Anthropic · 2026-07-19 · major
Claude Fable 5 disproves Jacobian conjecture — 3D polynomial counterexample
Mathematician Levent Alpöge used Claude Fable 5 to build a concrete 3D polynomial counterexample to the 1939 Jacobian conjecture, one of the central open problems in algebraic geometry.
Anthropic mathematician Levent Alpöge posts an explicit polynomial map that breaks a central 87-year-old conjecture in algebraic geometry, built with the help of Claude Fable 5.
Key specs
| Age of conjecture | 87 years (posed 1939) |
|---|---|
| Model used | Claude Fable 5 |
| Hn traction | 429 points, front page |
Quick facts
| Problem | Jacobian conjecture (posed by Ott-Heinrich Keller, 1939) |
|---|---|
| Mathematician | Levent Alpöge (Harvard Society of Fellows, now Anthropic) |
| Tool used | Claude Fable 5 |
| Counterexample | Explicit polynomial map C^3 → C^3, Jacobian determinant = -2 |
| Announced | X post by @__alpoge__, 2026-07-19 |
| Peer review | Not yet; endorsed on X by mathematicians Jared Duker Lichtman and Daniel Litt |
| Wikipedia | Article updated to list conjecture as 'disproved in 2026' |
What is it?
The Jacobian conjecture, posed by Ott-Heinrich Keller in 1939, asks whether every polynomial map from n-dimensional complex space to itself whose Jacobian determinant is a non-zero constant must have a polynomial inverse. On 2026-07-19 Levent Alpöge — a Harvard Junior Fellow now at Anthropic — posted an explicit polynomial map from C^3 to C^3 with Jacobian determinant -2 that sends three distinct points to the same image, meaning the map is not invertible. That single example is enough to disprove the conjecture.
How does it work?
Alpöge built the counterexample with Claude Fable 5, Anthropic's flagship model, and credits it in his post with 'working during the world cup final'. The polynomial map has three components in x, y and z, each a low-degree polynomial. Verification is direct arithmetic: compute the Jacobian determinant to confirm it equals -2 everywhere, then evaluate the map at the three named points and check they all land at (-1/4, 0, 0). No formal proof system or scaffolded proof search was reported; Alpöge treats Fable as a research collaborator that produced and tested candidate maps.
Why does it matter?
The Jacobian conjecture has resisted proof and counterexample since 1939, so a concrete disproof is a significant algebraic-geometry result on its own. It also lands weeks after OpenAI's May 2026 disproof of Erdős's 1946 unit-distance conjecture, suggesting frontier general-purpose reasoning models are now being used routinely by working mathematicians to attack open problems rather than only re-derive known results. Alpöge is a card-carrying research mathematician, so the credit and the workflow are far removed from earlier premature AI-math claims.
Who is it for?
Algebraic geometers, AI-for-math watchers, mathematicians using LLMs as research collaborators.
Frequently asked questions
- What is the Jacobian conjecture and why did it stand for 87 years?
- The Jacobian conjecture, posed by Ott-Heinrich Keller in 1939, states that any polynomial map from n-dimensional complex space to itself whose Jacobian determinant is a non-zero constant must have a polynomial inverse. It stood as one of the central open problems in algebraic geometry, resisting proof and counterexample despite decades of attention from top mathematicians.
- How did Claude Fable 5 help find the counterexample?
- Levent Alpöge used Claude Fable 5 to search for and verify candidate polynomial maps. His X post credits Fable with 'working during the world cup final' and gives the explicit map from C^3 to C^3 with Jacobian determinant -2 that sends three distinct points — (0,0,-1/4), (1,-3/2,13/2), (-1,3/2,13/2) — to the same image, proving the map is not invertible.
- Is the disproof considered official?
- Not yet. Alpöge announced the counterexample on X on 2026-07-19 and Wikipedia has updated the Jacobian conjecture article to note the disproof, but the result has not undergone formal peer review or publication. Mathematicians Jared Duker Lichtman and Daniel Litt validated it publicly on X; formal recognition typically waits for a refereed paper.
- Who is Levent Alpöge?
- Levent Alpöge is a Harvard-trained mathematician who was a Junior Fellow in Harvard's Society of Fellows and now works at Anthropic. His prior research includes work with Manjul Bhargava, Wei Ho, and Ari Shnidman on Hilbert's tenth problem over number fields and on rational cube sums.
- How does this compare to earlier AI-assisted math breakthroughs?
- If verified, it would join OpenAI's May 2026 disproof of Erdős's 1946 unit-distance conjecture as one of the highest-profile AI-assisted math results. Unlike specialized proof systems, Claude Fable 5 is a general-purpose model, and Alpöge — an active research mathematician — treats it as a collaborator rather than an autonomous prover.
Try it
Read the X post at xcancel.com/__alpoge__/status/2079028340955197566 for the exact polynomial and the three collision points.