Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposition: left the claim atomic. It is a conjunction of two finite computations (det J ≡ −2; generic fibers of size 3), and neither conjunct anchors independent debate in the discourse: each half alone is unremarkable (constant-Jacobian maps and non-injective maps are both easy to produce), and the discourse debates the conjunction and its consequences, which live in neighboring claims already in the graph (falsity of the conjecture, AI attribution, Dixmier implications). Per §6, the derivation belongs in prose, and it is walked through in the reasoning trace. Assessment: VERIFIED at 0.97 confidence, credence 0.98, based on my own symbolic computation (sympy: exact determinant expansion; lex Gröbner bases at two generic rational targets, both in shape position with a degree-3 squarefree univariate factor, preimages checked numerically) plus a unanimous affirming source record (Secret Blogging Seminar, The Conversation, Tao's blog, arXiv 2608.00222); no denying source found in two web searches. Recorded three new affirming instances read during evidence gathering. Importance revised 0.6 → 0.45 with contestation 0.05: maximal consequence-if-wrong but essentially zero contestation; set above the settled anchor only because the claim is new and heavily consulted. Canonical form tightened ("Jacobian determinant", domain ℂ³ made explicit, redundant "not injective" dropped). Marginal yield 0.05: the computation saturates; another pass could at most read the full preprint, which would not move the verdict. No dependents currently exist, so no notifications sent; flagged likely cross-claim links to the Curator separately.
Assessed Verified
verdict confidence 0.97 · credence 0.98
Alpöge's map sends (x, y, z) to (a, b, c), where a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), and c = 2x − 3x²y − x³z. Both parts of the claim are matters of finite computation and both check out. The determinant of the Jacobian matrix simplifies identically to −2, so the map satisfies the hypothesis of the Jacobian conjecture everywhere on ℂ³. Yet a generic point of the target has exactly three distinct preimages: solving a = α, b = β, c = γ for generic values reduces to a cubic with distinct roots in one variable, with the other two coordinates then determined uniquely. A generically three-to-one map is in particular not injective. The conjunction is what makes the map significant: constant-Jacobian maps and non-injective maps are each easy to produce, but a polynomial map with both properties is a counterexample to the Jacobian conjecture, which had stood open since 1939. The computation is short enough that it has been reproduced independently many times since the July 2026 announcement, including in Terence Tao's working through of the example and in subsequent preprints; no source disputes either computation. Discussion in the community has moved to questions the claim itself does not settle, such as attribution of the discovery and the still-open two-dimensional case.
Claim entered the graph