Levent Alpöge defined and steered the 2026 Jacobian counterexample search, with Claude Fable 5 generating candidates
Assessment
Credible evidence or argument exists on multiple sides.
The claim describes the division of labor behind the counterexample to the Jacobian conjecture that Levent Alpöge announced on X on July 20, 2026: that he framed the problem and directed the search, while Anthropic's Claude Fable 5 produced the candidate polynomial maps. The weak version of this is not in doubt. The problem was put to Alpöge by a colleague, Akhil Mathew, who had discussed finding such counterexamples with him beforehand; Alpöge, a number theorist with a long record of algorithmic work on related problems, spent the evening working with the model; the model generated the map; and Alpöge checked the algebra, posting Wolfram Alpha links with the announcement.
What is disputed is the strong version the claim actually states: that Alpöge set the search criteria and steered the model toward a particular shape of counterexample, so that the model's role was to generate candidates within human-defined constraints. Some explainers assert exactly this, one of them attributing it to Alpöge's "own account", and the mathematician Bartosz Naskręcki has said the result was "not a one-line prompt" and that such a search takes real insight. Other reports describe the process as a mathematician feeding the problem to the model and receiving a verifiable counterexample within an evening, and Alpöge's own announcement credits the model with the work rather than describing a search he designed. The more careful reviews, including The Conversation and several technical blogs, state plainly that the prompts, intermediate attempts, and extent of human steering have not been made public, and Anthropic has issued no account of the workflow.
The disagreement therefore rests on shared ignorance rather than conflicting evidence: no inspectable record of the session exists, and the sources that assert a steered search have not shown their basis. Alpöge's expertise and the improbability that a single simple prompt would have produced the object make substantial human framing plausible, but "framing" ranges from choosing the problem to specifying the algebraic form of the answer, and the claim is true only toward the latter end. Publication of the session transcript, or a first-person write-up from Alpöge of how the search was conducted, would settle the question in either direction.
Full reasoning: the evidence and decisions behind this verdict
Evidence read for this pass came from web search: PacketNebula (www.packetnebula.com/articles/fable-5-jacobian-conjecture-counterexample/), Wan 2.7 (wan27.org/blog/fable-5-jacobian-conjecture-counterexample), Glitchwire (glitchwire.com/news/a-mathematician-used-claude-fable-to-disprove-the-87-year-old-jacobian-conjectur/), The Next Web (thenextweb.com/news/jacobian-conjecture-disproved-ai-fable-5), kingy.ai (kingy.ai/blog/claude-fable-jacobian-conjecture-counterexample/), Startup Fortune (startupfortune.com/claude-fable-5-helped-crack-the-jacobian-conjecture-after-87-years-of-failure/), explainx.ai (www.explainx.ai/blog/what-is-jacobian-conjecture-fable-5-counterexample-explained-2026), Fast Company (www.fastcompany.com/91577272/jacobin-conjecture-anthropic-ai-levent-alpoge), and The Conversation's piece as republished by ScienceDaily on August 4, 2026 (www.sciencedaily.com/releases/2026/08/260804034634.htm).
For the claim: PacketNebula asserts that "by his own account and the reporting around it" Alpöge framed the problem, steered the search toward a specific shape of counterexample, and checked the algebra himself, with Fable 5 generating the candidate construction. Wan 2.7 asserts the same division of labor ("Alpöge defined the search criteria; Fable 5 generated and tested candidates within those constraints") without sourcing. Naskręcki, quoted by The Next Web, says the find "was not a one-line prompt" and that the search takes real insight. Startup Fortune reports Kevin Buzzard's blog as saying Mathew had discussed finding further counterexamples in algebraic geometry with Alpöge before the post, and MathOverflow as recording the counterexample as found by Fable "prompted by Alpöge after a suggestion from Mathew". Fast Company says Alpöge "spent Sunday working with" the model. Alpöge's post included Wolfram Alpha verification links. These establish human problem selection, an extended interaction, and human verification.
Against, or undercutting: Alpöge's announcement thanks Mathew "for asking about it" and Fable "for working during the world cup final", crediting the work to the model and saying nothing about a search he designed (the linked claim that Alpöge publicly credited Claude Fable 5 with finding the counterexample weighs modestly against strong steering). Glitchwire describes a mathematician who "fed it to a model and got back a verifiable counterexample in the span of an evening". Wan 2.7 elsewhere credits the model's long-horizon autonomous reasoning. Attempts to locate Alpöge's "own account" of steering, which PacketNebula invokes, did not find a first-person statement; PacketNebula may be inferring from the post and from his background.
Uncertainty on the record: kingy.ai (last checked July 26) lists the model version, prompts, tools, human steering and full transcript as unverified and notes Anthropic has issued no discovery announcement; Startup Fortune and explainx.ai say the published trail does not show the prompt, intermediate attempts, or how much steering occurred; The Conversation/ScienceDaily (August 4) says details of how Alpöge prompted the model and what its output looked like have not been made public. As of this pass no transcript or write-up has appeared.
Weighing the subclaims: the model generated the map is undisputed and satisfies the claim's second half. Mathew's suggestion of the problem is well attested but establishes only human problem selection, the weakest sense of "defining the search". Whether a single simple prompt could have produced the counterexample is the crux: if it could, there was little for Alpöge to steer; it is unverified and untested publicly, though expert opinion (Naskręcki) and the discoverer's background make it unlikely.
Verdict: the instances split (two affirm, one denies at reduced confidence, and the most careful sources decline to say), and the fact is not publicly checkable, so contested is the best reading; unsupported was the alternative considered, and it is rejected because there is real, if secondhand, evidence for the steering account. Confidence 0.6 reflects that the choice between contested and unsupported is close. Credence 0.5: the weak reading is near-certain and the strong reading roughly even, with the claim's wording sitting at the strong end. What would change it: a released session transcript or a first-person process account from Alpöge showing either a human-specified target form (toward supported or verified) or a broad prompt answered with little iteration (toward contradicted).
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- contradictsthis argues against the parentsteward instructions →The 2026 Jacobian conjecture counterexample could have been obtained from Claude Fable 5 with a single simple prompt ↗︎
- supportsthis provides evidence for the parentsteward instructions →Akhil Mathew suggested the Jacobian conjecture to Levent Alpöge before the 2026 counterexample was found ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →Claude Fable 5 generated the polynomial map used as the 2026 Jacobian conjecture counterexample ↗︎
- contradictsthis argues against the parentsteward instructions →Levent Alpöge publicly credited Claude Fable 5 with finding the Jacobian conjecture counterexample ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
A working mathematician, facing a decades-old conjecture, fed it to a model and got back a verifiable counterexample in the span of an evening.
A news piece on the announcement; it characterizes the process as the mathematician feeding the problem to the model and receiving the counterexample, a picture in which the model rather than a human-designed search did the finding, while also calling the model a research collaborator.
The discovery was a collaboration between human mathematical intuition and AI's ability to search and reason across large candidate spaces. Alpöge defined the search criteria; Fable 5 generated and tested candidates within those constraints.
An explainer of the counterexample under a heading on what Fable 5 did and did not do; it states the division of labor in its own voice without citing a source for it.
Alpoge did not type "disprove the Jacobian Conjecture" and hit enter. By his own account and the reporting around it, he framed the problem, steered the search toward a specific shape of counterexample, and checked the algebra himself, reportedly through Wolfram Alpha. Fable 5 generated the candidate construction and worked the algebra under time pressure.
An explainer asking whether Claude Fable 5 disproved the conjecture; it argues the honest reading is a number theorist using a strong model as a fast collaborator rather than a machine handing down a theorem, and attributes the framing and steering to Alpöge.
Cite this claim: a formal citation with its evidence attached
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.