Tiny Polynomial Map Ends Jacobian Conjecture for Higher Dimensions

Levent Alpöge, a mathematician at Anthropic, posted a three-variable polynomial map on 19 July 2026 whose Jacobian determinant is the constant -2 yet three distinct inputs collide at one output. The map, found with help from Claude Fable 5, falsifies the Jacobian conjecture for every dimension greater than two.

The two-dimensional case stays open. The discovery also collapses several long-linked conjectures and shows frontier models can surface simple unexpected objects that decades of human search missed.

The Map That Fits in One Post

Alpöge’s casual X announcement with the full map gave the explicit functions from C³ to C³:

Symbolic computation confirms the Jacobian determinant equals exactly -2 at every point. The same map sends the three points (0, 0, -1/4), (1, -3/2, 13/2) and (-1, 3/2, 13/2) to the single output (-1/4, 0, 0). Local invertibility holds everywhere by the inverse-function theorem, yet no global inverse exists, polynomial or otherwise.

The example extends immediately to all higher dimensions by padding with identity maps on the extra coordinates. Degree is seven, short enough that any computer-algebra system or even careful hand calculation verifies both the constant determinant and the collisions within minutes.

That brevity matters. A counterexample this compact can be checked by hand or machine without specialized hardware, so the claim does not rest on opaque computation. Once the three polynomials are written down, the rest is routine algebra.

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final

Alpöge wrote that line, crediting Akhil Mathew for the suggestion and Claude Fable 5 for the work done while the World Cup final played. The post drew tens of millions of views.

Eighty-Seven Years of Near Misses

Ludwig Kraus stated the two-variable version in 1884. Ott-Heinrich Keller generalized it to n dimensions in 1939. Stephen Smale placed it on his 1998 list of problems for the next century. Many claimed proofs appeared, among them attempts by Beniamino Segre and Wolfgang Gröbner; each later failed on a subtle gap.

1. 1884, Kraus formulates the planar case.
2. 1939, Keller states the general conjecture.
3. 1970s-1990s, reductions show it is enough to check cubic-homogeneous or cubic-linear maps; integer-coefficient examples with determinant 1 would suffice if any counterexample exists.
4. 1994, Sergey Pinchuk builds a real two-variable map that is locally invertible everywhere but not globally; its Jacobian is non-constant, so the classical conjecture survives.
5. up to 2025, computer checks confirm the planar case for degrees through 104.
6. 19 July 2026, the three-variable counterexample appears.

Special cases held: degree 2 is true, certain symmetric forms are true, birational or Galois cases are true. The general statement above dimension 2 is now false.

The long interval between Keller’s formulation and the counterexample was filled with partial positive results and failed full proofs. Each reduction narrowed the search but never closed it for dimensions three and higher. The 1994 real-plane example of Pinchuk showed that local invertibility alone does not force global invertibility once the constant-Jacobian condition is dropped; that distinction kept the classical statement alive until the new map appeared.

What Else Falls With It

Several conjectures are known to be equivalent to the Jacobian conjecture or to imply it. Their higher-dimensional versions therefore collapse at once.

• Dixmier conjecture (endomorphisms of Weyl algebras are automorphisms) fails for n > 2.
• Poisson conjecture fails in the same range.
• Related vanishing and moments statements that implied the Jacobian statement for large n are now known false, and smaller independent counterexamples have already appeared for some of them.

The two-variable Jacobian conjecture remains equivalent to the two-variable Dixmier and Poisson statements; those stay open.

Because the logical links run in both directions for many of these statements, a single explicit map in three variables is enough to settle an entire cluster. The same three points that collide under the polynomial map supply concrete witnesses against every equivalent higher-dimensional claim.

Search Space, Not Proof Length

Most recent AI math successes produced long, intricate arguments that combine distant techniques. This one is different. The counterexample is short. The hard part was locating any polynomial map that is simultaneously of constant non-zero Jacobian and non-injective.

Mathematicians had long noted that, in principle, a clever undergraduate formula might suffice. The space of candidate maps is enormous; exhaustive search is impossible. Claude Fable 5, released to the public only weeks earlier under Claude Fable 5 public release notes and the accompanying Mythos guardrails on the public Fable 5 model, navigated that space during a single evening.

Details of the prompts and intermediate outputs remain private. Independent geometric reconstructions appeared within a day. Andy Jiang posted a reformulation; Terence Tao later wrote a full Terence Tao geometric digestion of the map that explains the construction via multiplication of binary forms, resultant normalizations that kill scaling, and a special affine slice that turns out to be polynomially isomorphic to affine 3-space. The cancellations that keep the Jacobian constant still look miraculous when written out, yet they follow from the geometry once the right slice is chosen.

The contrast with earlier AI proofs is sharp. Those results grew by stitching lemmas across fields until the argument became long enough to be novel. Here the novelty lives entirely in the choice of object. Once found, the object needs almost no further argument: constant determinant plus three colliding points finish the disproof.

The Plane Case and the Missing Story

Padding cannot reduce a three-dimensional counterexample to two variables. The original planar Jacobian conjecture is therefore untouched. Computational evidence up to high degree continues to support it, and many specialists regard the two-variable question as the deeper core.

Verification of the new map is easy; understanding why it works, in the sense of a clean conceptual narrative one can rebuild without the computer, is still incomplete. Akhil Mathew and others noted that a verified answer without the accompanying story leaves a gap mathematicians usually fill. The process that produced the map is likewise only partly public: a human posed the question, the model generated candidates, a human checked and announced. How much steering occurred is unknown.

Crowd reaction split along familiar lines. Some treated the result as another casual flex (“now order the chicken tenders”). Others, including Qiaochu Yuan, recalled the conjecture’s long history of false proofs by serious mathematicians and observed that sheer age of an open problem is weaker evidence of intrinsic difficulty than it once seemed. Career anxiety surfaced too: if models can hand over verified objects overnight, the training path that builds human intuition may need redesign.

How the Geometry Makes the Cancellations Work

Tao’s digestion supplies the missing geometric frame. Multiplication of binary forms produces families of maps whose Jacobians are controlled by resultants. Normalizing those resultants removes scaling freedom and isolates a lower-dimensional slice. On that slice the map becomes polynomially isomorphic to ordinary affine 3-space, yet the same normalizations force the three chosen points to land on one another.

The constant value -2 is then no longer an accident of coefficient chasing. It is the residual determinant left after every scaling factor has been killed. The same mechanism explains why padding with identity coordinates preserves both the constant Jacobian and the collisions in every higher dimension: the extra variables never enter the resultant calculation.

What remains incomplete is a fully human-scale derivation that begins from the binary-form multiplication and ends at the three explicit polynomials without intermediate computer search. The geometric outline is now public; the shortest path through the outline is not yet written.

Object Discovery Joins Proof Construction

Earlier 2026 results already showed models can assemble novel proofs that draw on multiple fields. The Jacobian counterexample adds a complementary strength: locating simple but unexpected mathematical objects inside huge combinatorial spaces when an exact verifier exists.

That pattern matches other recent disproofs and constructions. It also matches the model’s design strengths. Fable 5 general availability details emphasize long-horizon autonomy and scientific research performance. Navigating a space of polynomials until a constant-Jacobian non-injection appears is precisely the kind of extended, checkable search the architecture supports.

• Human supplies the question and the verifier.
• Model explores the combinatorial space overnight.
• Human checks the candidate and announces.
• Community supplies geometric insight afterward.

Human mathematicians remain essential for posing the right questions, for geometric insight after the fact, and for the still-open planar case. The division of labor is shifting. Models now reliably surface candidates that would have taken years of manual trial; humans still supply the framing and the deeper explanation.

The Jacobian conjecture above dimension two is settled. A short map did the work. A cluster of related statements fell with it. The two-dimensional problem and the full conceptual story of the counterexample wait for the next round.

Frequently Asked Questions

What exactly does the Jacobian conjecture claim?

If a polynomial map from n-dimensional space over a characteristic-zero field to itself has Jacobian determinant equal to a non-zero constant, then the map admits a polynomial inverse. Local invertibility is already guaranteed by the inverse-function theorem; the conjecture asserted global polynomial invertibility.

What is the precise counterexample map?

The three components are f1 = (1 + xy)³z + y²(1 + xy)(4 + 3xy), f2 = y + 3x(1 + xy)²z + 3x y²(4 + 3xy), f3 = 2x – 3x²y – x³z. Its Jacobian determinant is identically -2, and it identifies at least the three points (0,0,-1/4), (1,-3/2,13/2), (-1,3/2,13/2).

Why does the two-dimensional case remain open?

A counterexample in three or more variables cannot be compressed into two variables by any simple padding or restriction that preserves both the constant-Jacobian property and the collisions. All known reductions and computer checks for the plane stay consistent with the conjecture.

How can anyone verify the result quickly?

Any computer-algebra system (SymPy, Mathematica, Sage, WolframAlpha) computes the symbolic Jacobian determinant and evaluates the map at the listed points in seconds; the arithmetic is exact and requires no floating-point approximation.

Which other conjectures are now known false in higher dimensions?

The Dixmier conjecture and the Poisson conjecture fail for n > 2 because each is equivalent to, or implies, the Jacobian conjecture in the corresponding dimension. Several vanishing and moments conjectures that implied high-dimensional Jacobian statements have also acquired independent small counterexamples.