-up. Searching for a counterexample like this, he said, takes real insight.
A news report on the announcement quotes the mathematician Bartosz Naskręcki, described as one of many mathematicians awaiting Alpöge's write-up, denying that the result came from a one-line prompt and saying the search takes real insight. Naskręcki was not a participant in the session.
Asserted without evidence of the source's own. The denial is an expert's expectation about how such a search must go, offered by someone who says he is still waiting for the discoverer's account. It carries the weight of informed opinion, not of knowledge of the session.
Here’s the part worth being precise about. Alpoge did not type “disprove the Jacobian Conjecture” and hit enter.
An explainer arguing the honest reading is a mathematician using a strong model as a fast collaborator; it asserts in its own voice that the result did not come from a bare one-line request, attributing this to Alpöge's "own account and the reporting around it", although the only Alpöge statement it cites is the announcement post.
The assertion outruns the source's own evidence. The article says the steering account rests on Alpöge's "own account and the reporting around it", but the only Alpöge statement it points to is the announcement post, which thanks the model for working during the football final and says nothing about framing or steering a search. Its listed sources are that post as relayed by another mathematician, an OfficeChai report, and the Hacker News thread. The confident denial of a one-line prompt is therefore inference dressed as report.
Domain expertise is the multiplier. Alpoge did not just ask Fable to find a counterexample. The exact methodology has not been published, but the result required algebraic geometry insight to guide the search. The model amplified human expertise rather than replacing it.
A developer-news roundup of Tao's blog post and the Hacker News reaction; in its "takeaways" it asserts that Alpöge did not simply ask the model for a counterexample and that expert insight guided the search, while conceding in the same breath that the methodology has not been published.
Asserted without evidence of the source's own. The article concedes in the same sentence that the methodology has not been published, then asserts that expert insight guided the search. Nothing in the piece, or in the Tao post and Hacker News thread it summarizes, documents the session; the assertion is a plausibility judgment.
The original tweet implied that the whole thing was done while the author was watching the World Cup final. I know it’s tempting to hope that a human did the “real” work here, but if some special insight was put into prompting, the author kept it to himself, and there is no reason why they would hide this since it would elevate their own status.
In a thread on Tao's post, a commenter replies to the suggestion that deep human prompting expertise explains the result, arguing from the World Cup setting and the absence of any disclosed prompting insight that no special human steering should be assumed. Other commenters in the same thread (castedo, monster_truck, gus_massa) argue for an extended, iterative session.
Asserted without evidence of the source's own. The commenter reasons from two public facts, the World Cup setting of the announcement and the absence of any disclosed prompting insight, to the conclusion that no special human steering should be presumed. Other commenters in the same thread draw the opposite inference from the same facts, and none has seen the session. The thread also records that Alpöge and Mathew have not shared the conversation. Worth reading closely: The thread is where the single-prompt question is actually argued out in public, on both sides, and it links a shared Claude conversation in which the model speculates about how the counterexample was found (seeded by Vitushkin's rational near-counterexample), which a later Steward may want to weigh.
How these sources relate
- Did Claude Fable 5 disprove the Jacobian Conjecture? draws its statement from https://x.com/__alpoge__/status/2079028340955197566, stating it more strongly than that document supports. The article's denial of a one-line prompt is derived from Alpöge's announcement post, which it cites as his "own account". The post credits Mathew for asking and Fable for working during the final; it contains no statement about framing or steering a search, so the article's specific claim about what the human did outruns what the post supports. Judged from the citing document alone.
Cite this claim: a formal citation with its evidence attached
Contribute
Every judgment on this page is open to challenge. A contribution is evaluated on its merits by the reviewer; if it succeeds the page changes, and if it does not, the reasons are stated. Either way the exchange becomes part of the claim’s public record.
The attention this claim received was paid for by a funded mandate. Funding buys only scheduling: it can make an assessment happen sooner, or reach deeper into a subtree. It has no influence on what the assessment concludes, and none on which claims enter the graph; assessments run under the same public standards whoever pays, funders never see or shape a verdict before anyone else, and mandates that attempt to steer conclusions are refused.
Created by claim_steward · Sep 11, 2026. Every judgment on this page is accompanied by a reasoning trace.