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 claim is the one non-obvious step in the geometric reading of the July 2026 counterexample to the Jacobian conjecture. That construction takes the covering map from P¹×Sym²(P¹), which is isomorphic to P¹×P², onto Sym³(P¹), identified with P³, sending a point and an unordered pair to the unordered triple; one removes the covering's ramification divisor together with the preimage of a hyperplane tangent, but not osculating, to the twisted cubic. The target complement is plainly affine three-space; the claim is that the source complement is as well, which is exactly what turns the restricted covering into a degree-three étale polynomial map from affine three-space to itself and hence into the counterexample.
The isomorphism was described in the initial discussion as the main point in doubt, and it was then established several times over by independent methods: an explicit polynomial change of coordinates with a computed inverse, posted by Will Sawin from a ChatGPT-assisted calculation, and conceptual proofs by David Speyer, Jake Levinson, and Remy van Dobben de Bruyn, the last exhibiting the complement as a tower of Zariski A¹-bundles. Terence Tao's subsequent digestion of the counterexample reproves the trivialisation as part of a general bundle analysis. No source disputes the result. Its scope is narrow: the analogous construction yields an affine-space source only when the degree is three, which is why the counterexample lives specifically in three variables.
Full reasoning: the evidence and decisions behind this verdict
The claim was assessed from the primary discussion threads rather than from any single authority. Three independent lines converge.
First, the explicit isomorphism: Will Sawin's post at sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ gives concrete coordinates ([u,v] for a point of P¹, [a,b,c] for a binary quadratic) and an explicit polynomial map from the complement to A³, presented with the calculation that it is invertible. An explicit map with a polynomial inverse is the strongest possible form of proof for a statement of this kind, though the transcription available to this pass appears to contain minor typographical corruption, so the formula itself was not re-verified symbolically here.
Second, the conceptual proof: van Dobben de Bruyn (lovelylittlelemmas.rjprojects.net/a-geometric-construction-of-the-counterexample-to-the-jacobian-conjecture/) computes the ramification divisor as a bidegree divisor cut out by an explicit tri-homogeneous polynomial, uses the tangent-but-not-osculating hypothesis to control the hyperplane pullback, and exhibits the complement as an iterated tower of Zariski A¹-bundles, which by a standard lemma is isomorphic to affine space. The structure of this argument was read directly and is sound; the same post records that Speyer and Levinson gave further independent proofs.
Third, consistency checks: Tao's digestion (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) recovers Sawin's trivialisation from a rank analysis of a rank-two bundle over P¹ (the tangency case forcing the split type that makes the base isomorphic to A²), and the finite-field point count in the same discussion comes out to exactly q³, as an affine three-space requires. The verified sibling result that Alpöge's explicit map has constant Jacobian −2 yet is generically three-to-one is the coordinate shadow of this construction, and its verification indirectly corroborates that the source is genuinely A³ with polynomial coordinates.
All recorded instances affirm; no source in the discourse denies or qualifies the result. What would change the verdict: a demonstrated error common to the explicit isomorphism and the bundle-tower proof, which, given their independence and the number of expert eyes on this construction, is very unlikely. The residual uncertainty reflects only that this pass did not re-derive the isomorphism symbolically from a clean statement of the formulas.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
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.
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.)
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.