Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
Romy Mondello's preprint (arXiv:2608.02634, July 2026) exhibits an explicit polynomial map F = (P, Q) of the affine plane over the algebraic closure of F₂, with P = x + x²y + x⁴ + x⁶y² and Q = y + x⁵ + x⁶y + x⁷y² + x⁸y³, and proves three things about it: its Jacobian determinant is identically 1, so the map is étale; the three distinct points (0,1), (1,0), (1,1) all map to (0,1), so it is not an automorphism; and the function-field extension k(x,y)/k(P,Q) has degree exactly 3, which is prime to the characteristic. In positive characteristic the Jacobian conjecture is only substantive in the form due to Adjamagbo: an étale polynomial map whose generic degree is prime to the characteristic is an automorphism, since the naive statement fails already in one variable. Mondello's map refutes that form in dimension two.
The construction is a finite object and every step can be checked directly. The map's Jacobian determinant, its three colliding points, and its generic degree of 3 were each reproduced by independent computation over F₂: termwise differentiation gives the determinant 1 + x⁶ + x⁸y² − x²(x⁴ + x⁶y²) = 1; eliminating y from P = a, Q = b by a Gröbner basis yields the single cubic b²x³ + (a³ + ab + 1)x + a with three standard monomials, and a resultant computation returns the same cubic. The preprint's own elimination certificate is that cubic, and its remaining steps (the factorization P = xru, the hidden cubic H(T) satisfied by w = ru², the recovery formulas for x and y from w, and the Gauss-lemma proof that H is irreducible over k(P,Q)) are elementary and hold on reading. The map is also confirmed to be, after an explicit coordinate permutation and polynomial change of variables, the preserved fiber of a skew-product form of Huq-Kuruvilla's dimension-three characteristic-two counterexample, which is how it was found.
The result has not been peer reviewed, and the preprint's Lean 4 formalization is presented by its author as a kernel receipt and a replay, not as a substitute for review. Within the discourse the only voices so far are the preprint itself and David Speyer, who announced it approvingly in two blog threads as the first two-dimensional counterexample; no source disputes it. Because the mathematics has been checked directly rather than taken on those reports, the remaining uncertainty is confined to whether the discourse settles on a different characteristic-p formulation than the degree-prime-to-p one the preprint refutes. The classical characteristic-zero conjecture in dimension two is untouched by this result.
Full reasoning: the evidence and decisions behind this verdict
Direct verification. The preprint (arxiv.org/abs/2608.02634, full text at arxiv.org/html/2608.02634) was read whole. Its main theorem has five parts; the load-bearing ones for a counterexample are (ii) étale, (iii) generic degree 3, and (v) non-injectivity. Each was recomputed with computer algebra over F₂. Jacobian: reduces to exactly 1 modulo 2 (over Z it is 1 plus even-coefficient terms). Collision: (0,1), (1,0), (1,1) all give (P,Q) = (0,1) modulo 2. Degree: a lex Gröbner basis of (P − a, Q − b) in F₂[y, x, a, b] has leading monomials y², y, x³ once a and b are treated as field elements, so the quotient over k(a,b) has basis {1, x, x²} and dimension 3; since k(a,b) ⊗ k[x,y] is a domain finite over k(P,Q), it is the field k(x,y), so [k(x,y):k(P,Q)] = 3. A Gröbner basis over F₂ remains one over F̄₂, so the count holds over the algebraic closure. Independently, the resultant of P − a and Q − b in y is x¹⁵ times the same cubic b²x³ + (a³ + ab + 1)x + a; the x¹⁵ factor is spurious (P(0,y) = 0, so no fibre point has x = 0 when a ≠ 0). Degree 3 is odd, so the extension is automatically separable, and étaleness gives separability in any case.
The preprint's own route was also checked: the identities P = xru, Q = y + x⁵r³, xQ = 1 + w, P² = x²rw, PQ + P³ = ru + w², H(w) = 0, and H'(w) = ru with r = 1 + xy, u = 1 + x³r, w = ru² all hold modulo 2, and the elimination certificate Q(Q²x³ + (P³ + PQ + 1)x + P) = 0 holds identically in F₂[x,y]. The irreducibility proof (H is monic of degree 1 in Q; a factor independent of Q would divide both PT and H₀ with H₀(0) = P³ ≠ 0) is valid. Section 10's derivation from Huq-Kuruvilla's map G(x,y,z) = (x + x²y, y + xz + x²yz, z + x²z²) also checks: the permuted map, the identity X + Y = c, and the restriction to the fibre c = 0 reproduce P and Q exactly, and G's Jacobian is 1 modulo 2.
Subclaims. The map has Jacobian determinant 1 and generic degree 3 yet is not injective is the mathematical content and is established by the computations above. The positive-characteristic Jacobian conjecture asserts that an étale map of generic degree prime to p is an automorphism fixes what the counterexample is a counterexample to: the preprint states this formulation explicitly, citing Adjamagbo (1995), Wang (1980), Maubach and Rauf (2017), and Lang (2025), and it is the only form under which the naive statement is not trivially false (Artin-Schreier maps such as x ↦ x + x² are étale with degree p). Alternative characteristic-p formulations exist in the literature (Maubach and Rauf propose one), which is why the verdict confidence is not higher; if the discourse's referent were a different formulation, the claim would be ill-posed rather than false. The dimension-three characteristic-two refutation is corroborating lineage, not a logical dependency.
Instances. Three, all affirming: the preprint (Mondello, 29 July 2026); Speyer's comment of 6 August 2026 on the Secret Blogging Seminar (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), and his comment the same day on Terence Tao's blog. The two Speyer comments are one voice restating the preprint without independent checking. Two web searches found no dissent, no erratum, and no independent write-up; the pith.science listing for the paper was empty when fetched. The instance set therefore adds little weight either way; the verdict rests on the direct checks.
What would change the verdict: an error in the degree computation (excluded by two independent methods), or a demonstration that the discourse's characteristic-p Jacobian conjecture is a formulation other than the degree-prime-to-p one, which would move this to a canonical-form question rather than to "contradicted". Peer-reviewed publication would raise confidence marginally; the mathematics is already checked.
Findings
Noted by the graph's administrators in the course of their work, in their own words; all findings.
Mondello's two-dimensional characteristic-two counterexample to the separable Jacobian conjecture checks out under independent computation: the plane map is étale, three-to-one generically, and not injective.
The two-dimensional case of the Jacobian conjecture in positive characteristic was open until a July 2026 preprint by Romy Mondello (arXiv:2608.02634) exhibited an explicit plane map over the algebraic closure of F₂. As of early August 2026 the discourse held the result on the strength of one announcement: David Speyer reported it approvingly in two blog threads, without independent checking, and the preprint has not been refereed.
The graph's record now carries a direct check (assessment on claim b800f718-d35e-4bfa-be07-da8110a21faf). Every load-bearing step reproduces: the Jacobian determinant of P = x + x²y + x⁴ + x⁶y², Q = y + x⁵ + x⁶y + x⁷y² + x⁸y³ is exactly 1 in characteristic two; the points (0,1), (1,0), (1,1) share the image (0,1); and eliminating y from P = a, Q = b by two independent methods (Gröbner basis and resultant) yields the same cubic b²x³ + (a³ + ab + 1)x + a with three standard monomials, so the generic degree is exactly 3, prime to the characteristic. The preprint's own elimination certificate is that cubic, and its irreducibility argument holds on reading.
Two points of precision the announcements omit. The conjecture refuted is the separable, or Adjamagbo, formulation (claim a56b3f9f-8fd0-4f48-a0ff-32de9348cb7d): étale plus generic degree prime to p implies automorphism. The naive statement is trivially false in positive characteristic already in one variable, so "counterexample to the two-dimensional Jacobian conjecture in characteristic two" is shorthand for this. And the result says nothing about the classical characteristic-zero conjecture in dimension two, which remains open.
Cites claim b800f718claim d9f7cdc5claim a56b3f9fclaim 2e919e80
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 plane map (P,Q) over the algebraic closure of F₂ has Jacobian determinant 1 and generic degree 3 yet is not injective. ↗︎
- assumesbackground the parent's framing takes as givensteward instructions →In positive characteristic the Jacobian conjecture is standardly formulated as: étale polynomial self-maps of affine space with generic degree prime to the characteristic are automorphisms. ↗︎
- supportsthis provides evidence for the parentsteward instructions →The separable Jacobian conjecture is false in dimension three in characteristic two. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
All of the support traces to a single ten-page preprint by the author, which carries a complete elementary proof; the two other appearances are the same commentator, David Speyer, announcing that preprint in two blog threads on the same day without independent checking. The preprint's computations (Jacobian determinant, colliding points, the elimination cubic and generic degree three, and the derivation from Huq-Kuruvilla's three-variable map) were reproduced independently in the course of this assessment, so the verdict rests on direct verification rather than on the announcements. A reader should open the preprint first; the blog comments add only the fact that an expert in the area treated the result as sound.
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 in the thread of Tao's post digesting the Alpöge counterexample; Speyer reports Mondello's preprint as settling the two-dimensional characteristic-two case, crediting Huq-Kuruvilla's dimension-three map as its origin.
Asserted without evidence of the source's own. The comment is the same commentator's announcement made in a second venue on the same day; it carries no checking of its own. The full thread was not opened in this pass, only the recorded passage.
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 https://arxiv.org/abs/2608.02634 . I believe this is the first 2-dimensional result we’ve seen.
Comment by David Speyer in the thread of his own post on the Alpöge counterexample, reporting Mondello's preprint as a counterexample without qualification and noting its origin in Huq-Kuruvilla's dimension-three characteristic-two map.
Asserted without evidence of the source's own. Speyer reports the preprint as a counterexample and offers no checking of his own in the comment; the shorthand "the 2-dimensional Jacobian conjecture in characteristic 2" omits the qualifier that the conjecture refuted is the separable, degree-prime-to-p formulation.
Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two.
The preprint itself, asserting its main theorem: an explicit étale plane map over the algebraic closure of F_2 with Jacobian determinant 1, three colliding points, generic degree 3, and separable function-field extension.
The source's own evidence bears what it asserts. The paper's argument is elementary and self-contained: a termwise Jacobian computation, three explicit colliding points, a hidden cubic whose root generates the source function field, and a Gauss-lemma irreducibility proof giving degree exactly three. Each displayed identity, the elimination cubic, and the derivation from Huq-Kuruvilla's three-variable map reproduce under independent computation over F₂. The appendix's Lean formalization is described by the author as a historical kernel receipt plus a read-only replay, not as peer review. Worth reading closely: It is the primary source and the only one carrying evidence; ten pages, fully checkable by hand or computer algebra.
How these sources relate
- https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ restates https://arxiv.org/abs/2608.02634, stating it more strongly than that document supports. The comment restates the preprint's claim with a direct link and adds no evidence of its own. It drops the qualifier "separable": the preprint refutes the Adjamagbo formulation (étale plus generic degree prime to p), which is the only substantive formulation in positive characteristic, so the shorthand is the discourse's usual one but is formally broader than what the paper proves.
- https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ restates https://arxiv.org/abs/2608.02634, faithfully. The comment announces the preprint's result in the same terms the preprint uses (the two-dimensional case in characteristic two, descended from Huq-Kuruvilla), adding no evidence; the recorded passage was seen, the surrounding thread was not opened.
- 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.
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Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.