Reporting status, July 20, 2026: The public polynomial map attributed to Claude Fable passes exact, independently reproducible checks and appears to disprove the general Jacobian conjecture. No formal paper, complete model transcript or official Anthropic announcement was publicly available when this article was prepared. The famous two-variable case remains open.
A striking claim moved through mathematics and AI circles this weekend: Claude Fable may have found a counterexample to the Jacobian conjecture, a problem that has resisted mathematicians for generations. The initial announcement arrived not as a journal paper or corporate press release, but as an X post by mathematician Levent Alpöge. It contained something more valuable than a sweeping assertion: an explicit polynomial map in three variables and three distinct points that allegedly collide under that map.
The crucial distinction is this: the displayed mathematics and the discovery story do not have the same evidentiary status. The algebra is finite, exact and reproducible. We independently checked it in symbolic computation, and other mathematicians quickly published their own checks. The map has constant Jacobian determinant -2, yet it sends three different inputs to the same output. If the formula was transcribed correctly, that is a counterexample. The universal, all-dimensional Jacobian conjecture is false.
What remains uncertain is how Claude Fable found it, which exact Fable version was used, what prompts and tools were supplied, how much human steering occurred, and how credit should ultimately be apportioned. Those are important questions about AI-assisted discovery. They do not alter the two short algebraic tests.
Key takeaways
- The public map checks out. Its Jacobian determinant simplifies identically to the nonzero constant
-2. - It is not one-to-one. Three distinct rational input points all map exactly to
(-1/4, 0, 0). - That refutes the general conjecture. A counterexample in three variables is enough to make the universal statement false, and it extends to every dimension above three by adding untouched coordinates.
- The plane case survives. The two-variable Jacobian conjecture—the version closest to the historical starting point—remains open.
- The AI provenance is not yet auditable. Alpöge credits “Fable,” but the full prompt history, model version and human workflow have not been released.
What the Jacobian conjecture actually says
Take a polynomial map F: C^n → C^n. It accepts n complex numbers and returns n polynomial expressions. The Jacobian matrix records all first partial derivatives of those output polynomials with respect to the inputs. Its determinant describes how the map behaves infinitesimally.
If that determinant never vanishes, the inverse function theorem says the map is locally reversible: near each point, it behaves like a one-to-one change of coordinates. The Jacobian conjecture asked whether a much stronger global conclusion follows for polynomial maps. If the determinant is a nonzero constant, must the map be globally invertible, with a polynomial inverse?
That leap from local control to global control is the heart of the problem. A road can look perfectly regular in every short segment while looping back to a place it visited earlier. The conjecture said polynomial structure plus a constant, nonzero Jacobian was rigid enough to rule out the analogous global collision.
Stephen Smale included the question as Problem 16 in his 1998 list of mathematical problems for the next century. A 2026 paper in the Journal of the London Mathematical Society, written before this counterexample surfaced, still called it one of mathematics’ outstanding open problems and noted that even the case n = 2 remained unresolved.
How the Claude Fable claim emerged
Alpöge posted the example late on July 19 Pacific time, or early July 20 UTC. His post said that he and a close friend identified only as Akhil spent the morning asking Fable about the problem. The post then gave a map F: C^3 → C^3, stated that its determinant is -2, and listed three inputs with the same output.
Alpöge is not an anonymous account chasing engagement. His Harvard mathematics page documents a substantial research record and currently lists Anthropic. That makes the source credible, but it should not be misreported as an Anthropic announcement. As of publication, we found no company newsroom post or technical report claiming the discovery.
There are two additional naming cautions. Alpöge’s post says “Fable,” not definitively “Fable 5,” even though the latter is Anthropic’s current model and the subject of Kingy.ai’s earlier Claude Fable 5 explainer. It also does not give Akhil’s surname. Secondary posts have filled in both blanks, but Kingy.ai is not treating those inferences as confirmed facts.
The explicit three-variable map
Here is the counterexample as posted, with x, y and z as complex variables:
F1 = (1 + xy)^3 z
+ y^2(1 + xy)(4 + 3xy)
F2 = y + 3x(1 + xy)^2 z
+ 3xy^2(4 + 3xy)
F3 = 2x - 3x^2y - x^3zThe three component polynomials have total degrees 7, 6 and 4. That is compact enough to copy into Sage, SymPy, Mathematica or another computer algebra system. More importantly, every coefficient is an integer and every test point is rational. There is no numerical approximation, floating-point tolerance or interpretive judgment hiding in the result.
This is why the announcement is fundamentally different from the many purported “proofs” that have surrounded the Jacobian conjecture. A long proof can contain a subtle gap in an argument about infinity, algebraic extensions or a reduction step. A proposed counterexample only has to survive the definition. Compute the determinant. Evaluate the map at the listed points. If both results are exact, the conjecture loses.
The evidence: two exact checks
Check 1: the Jacobian determinant is a nonzero constant
Differentiate the three components with respect to x, y and z, assemble the resulting 3-by-3 matrix, and take its determinant. Exact symbolic simplification gives:
det(JF) = -2Kingy.ai reproduced that identity in SymPy. An independent Sage verification script linked from Qiaochu Yuan’s MathOverflow answer reaches the same result. This is an identity of polynomials, not a sample taken at a few points.
There is also a compact human-readable route. Set u = 1 + xy, w = 2 - 3xy - x^2z and q = u^2z + y^2(4 + 3xy). Then the map becomes F = (uq, y + 3xq, xw). On the dense region where x ≠ 0, changing coordinates to (x,u,w) produces Jacobian determinants -1/x^3 for the input transformation and 2/x^3 for the output transformation. Their ratio is -2. Because the original determinant is polynomial, equality on that dense region establishes the identity everywhere.
Some online comments object that the traditional statement uses determinant 1, not -2. That is only a normalization. “Nonzero constant” is the standard hypothesis. If determinant one is desired, multiply the first output coordinate by -1/2. The new determinant is exactly 1, and the collision remains.
Check 2: three distinct inputs have one output
Now evaluate the map at the three points supplied by Alpöge:
F( 0, 0, -1/4)
= (-1/4, 0, 0)
F( 1, -3/2, 13/2)
= (-1/4, 0, 0)
F(-1, 3/2, 13/2)
= (-1/4, 0, 0)These inputs are plainly different, while their outputs agree coordinate by coordinate. The map is therefore not injective. A function that assigns the same output to multiple inputs cannot have an inverse, polynomial or otherwise.
Together, the two checks are decisive: the map satisfies the conjecture’s premise and violates its conclusion. Stanford mathematician Jared Duker Lichtman publicly walked through the collision, while Yuan described the example as symbolically easy to verify. These reactions are early expert scrutiny, not journal peer review, but they are backed by reproducible algebra rather than reputation alone.
Does non-properness invalidate the counterexample?
No. This is the most persistent technical misunderstanding in the online discussion.
A proper map cannot send a sequence escaping to infinity into a bounded region. The proposed map is not proper. For example, along the inputs (t, -1/t, 5/t^2), the norm of the input grows without bound as t grows, while the output is (0, 2/t, 0), which approaches the origin.
But properness is not a hypothesis of the Jacobian conjecture. It is an extra global condition known to help guarantee invertibility. The Encyclopedia of Mathematics overview describes properness as a sufficient additional condition; Smale’s formulation does not assume it. Saying this map is non-proper explains where its global behavior escapes local control. It does not disqualify the example. In fact, a genuine counterexample must fail some global condition of this kind.
What is verified—and what remains unverified
| Claim | Assessment as of July 20, 2026 |
|---|---|
| The displayed formula is publicly available | Confirmed. It appears in Alpöge’s original X post and in the MathOverflow record. |
The Jacobian determinant equals -2 |
Exactly reproducible. Independent Sage and SymPy calculations agree. |
| Three distinct inputs collide | Exactly reproducible. Direct substitution uses rational arithmetic. |
| The general Jacobian conjecture is false | Logical consequence of the two checks, assuming the posted formula is the intended map. |
| Claude Fable found the map | Credible attribution, not independently auditable. Alpöge credits Fable, but the full model session is not public. |
| The exact model was Fable 5 | Unconfirmed. The original announcement names only Fable. |
| The work was autonomous | Unknown. We do not have the prompts, tool calls, search process, retries or extent of human steering. |
| Anthropic officially announced the result | No. This was a personal announcement by a mathematician whose page lists Anthropic. |
| A formal paper has passed journal peer review | No evidence found. Early public verification is not the same as refereed publication. |
The missing transcript matters for the history of AI, not for evaluating the written map. Was the model asked to search a constrained family? Did Alpöge or Akhil supply a promising ansatz? Did Fable write and run symbolic code? How many failed candidates came first? Did a human simplify the output? A discovery record would help scholars understand the contribution and reproduce the workflow.
This is closely related to the broader distinction Kingy.ai made in its analysis of Anthropic’s claim that AI is helping build AI: automation of a measurable research loop is meaningful, but it is not evidence that every part of open-ended research has become autonomous.
What remains open after the counterexample
The dimensional qualifier is essential.
- Dimension 1: the conjecture is true for an elementary reason. A one-variable polynomial with constant nonzero derivative is linear.
- Dimension 2: the plane Jacobian conjecture remains open.
- Dimension 3: the displayed map supplies a counterexample to the general statement.
- Dimensions above 3: append identity coordinates to the three-variable map. The determinant stays nonzero and the original collision remains, so the conjecture is false there too.
The history also needs precision. The conjecture is commonly associated with Ott-Heinrich Keller’s 1939 work, which is why “87-year-old problem” is circulating. A 2026 historical study in Comptes Rendus Mathématique traces the precise statement to Ludwig Kraus in 1884 and identifies a flaw in Kraus’s attempted proof. Either way, saying that “Keller’s original plane problem has been disproved” would be wrong. The counterexample ends the universal formulation while leaving the two-variable frontier intact.
Researchers may now ask what additional hypotheses rescue a useful theorem, whether the three-variable example can be simplified further, and how it interacts with the major reduction frameworks built over decades. A false conjecture does not erase that body of work. It changes the destination.
What this shows—and does not show—about AI mathematical discovery
If the attribution holds, this is an important AI-assisted mathematical discovery precisely because the output is externally checkable. The strongest AI results are not those that merely sound sophisticated; they are those that produce an artifact humans can test independently. Here, the artifact is a small set of integer-coefficient polynomials and rational points. Verification does not require trusting Anthropic, Fable, Alpöge, Kingy.ai or a benchmark.
That does not prove that language models are generally reliable theorem provers. It does not establish autonomy, erase the contribution of expert prompting, or show that a model can navigate every hard research problem. It also has no immediate demonstrated impact on robotics, physics, cryptography or commercial software. The immediate significance is in pure mathematics and in the methodology of AI-assisted discovery.
The episode strengthens a narrower claim: frontier models can sometimes be useful search partners in spaces where success has an exact verifier. That is consistent with recent systems that generate code, run symbolic algebra and iterate against crisp tests. It also reinforces the case for access, reproducibility and independent evaluation raised in Kingy.ai’s coverage of the Claude Fable research-access debate. A spectacular answer is more scientifically valuable when the path to it can be inspected.
Kingy.ai verdict
The careful headline is not “Anthropic proved the Jacobian conjecture.” A counterexample disproves a conjecture, and Anthropic has not officially claimed this result. The defensible conclusion is stronger and more precise:
An explicit three-variable map attributed to Claude Fable satisfies the Jacobian conjecture’s constant-determinant hypothesis and fails injectivity. The algebra is exact, short and independently reproducible. With very high confidence, the general Jacobian conjecture is false. The two-variable case remains open, while the exact AI discovery process and final credit remain unverified.
Kingy.ai will update this article if a full prompt transcript, formal mathematical note, clearer attribution or official Anthropic statement becomes available.
Frequently asked questions
Has the Jacobian conjecture been disproved?
The general, all-dimensional statement appears to be disproved by the explicit map in three variables. Its determinant is the nonzero constant -2, but it is not injective. The two-variable or plane case remains open.
Did Claude Fable discover the counterexample?
Levent Alpöge attributes the discovery to Claude Fable. That is credible first-party testimony, but no complete prompt transcript, model log or research notebook has been released, so the end-to-end discovery process cannot yet be independently audited.
Why is a Jacobian determinant of -2 enough?
The conjecture assumes a nonzero constant determinant. Determinant 1 is a convenient normalization, not an extra mathematical requirement. Scaling one output coordinate converts this example to determinant 1 without removing the collision.
Does the map’s non-properness invalidate it?
No. Properness is not among the conjecture’s hypotheses. Its failure explains how the map can be locally reversible everywhere but still fail globally.
Has the result been peer reviewed?
Not in the journal sense as of July 20, 2026. Mathematicians have independently reproduced the short symbolic checks in public, which strongly validates the displayed map, but a formal paper and conventional refereeing process have not yet appeared.