In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The geometric form of the 2026 counterexample to the Jacobian conjecture takes the map P¹ × Sym²(P¹) → Sym³(P¹) sending (p, {q, r}) to {p, q, r}, removes a hyperplane H tangent but not osculating to the small diagonal from the target, and removes the ramification divisor R together with the preimage of H from the source. The target complement is obviously affine three-space; the claim is that the source complement is too, which is what makes the restricted map a polynomial self-map of A³ with constant Jacobian that fails to be injective.
The claim is established by several independent arguments that agree. In coordinates [u:v] on P¹ and [a:b:c] for the binary quadratic ax² + bxy + cy², the hyperplane pullback is av + bu = 0 and the ramification divisor is av² − buv + cu² = 0; the complement is the affine variety where both equal one. Writing D = 2vb − uc, the second equation reduces to v = 1 + uD, and the map (u, v, a, b, c) ↦ (u, D, −2D²a − 4Db + c) has a polynomial inverse, so the complement is isomorphic to A³ by an explicit polynomial change of coordinates. This has been checked symbolically. Independently, the complement can be written as an affine-line bundle over an affine-line bundle over the affine line, hence affine space; a count over a finite field with q elements gives exactly q³ points; and a September 2026 preprint by Remy van Dobben de Bruyn derives the isomorphism from a general criterion for complements of divisors in projective bundles over P¹ and proves the converse, that the complement is affine three-space only when H is tangent but not osculating.
No source in the discourse disputes the result. One caution: the ChatGPT-generated coordinate formula first circulated in the Secret Blogging Seminar comment thread does not define an isomorphism as written, though the corrected coordinates given later in the same thread do, and the result never depended on that formula alone.
Full reasoning: the evidence and decisions behind this verdict
This pass re-examined the claim after the previous assessment and found the landscape moved in one respect: a preprint, arXiv:2608.27341 (v1 27 Aug 2026, v2 14 Sep 2026, arxiv.org/html/2608.27341), now proves the isomorphism as Theorem A.7 from a general criterion (Theorem 3.1: for two irreducible divisors of classes df + h and ef + h in a projective bundle over P¹, the complement is affine n-space iff |e − d| = 1 and the fibres coincide over exactly one point). The appendix shows R has bidegree (2,1), the pullback of H has bidegree (1,1), and the fibres coincide exactly over the tangency point when H is tangent to order 2. Version 2 corrected the statement and proof of the general theorem after Charles Newton pointed out a mistake; the counterexample case in the appendix is unchanged in v2 and is in any case independently established, so the correction does not weaken the claim.
The previous pass had not re-derived the explicit isomorphism symbolically. This pass did, from the formulas in the Secret Blogging Seminar thread (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/). With H = av + bu and R = av² − buv + cu², the identity R = vH − uD with D = 2vb − uc holds, so on the slice H = R = 1 one has v = 1 + uD. Solving the linear system av + bu = 1, 2vb − uc = D, −2D²a − 4Db + c = E for (a, b, c) gives a 3×3 matrix of determinant exactly 2, hence polynomial solutions: a = (2 − 3Du − Eu²)/2, b = (D + Eu + 3D²u + DEu²)/2, c = E + 4D² + 2DEu + 3D³u + D²Eu². Composing forward and inverse returns (u, D, E) identically, so the slice, and hence the complement of R ∪ φ⁻¹(H), is isomorphic to A³. The slice is a section of the two-dimensional torus action (H has weights (1,1) and R weights (2,1), so H = R = 1 forces both scalars to be 1), which is why the affine model represents the complement.
The same check shows the ChatGPT formula relayed by Will Sawin, with third coordinate 12vb² − 6ubc + 4b² + 2a − 3ac, does not define an isomorphism as transcribed: expressed in the coordinates (u, D, E) it is quadratic in E (for u = 1, D = 0 it equals (5E² − 8E + 4)/2), and the Jacobian of the map (u, D, F) is non-constant. The recorded instance from that thread is therefore an overstatement as to its specific formula, though the conclusion is right and the same thread supplies the correct coordinates (the comment by "bruno"), David Speyer's finite-field count (q² + q(q² − q) = q³), and Sawin's own A¹-bundle-tower argument.
Instances: all four recorded instances affirm (Speyer/Sawin thread, Tao's digestion terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/, van Dobben de Bruyn's blog post, and the arXiv preprint). Two of these are one author; the remaining voices are independent and use different methods. No denial or qualification was found in a web search of the discourse. The verified sibling result that Alpöge's explicit map has constant Jacobian −2 yet is generically three-to-one remains consistent with this.
What would change the verdict: an error common to the explicit coordinate isomorphism (now checked directly), the bundle-tower argument, the point count, and the preprint's criterion. Given the direct symbolic verification, that is now very unlikely; the residual uncertainty is essentially the possibility of a shared misreading of which divisors the construction removes.
Findings
Noted by the graph's administrators in the course of their work, in their own words; all findings.
The ChatGPT-derived coordinate formula first circulated as the explicit isomorphism between the Jacobian-counterexample source variety and affine three-space does not define an isomorphism, though a corrected formula from the same thread does.
In the July 2026 discussion of the counterexample to the Jacobian conjecture, the source variety of the geometric construction, the complement of the ramification divisor and the hyperplane preimage in P¹ × P², was widely reported to be affine three-space by an explicit change of coordinates relayed by Will Sawin from ChatGPT: on the affine slice where av + bu = 1 and av² − buv + cu² = 1, the map (v, u, a, b, c) ↦ (u, 2vb − uc, 12vb² − 6ubc + 4b² + 2a − 3ac) was said to "give an isomorphism from the complement of R and H to A³." That statement is the recorded instance on claim a21e1db1-f393-4eb0-9543-7411544c2015.
Direct symbolic computation, recorded in the claim's current assessment, shows the formula as transcribed is not an isomorphism. Writing D = 2vb − uc, the two defining equations force v = 1 + uD, and the coordinates (u, D, E) with E = −2D²a − 4Db + c (proposed by the commenter "bruno" later in the same thread) do give a polynomial isomorphism with polynomial inverse, the relevant 3×3 matrix having determinant exactly 2. Expressed in these coordinates, Sawin's third component is quadratic in E (for u = 1 and D = 0 it equals (5E² − 8E + 4)/2), so the map (u, D, F) is neither injective nor of constant Jacobian.
The conclusion that the complement is affine three-space is nonetheless correct and verified, resting on the corrected coordinates, on the bundle-tower arguments of Sawin, Litt, and van Dobben de Bruyn, on Speyer's finite-field point count, and on Theorem A.7 of arXiv:2608.27341. The point is narrow: anyone taking the first circulated formula as the explicit polynomial change of variables should use the corrected one instead.
Cites claim a21e1db1claim 2fb3c345
Decomposition
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Provenance
Where this claim has been said, linked to its canonical form.
Support comes from the July 2026 discussion threads and a later preprint, and it is not a chain of restatements: the Secret Blogging Seminar thread contains a point count, an explicit coordinate change, and a bundle-tower argument; Terence Tao's post recovers the trivialisation by a rank analysis; and Remy van Dobben de Bruyn gives a bundle-tower proof on his blog and, in a September 2026 arXiv preprint by the same author, a proof from a general criterion together with the converse. One caution for readers: the ChatGPT-derived formula quoted in the Secret Blogging Seminar comment does not itself define an isomorphism as written, though a corrected coordinate change later in the same thread does, and the conclusion rests on several independent arguments rather than on that formula.
There exists an affine variety that is isomorphic to [A³] by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.
Tao's reformulation of the counterexample (his Theorem 3) rests on the source variety of the covering construction being isomorphic to affine three-space by polynomial changes of variable; the post walks through why this holds, recovering Sawin's trivialisation via a rank analysis of the bundle structure.
Then (v,u,a,b,c) => (u, 2vb – uc, 12 v b^2 – 6 ubc + 4 b^2 +2a – 3ac) gives an isomorphism from the complement of R and H to A³.
Will Sawin, relaying a ChatGPT-assisted proof that the source X of the geometric construction of the counterexample map is isomorphic to affine three-space; several independent proofs are later given by Speyer, Levinson, and van Dobben de Bruyn.
The assertion outruns the source's own evidence. The comment gives the correct affine model of the complement (the slice where the equations of the hyperplane pullback and of the ramification divisor both equal one) but the specific third coordinate it states, 12vb² − 6ubc + 4b² + 2a − 3ac, does not define an isomorphism: on the chart it is quadratic in the remaining fibre variable and the resulting map has a non-constant Jacobian. The conclusion is nonetheless established within the same thread by other means: a point count over finite fields giving exactly q³, the same commenter's later argument writing the complement as an affine-line bundle over an affine-line bundle over a line, and a corrected coordinate change in a subsequent comment (u, 2vb − uc, −2D²a − 4Db + c with D = 2vb − uc) which does have a polynomial inverse. Worth reading closely: The thread contains the corrected explicit isomorphism and the bundle-tower argument, both of which independently settle the claim; the quoted formula itself should not be relied on.
To construct an isomorphism, it suffices to construct iterated (Zariski) A¹-bundles, by Lemma 3.5 of this preprint. [...] This produces the tower of A¹-bundles, finishing the proof.
A blog post giving a self-contained geometric account of the counterexample construction; it notes that the isomorphism of the source with affine three-space "was the main point of discussion" and then proves it by exhibiting the complement as a tower of Zariski A¹-bundles. (Formulas were stripped in retrieval; the quoted text elides them.)
The complement U = 𝐏 1 × 𝐏 2 ∖ ( R ∪ φ − 1 ( H ) ) U=\mathbf{P}^{1}\times\mathbf{P}^{2}\setminus(R\cup\varphi^{-1}(H)) is isomorphic to 𝐀 3 \mathbf{A}^{3} if and only if H H is tangent but not osculating to the small diagonal.
Theorem A.7 of the appendix, which derives the isomorphism from a general criterion (Theorem 3.1) for complements of two divisors in a projective bundle over P¹ to be affine space, and supplies the converse. Version 2 corrected the statement and proof of the main theorem after an error was pointed out; the appendix conclusion is stated unchanged.
The source's own evidence bears what it asserts. The preprint proves the isomorphism from a general criterion rather than by coordinates, and adds the converse: the complement is affine three-space exactly when the hyperplane is tangent but not osculating to the small diagonal. The ramification divisor is shown to have bidegree (2,1) and the hyperplane pullback bidegree (1,1), and the tangency condition is shown to make their fibres over the tangency point coincide. The second version corrected the statement and proof of the general theorem after a reader pointed out a mistake; the appendix theorem for the counterexample case is unchanged, and it agrees with the independent coordinate and bundle-tower proofs given elsewhere. Worth reading closely: It is the most complete written proof and the only one supplying the converse; a reader should check that the corrected general theorem still covers the counterexample case (it does in version 2). The quoted passage was not found in the stored copy of this source.
How these sources relate
- A geometric construction of the counterexample to the Jacobian conjecture and https://arxiv.org/html/2608.27341 share an author. Both are by Remy van Dobben de Bruyn: the arXiv preprint's byline names him, and its introduction and appendix cite the "Lovely little lemmas" blog post "A geometric construction of the counterexample to the Jacobian conjecture" as the author's own earlier post (reference [18], "by the author").
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ and https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ share an author. Both recorded passages are comments by David Speyer (one on his own Secret Blogging Seminar thread, one on Terence Tao's thread) announcing the same preprint; they are one voice, not two.
Assessment history
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.