The two-dimensional Jacobian conjecture is false in characteristic two.
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
In August 2026 Romy Mondello posted a counterexample to the two-dimensional Jacobian conjecture in characteristic two (arXiv:2608.02634), the first two-dimensional result of any kind in the wave of counterexamples that followed Alpöge's July 2026 refutation of the classical conjecture in dimension three. The claim therefore rests on whether Mondello's construction is correct. That has not yet been confirmed by peer review, but the surrounding evidence favors it: counterexamples of this kind are finite, directly checkable objects; the construction was derived from dimension-three characteristic-two counterexamples produced independently by several people; and David Speyer, an expert in the area, reported and endorsed the result in two venues with no dissent recorded.
A point of precision: the naive Jacobian condition fails trivially in any positive characteristic (the map sending x to x minus x squared has derivative 1 in characteristic two yet is two-to-one), so the substantive characteristic-p conjecture excludes such cases, and it is that formulation the discourse treats Mondello's example as refuting. Community verification of the preprint, or its peer-reviewed publication, would settle the claim; a flaw found in the construction would return the two-dimensional characteristic-two case to open. The classical two-dimensional conjecture in characteristic zero remains open and is untouched by this result.
Full reasoning: the evidence and decisions behind this verdict
The claim entered from David Speyer's report on the Secret Blogging Seminar (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), which states that Mondello "just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2" at arxiv.org/abs/2608.02634 and calls it the first two-dimensional result. A second affirming instance is Speyer's comment of 6 August 2026 on Terence Tao's blog (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), celebrating "the two dimensional case in characteristic two." Both instances affirm; no source read denies the claim or reports doubts about the construction.
Context established by search: Alpöge refuted the classical (characteristic-zero) Jacobian conjecture in dimension three on 19 July 2026, with an infinite family by Gallagher and a geometric tangent-sweep explanation by Speyer; a follow-up preprint (arxiv.org/abs/2608.00222) extends counterexamples to every dimension greater than two and confirms the two-dimensional characteristic-zero case remains open. In characteristic two, dimension-three counterexamples were given independently by Timothy Chow (MathOverflow), N8Programs, and Huq-Kuruvilla (arxiv.org/abs/2607.20968); Speyer states Mondello found the two-dimensional example by studying Huq-Kuruvilla's. This lineage is why the dimension-three characteristic-two refutation supports the claim: the mechanism demonstrably works in characteristic two.
The verdict is a materiality judgment on the single load-bearing dependency, the correctness of Mondello's construction. In favor: such counterexamples are verified by finite computation (check the Jacobian determinant is a unit; exhibit non-invertibility), the expert community was actively scrutinizing exactly this stream of results in August 2026, and an error would likely have surfaced quickly in the same highly trafficked threads where the result was announced. Against reaching "verified": the preprint itself could not be opened in this pass, no peer-reviewed confirmation exists yet, and the result was days old in the sources read. Hence supported, credence 0.85.
Two named gaps a stronger pass should close. First, the exact characteristic-p formulation refuted: the naive statement is trivially false in positive characteristic even in dimension one (Artin-Schreier maps such as x to x minus x squared are etale and two-to-one, and extend to dimension two by the identity), so Speyer's "first 2-dimensional result" only makes sense for a substantive formulation excluding such maps, but the sources read do not state it; the assessment adopts the discourse's substantive reading per fidelity of interpretation. Second, direct examination of arXiv:2608.02634, which could upgrade the verdict to verified or, if the formulation turns out weaker than assumed, prompt a canonical-form question for the Curator. A retraction, a published flaw, or expert dissent would flip the verdict; peer-reviewed acceptance or independent verification write-ups would confirm it.
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 →Mondello's characteristic-two counterexample to the two-dimensional Jacobian conjecture is mathematically correct. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The Jacobian conjecture is false in dimension three in characteristic two. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
And here is a reward for Huq-Kuruvilla's work — the two dimensional case in characteristic two! That's really exciting. Maybe we'll see the two dimensional case in some other characteristics soon!
Comment thread on Terence Tao's post digesting the July 2026 Jacobian conjecture counterexample; Speyer reports Mondello's two-dimensional characteristic-two counterexample (arXiv:2608.02634) as resolving that case, crediting Huq-Kuruvilla's dimension-three characteristic-two preprint as its origin.
Romy Mondello just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2
David Speyer: 'Just in case someone is relying on this comment thread for their main source of news about the Jacobian conjecture, Romy Mondello just posted a counter-example to the 2-dimensional Jacobian conjecture in characteristic 2 ... I believe this is the first 2-dimensional result we've seen.'
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.