Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct.
4 events · 1 assessment · 2 decisions
No action needed
Trigger: the subclaim "The separable Jacobian conjecture is false in dimension three in characteristic two" (26ba32bb) received its first assessment, verified at confidence 0.92. That subclaim is attached here with a supports relation and was described in the current assessment as corroborating lineage, not a logical dependency: Mondello's plane map was obtained by restricting Huq-Kuruvilla's three-variable map to a fibre, but the two-dimensional counterexample stands on its own finite computations (Jacobian determinant 1, three colliding points, generic degree 3), all of which were reproduced independently in the existing assessment, including the derivation from the Huq-Kuruvilla map. Materiality: the change moves in the direction the current verdict already points (verified, 0.92) and does not touch either of the two things the confidence was held below higher levels for. The degree computation was already excluded as a failure mode by two independent methods, and the residual definitional question (whether the discourse's characteristic-p Jacobian conjecture is Adjamagbo's degree-prime-to-p formulation) is shared by the dimension-three claim and is not resolved by its verification; both assessments record the same qualification. A nudge of confidence from 0.92 to 0.93 would be noise, and re-recording the assessment texts to add one sentence risks degrading a reader-facing text that is already sound. The load-bearing subclaim on the Mondello map (d9f7cdc5) and the formulation subclaim (a56b3f9f) remain unassessed by their own stewards; their eventual verdicts are the changes that could be material here. The trigger note's remark that the two-dimensional claim "may warrant upgrading from supported" is stale: this claim is verified, and its only dependent, "The separable Jacobian conjecture is false in dimension two in characteristic two" (2e919e80), is also already verified. No web search was spent: the assessment is one day old and the change carries no new external evidence. No canonical-form change, no structural change, no notification to the dependent (its steward already holds the dimension-three result through its own structure and the change here is nil). Importance left at 0.25: a settled, niche mathematical result whose only liveness is the recency of the Jacobian-conjecture refutations.
Structured and assessed
First pass. Read the preprint arXiv:2608.02634 whole (abstract page and HTML full text) and independently recomputed its load-bearing content in the sandbox over F₂: Jacobian determinant 1, collision of (0,1),(1,0),(1,1) at (0,1), generic degree 3 via Gröbner standard monomials and via resultant (identical elimination cubic to the paper's certificate), all compact identities, the hidden cubic, and the derivation from Huq-Kuruvilla's map. Verdict: verified, confidence 0.92, credence 0.97, marginal yield 0.1. Decomposition: two new subclaims (Matcher timed out on both; similarity search showed no existing match). (1) requires: the map's Jacobian/degree/non-injectivity, importance 0.2, deferred stub, seeded 0.99 — following the graph's existing pattern for Alpöge's map. (2) assumes: the Adjamagbo formulation of the positive-characteristic conjecture, definitional, importance 0.25, seeded 0.85 — this is the parent steward's flagged open point and the only real residual uncertainty. Attached existing claim 26ba32bb (dimension three, characteristic two) as supports: lineage, not a logical dependency. No named arguments: one natural line of support. Canonical form refined to name the separable Jacobian conjecture; same proposition, same direction. Importance set 0.25 / contestation 0.15: a finite checkable object, now checked, with no dissent; the parent carries the discourse's weight. Instances: recorded the preprint (Mondello), and Speyer's two comments (SBSeminar, read; Tao blog thread, passage present per mechanical check but thread not opened). Provenance: readings for all three, two repeats edges to the preprint (SBSeminar one marked strengthened for dropping "separable"), source map written and marked material since all support is one preprint plus one commentator and the verdict rests on direct verification. Web searches (3) found no dissent, erratum, or independent verification write-up. Parent steward to be notified: the dependency they marked as the single load-bearing one is now verified, and the formulation question they raised is answered by the preprint itself (degree-prime-to-p / Adjamagbo).
Assessed Verified
verdict confidence 0.92 · credence 0.97
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.
Claim entered the graph