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.