The 2026 Jacobian conjecture counterexample closely parallels Vitushkin's 1999 rational construction lifted to three dimensions
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The parallel is real and can be checked by hand. Example 1 of Vitushkin's 1999 note is a rational self-map of the plane with constant Jacobian determinant −2 that is generically two-to-one: it is built in two steps, a polynomial map (t, v) ↦ (t² + 2tv, t + v) whose fibres are the two roots of a quadratic in t and whose Jacobian is 2v, followed by a change of variables t = rs³, v = 1/s with Jacobian −1/v, which sends the ramification locus v = 0 to infinity and leaves the product −2. Because the second step needs 1/s, the result is rational rather than polynomial.
Alpöge's 2026 map on ℂ³ decomposes in exactly the same way one degree up. Writing v = 1 − Bt + 3At² and C = 2t − Bt² + 2At³, the map (A, B, t) ↦ (A, B, C) has fibres given by the three roots of a cubic in t and Jacobian 2v; the rational change of variables t = x/(1 + xy), v = 1/(1 + xy), with A and B chosen as polynomials in x, y, z, has Jacobian −(1 + xy) = −1/v, and composing the two gives Alpöge's three polynomials exactly, with determinant −2. This derivation, first published by Francisco Villatoro two days after the announcement, reproduces symbolically. The shared determinant is therefore not a coincidence but the same factorisation 2v · (−1/v), and the passage from two to three variables adds one parameter and raises the fibre equation from quadratic to cubic. Tao's geometric account of the counterexample, as a marked factorisation of a binary cubic into a linear and a quadratic form, contains the same correspondence: Vitushkin's example is the linear-times-linear case.
Two things the claim does not say should be kept apart from it. It does not establish that the counterexample was found by lifting Vitushkin's example; how the discovering system reasoned is unknown, and Villatoro's derivation was made after the fact. And the parallel in structure leaves open what is genuinely new in three dimensions, namely that the reparametrisation there can be made polynomial where in the plane it cannot; that question is treated on its own page. The one gap in the record is that Vitushkin's paper has been read only through others' reproductions of its example, which agree with one another and with the published map.
Full reasoning: the evidence and decisions behind this verdict
Direct verification. Taking Alpöge's polynomials A = (1+xy)³z + y²(1+xy)(4+3xy), B = y + 3x(1+xy)²z + 3xy²(4+3xy), C = 2x − 3x²y − x³z and Villatoro's substitutions t = x/(1+xy), v = 1/(1+xy), the identities v = 1 − Bt + 3At² and C = 2t − Bt² + 2At³ hold as rational-function identities; the Jacobian of (x,y,z) ↦ (A,B,t) is −(1+xy) = −1/v; the Jacobian of (A,B,t) ↦ (A,B,C) is 2(3At² − Bt + 1) = 2v; and the full determinant is −2, all confirmed by exact symbolic computation in this pass. For Vitushkin's map V(r,s) = (r²s⁶ + 2rs², rs³ + 1/s) the determinant is likewise −2 and V(−3,−1) = V(1,1) = (3,2). With t = rs³, v = 1/s the map becomes (t² + 2tv, t + v) with Jacobian 2v, and the fibre equation is t² − 2Qt + P = 0. The two constructions thus share the scheme: a d-to-one polynomial map whose fibre is the root set of a degree-d polynomial in one variable and whose Jacobian is 2v, composed with a reparametrisation of Jacobian −1/v that removes the ramification divisor v = 0. The 2026 map is the d = 3 case with one extra parameter; Vitushkin's is d = 2. This is precisely what "lifted to three dimensions" describes.
Sources. Villatoro's blog post (francis.naukas.com/2026/07/22/fable-5-refuta-la-conjetura-del-jacobiano/, 2026-07-22) and his X thread (x.com/emulenews/status/2080184306324984059, 2026-07-23) give the derivation and are one voice; he presents it as intuition about how the AI might have reasoned but the algebra is explicit. In the comment thread on Tao's digestion (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), an anonymous commenter on 29 July asserts the parallel; a reply calls the "nothing new" framing extreme but "the general idea is correct"; a comment of 8 August frames Vitushkin's idea as the k = 1 case of Tao's linear-times-degree-k factorisation, and Michael M. Ross agrees that the 2D case yields a rational map with a 1/s term while k = 2 yields a polynomial one. A note at mahalex.net (2026-07-22) says similar constructions appeared in Vitushkin's paper on rational maps. A GitHub gist independently reproduces Vitushkin's map in the same form and calls the −2 mechanism the gadget the 2026 family is built around. No source found denies the structural parallel; disagreement in the discourse is about what it implies for novelty, which is the parent claim's question, not this one.
Subclaims. Both dependencies stand: Alpöge's map has constant Jacobian −2 and is generically three-to-one is verified by exact computation and two proof-assistant formalisations; Vitushkin's 1999 paper constructs a non-injective rational self-map of ℂ² with constant nonzero Jacobian is not yet assessed on its own page but the map attributed to it is reproduced identically by independent sources and checks out.
Adversarial check. Could the parallel be superficial? The strongest objection is that the essential new step in three dimensions, a polynomial rather than Laurent reparametrisation (Tao's affine-space "miracle"), has no planar analogue, so the lift is not routine. That bears on novelty, not on whether the structures correspond; the claim as worded is about structure. A second objection is that Vitushkin may not present his example as a "construction method" at all. His paper could not be opened (Russian original; English version paywalled; the zbMATH review returned 403), so the reading here depends on the reproductions by Villatoro and the gist, which agree. That is the only reason confidence is not higher. Credence 0.9 reflects a small residual chance that the reproductions misrepresent the example or that the correspondence is judged looser than "closely parallels" by a specialist reading. What would change the verdict: a reading of Vitushkin's paper showing the example arises from a different mechanism, or a demonstration that the 2v · (−1/v) factorisation of Alpöge's map is an artefact of one coordinate choice rather than intrinsic (Tao's coordinate-free account argues against this).
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.
- requiresa load-bearing premise: the parent is false without itsteward instructions →Vitushkin's 1999 paper constructs a non-injective rational self-map of C² with constant nonzero Jacobian ↗︎
- requiresa load-bearing premise: the parent is false without itsteward instructions →Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The support comes from a handful of informed commentators writing in the weeks after the July 2026 announcement, none of them in a refereed venue, and two of the four recorded sources are one voice (Villatoro's blog post and his X thread of the next day). Only Villatoro does the work: his post exhibits an explicit derivation of Alpöge's map from a three-dimensional, cubic version of Vitushkin's two-step construction, and that derivation can be checked line by line. The remaining sources state the parallel without working it out. Every source reproduces Vitushkin's 1999 example at second hand; the paper itself has not been opened for this claim, and readers should start with Villatoro's post.
Contrary to all the hype, the Jacobian conjecture counterexample is not that exciting because it relies on techniques established by Vitushkin in his 1999 paper. So there really is nothing new here. The only thing different is that here it’s applied to three dimensions and one gets a polynomial map, whereas in two dimensions one only gets a rational map.
Comment on Tao's digestion of the counterexample arguing the result is Vitushkin's 1999 technique applied in three dimensions. A reply on 30 July calls this "probably a bit extreme but the general idea is correct"; a later comment (8 August) frames Vitushkin's idea as the k = 1 case of multiplying a linear form by a degree-k form, and Michael M. Ross (9 August) notes Vitushkin's 2D construction gives a rational map with a 1/s term while k = 2 gives a genuine polynomial one.
Asserted without evidence of the source's own. An anonymous comment states the parallel to Vitushkin's 1999 paper without working it out; replies in the thread accept the general idea while calling the "nothing new" framing extreme, and a later comment locates Vitushkin's example as the linear-times-linear case of the factorisation construction that Tao's post develops for linear-times-quadratic. The thread thus records agreement on the structural parallel and disagreement only about what it implies for novelty.
Mi intuición me dice que puede haber transformado dicho ejemplo en dos dimensiones (t, v) en un ejemplo en tres dimensiones (A, B, C). Definiendo v = 1 − B t + 3 A t², integrando C = 2 ∫v dt = 2 t − B t² + 2 A t³, se tiene ∂C/∂t = 2 − 2 B t + 6 A t² = 2 v. Luego la función (A,B,t) ↦ (A,B,C), cumple que det[∂(A,B,C)/∂(A,B,t)] = 2 v. El análogo a la ecuación cuadrática de Vitushkin es la ecuación cúbica 2 A t³ − B t² + 2 t − C = 0.
Villatoro notices that the Alpöge map and Example 1 of Vitushkin's 1999 paper share Jacobian determinant −2, reproduces Vitushkin's two-step construction (a map with Jacobian 2v followed by a reparametrisation with Jacobian −1/v), and then, with ChatGPT assistance and explicitly a posteriori, exhibits the three-dimensional analogue: the rational change of variables t = x/(1+xy), v = 1/(1+xy) turns the cubic-fibre map (A,B,t) ↦ (A,B,C) into exactly Alpöge's three polynomials. He stresses that whether the discovering system followed this route is unknown.
The source's own evidence bears what it asserts. The post frames its conclusion as intuition, but then does the work: it reproduces Vitushkin's example, writes the three-dimensional analogue with a cubic fibre equation, and shows that a rational change of variables turns that analogue into Alpöge's three polynomials exactly. The algebra is elementary and checks out. What the post does not and cannot show is that the counterexample was actually found this way; the author says so explicitly and notes the derivation was done after the fact with an AI assistant. Worth reading closely: It contains the only explicit derivation on record of the 2026 map from a Vitushkin-style construction; reproducing its identities settles the structural parallel directly.
My intuition is that the two-dimensional construction in the variables (t,v) may have been lifted to a three-dimensional construction in variables (A,B,C). Resulting in a couterexample with |Jac(F)|= −2.
English thread restating the argument of Villatoro's 22 July blog post: Vitushkin's Example 1, its two-step Jacobian factorisation 2v · (−1/v) = −2, and the cubic analogue 2At³ − Bt² + 2t − C = 0 reproducing Alpöge's map. Hedged as intuition, but the derivation is given explicitly.
Only search excerpts of this thread were seen; its content matches the author's blog post of the previous day, which was read in full.
The `second wave' (that generated a conceptual explanation) seems more surprising; but then again, similar constructions appeared previously in a paper by A. G. Vitushkin (on rational maps, not polynomial).
Blog reflection on the counterexample and on the role of AI; in passing, it states that constructions similar to the conceptual explanation of the counterexample appeared earlier in Vitushkin's paper on rational maps. A weaker statement ("similar constructions") than a close structural parallel.
Asserted without evidence of the source's own. Only an excerpt was seen. The remark offers no working; it states that similar constructions appear in Vitushkin's paper on rational maps and moves on.
How these sources relate
- https://francis.naukas.com/2026/07/22/fable-5-refuta-la-conjetura-del-jacobiano/ draws its statement from Evaluation of the Jacobian of a rational transformation of C² and some applications (A. G. Vitushkin, Math. Notes 66, 1999). The post's whole argument starts from Example 1 of Vitushkin's paper, which it reproduces and then generalises. Vitushkin's paper itself could not be opened (Russian original; the English version is paywalled and the zbMATH review returned 403), so whether the reproduction is faithful to how Vitushkin presents and uses the example is not established here. The map itself is independently reproduced in the same form by another 2026 note, which lends the reproduction credibility. Judged from the citing document alone.
- From Vitushkin |Jac(V)| = −2 counterexample to Fable 5 Jacobian Conjecture |Jac(F)| = −2 counterexample and https://francis.naukas.com/2026/07/22/fable-5-refuta-la-conjetura-del-jacobiano/ share an author. Both are by Francisco R. Villatoro: the Naukas blog post of 22 July 2026 and his X thread of 23 July 2026 restating the same argument in English.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.