An explicit three-variable polynomial map announced by mathematician Levent Alpöge appears to provide a counterexample to the Jacobian conjecture: its Jacobian determinant is a non-zero constant, yet three different inputs produce the same output, so the map cannot have a polynomial inverse. That is the decisive mathematical structure a counterexample needs. Public symbolic checks and Lean formalisation projects have since reproduced the two core facts.
The important qualification is procedural, not cosmetic. As at 22 July 2026, the announcement is only days old. A formalisation pull request to Google DeepMind’s Formal Conjectures repository remains open, and the result has not travelled through the ordinary cycle of a submitted paper, referee reports and mature community scrutiny. The available evidence is unusually strong for such a young claim, but “independently checkable” and “fully absorbed by the field” are not the same status.
The AI angle also needs precision. Alpöge’s public announcement credited an Anthropic model identified as Fable 5 with finding the example after a question suggested by Akhil Mathew. The durable evidence is not the model’s reputation or its explanation. It is the concrete polynomial map and the checks that humans and software can perform without trusting the model.
The conjecture in plain English
The Jacobian conjecture concerns polynomial maps from complex space to itself. Imagine a function that accepts several complex-number coordinates and returns the same number of coordinates, with each output built from polynomials in the inputs.
The Jacobian matrix records how each output changes as each input changes. Its determinant tells mathematicians whether the map is locally squashing directions together. If that determinant is non-zero at a point, the inverse function theorem says the map has a local inverse near that point.
Ott-Heinrich Keller’s 1939 conjecture made a much stronger global claim: if a polynomial map has a constant, non-zero Jacobian determinant everywhere, then it should have a polynomial inverse everywhere. Local non-degeneracy, in other words, was conjectured to force global algebraic invertibility.
That gap between local and global behaviour is the heart of the problem. A map can behave perfectly well in every tiny neighbourhood while still wrapping the larger space in a way that causes different points to share an image. The conjecture said polynomial maps over complex numbers could not do that under the constant-determinant condition.
The newly announced map says they can—in dimension three.
What the counterexample has to show
A counterexample does not need an elaborate philosophical argument. It needs one object satisfying the conjecture’s premise and violating its conclusion.
For this case, there are two decisive tests:
Constant-Jacobian test: calculate the determinant of the map’s Jacobian matrix and show that it is the same non-zero constant for every input.
Non-invertibility test: show that the map is not one-to-one, for example by finding two distinct inputs with the same output. A function with such a collision cannot have an inverse.
The announced example is stronger than the minimum on the second test: it gives three distinct rational points in one fibre, meaning three inputs share an output.
Public accounts record the map (F=(a,b,c):\mathbb{C}^3\to\mathbb{C}^3) as:
\[
a=(1+xy)^3z+y^2(1+xy)(4+3xy),
\]
\[
b=y+3x(1+xy)^2z+3xy^2(4+3xy),
\]
\[
c=2x-3x^2y-x^3z.
\]
Its Jacobian determinant is reported and formally checked as the constant \(-2\). The three distinct inputs
\[
(0,0,-1/4),\quad (1,-3/2,13/2),\quad (-1,3/2,13/2)
\]
all map to
\[
(-1/4,0,0).
\]
The collision alone rules out injectivity. A polynomial inverse would imply injectivity, so no polynomial inverse can exist. If both calculations are correct—and multiple public checks now support them—the universal conjecture is false.
Evidence ladder: from announcement to acceptance
The most useful way to read a very new mathematical result is as an evidence ladder. Each rung answers a different question.
| Evidence rung | What exists on 22 July 2026 | What it establishes | What it does not establish |
|---|---|---|---|
| 1. Public claim | Alpöge posted the explicit map and credited Mathew and Fable | Provenance of the announcement and a concrete claim to inspect | Correctness by itself |
| 2. Direct substitution | The three listed points can be substituted into the displayed polynomials | A reproducible collision if the transcription is accurate | The constant determinant |
| 3. Symbolic algebra | Public repositories provide exact symbolic scripts checking the determinant and collision | Independent computational confirmation without relying on a model’s prose | That every implementation and transcription is bug-free |
| 4. Formal proof | Lean developments encode the determinant and non-injectivity claims; one PR is open in Formal Conjectures | Machine-checked derivations relative to the definitions, theorem prover and trusted foundations used | Automatic peer review, explanatory completeness or immediate merge into a canonical library |
| 5. Mathematical response | Mathematicians have published explanations and derived consequences; a same-day arXiv preprint explicitly cites the announced map | Serious expert engagement and evidence that the object is usable in further mathematics | A settled historical account of credit and significance |
| 6. Mature field acceptance | Not yet available after only a few days | Would include sustained checking, stable publication and broad expert agreement | Cannot be compressed into a viral-news cycle |
This ladder explains why the cautious formulation is “a reported counterexample with strong independent verification”. It is more informative than either extreme: treating a social post as proof, or pretending the claim has no evidential support until a journal issue appears.
Claim-confidence table
| Claim | Confidence on 22 July 2026 | Reason |
|---|---|---|
| Alpöge publicly announced the map and credited Fable | High | The original post is linked by the formalisation PR, contemporary mathematical accounts and reporting |
| The displayed map has constant Jacobian determinant \(-2\) | High | Exact symbolic checks and multiple Lean formalisation efforts report this identity |
| Three distinct listed points have the same image | High | Directly checkable arithmetic and formalised non-injectivity results support it |
| The map is a counterexample to the conjecture in dimension three | High, subject to continuing transcription and definition checks | The two previous facts logically satisfy the counterexample test |
| The conjecture is therefore false in every dimension at least three | High if the three-dimensional example stands | Identity coordinates can be appended to extend a counterexample to higher dimensions |
| The two-dimensional case is resolved | Low / unsupported | A three-dimensional counterexample does not settle dimension two |
| Fable independently originated every key idea without human steering | Unknown | Public attribution describes assistance, not a complete reproducible interaction log |
| The field has completed conventional peer review and reached final consensus | Not yet | The public record is only days old and a prominent formalisation PR remains open |
The strongest claims are strong because they reduce to finite algebraic checks, not because a company or news outlet says they are true.
Timeline of the first public checks
1939: Keller formulates the conjecture in its modern setting.
19–20 July 2026: Alpöge announces an explicit three-dimensional map, thanks Akhil Mathew for raising the problem and credits Fable for working on it. The date appears as 19 July in some local-time accounts and 20 July in UTC references.
20 July 2026: Mathematicians begin circulating hand and computer-algebra checks. Kevin Buzzard publishes a contemporaneous account describing the example and the rapid verification effort.
20 July 2026: Paul Lezeau opens pull request 4474 in Google DeepMind’s Formal Conjectures repository to add a Lean disproof. The pull request links the original announcement and, as checked on 22 July, remains open.
20–21 July 2026: Additional public repositories develop formal proofs and structural results around the map. One Lean repository describes checks of the constant determinant, the three-point fibre, non-injectivity and further geometry, while clearly listing claims it has not formalised.
20 July 2026: Christopher D. Long posts an arXiv preprint on the Gaussian Moments Conjecture. It cites the announced Jacobian map, tracks one consequence through known reductions and separately gives smaller explicit counterexamples to the Gaussian Moments Conjecture. This is evidence of rapid mathematical follow-through, not a substitute for checking the original map.
21 July 2026: Mainstream reporting, including Fortune, documents the claim, the AI attribution and reactions from mathematicians.
22 July 2026: The public evidence supports the core algebraic counterexample strongly, while formal repository review and longer-term scholarly assessment continue.
What was actually proven—and what was not
If the formal and symbolic checks stand, the result proves that the Jacobian conjecture, stated across all dimensions, is false. A universal statement needs to survive every allowed case; one valid three-dimensional counterexample defeats it.
It also proves more narrowly that constant non-zero Jacobian determinant does not force a polynomial map of \(\mathbb{C}^3\) to be globally one-to-one. Appending unchanged coordinates extends the failure to every dimension greater than three.
It does not prove that the two-dimensional case is false. Dimension two can remain open even when the general all-dimensions conjecture has fallen. It does not imply that the inverse function theorem is wrong; that theorem provides local invertibility under its hypotheses, not the global polynomial inverse claimed here. Nor does it mean every problem bearing the word “Jacobian” has been resolved.
The result also does not show that an AI system can reliably settle arbitrary open problems on demand. This example had an unusually crisp acceptance test. Once a candidate map appeared, its determinant and collision could be checked exactly. Many research questions do not offer such a compact certificate.
Why formal verification matters here
The Jacobian conjecture has a long history of incorrect proofs. A surprising claim therefore deserves more than social proof, screenshots or repeated assertions by other language models.
A proof assistant such as Lean forces the author to state definitions and justify steps within a formal system. It can catch algebraic omissions that persuasive prose hides. Public code also lets others inspect the statement being proved: a perfectly checked theorem is useful only if it represents the intended mathematical claim.
The current status illustrates both the power and the limit of formalisation. The public Lean repositories report machine-checked proofs that the determinant is constant and that the map is not injective. Those are exactly the core obligations. At the same time, one prominent contribution is an open pull request rather than a merged, stable library result. Repository authors also distinguish formalised statements from unproved structural extensions. That honesty is part of the evidence.
Formal verification does not decide attribution, explain why the map was discoverable, or measure how important the result will look in ten years. It does something narrower and crucial: it makes a mathematical claim testable at the level of exact definitions and derivations.
What the AI did—and what the record cannot yet tell us
The public account says Alpöge used Fable 5 and credited it with finding the counterexample. That is significant. Producing a viable candidate in a search space that resisted mathematicians for decades is different from merely polishing a proof somebody already supplied.
But the evidence available to readers does not provide a complete laboratory record of the interaction. We do not have, in the sources reviewed here, a fully reproducible transcript establishing how the problem was framed, what intermediate hints were supplied, how many attempts failed, which checks the model performed, or which human decisions redirected the search.
The responsible attribution is therefore collaborative and bounded: Mathew is credited with suggesting the problem, Alpöge with directing and announcing the work, Fable with producing the candidate according to that announcement, and independent mathematicians and formalisation contributors with checking and extending it. Claims that the model acted wholly alone go beyond the public evidence.
The practical lesson for AI users and operators
This episode offers a better evaluation pattern than asking whether a model “sounds intelligent”. Start with a task that can produce an external certificate. Separate generation from verification. Give the verifier the exact object, not the model’s confidence. Use more than one checking path where the stakes justify it, and publish the artefacts so outsiders can reproduce the result.
For a maths problem, that certificate may be a short counterexample, symbolic identity and formal proof. For software, it may be a failing test followed by a patch that makes the test pass. For finance, it may be reconciled source data and an auditable calculation. For publishing, it may be primary-source links and a claim ledger. The domain changes; the evidence discipline does not.
The weaker lesson would be “AI has solved mathematics”. The stronger one is that advanced models can sometimes search creatively while humans design the question, recognise a useful answer and build independent verification around it. The combination is more consequential than either uncritical celebration or reflexive dismissal.
How to read the headline a month from now
Three status changes would matter. First, whether the formalisation pull requests are reviewed and merged or materially corrected. Second, whether a stable manuscript presents the example, provenance and proof in a form the algebraic-geometry community can cite. Third, whether independent experts identify an error in the map, definitions or claimed implications.
Until then, the core counterexample deserves high confidence because it is explicit and multiply checkable. The historical narrative deserves more restraint because it is still forming. That is not hedging for its own sake. It is the difference between reporting evidence and reporting momentum.
For a separate example of evaluating model claims against cost and benchmark evidence, the cloud-versus-on-device compute guide uses the same discipline: identify what was measured, what remains unverified and what decision the evidence can support.
Sources and verification
Google DeepMind Formal Conjectures: open Lean formalisation pull request #4474
Formal verification and structure theory of the Alpöge Keller map
Christopher D. Long, “Small Counterexamples to the Gaussian Moments Conjecture”, arXiv:2607.18186
Kevin Buzzard, “Human mathematicians are being outcounterexampled”
Fortune: mathematicians react to the reported AI-assisted Jacobian counterexample
Verification note: The original announcement, repository status, formalisation claims, preprint and reporting were checked on 22 July 2026. The result is very recent; repository review, peer review and the scholarly consensus may change after publication.
