Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumvents
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposed under one named argument ("The graded obstruction") with four subclaims: two existing (Vitushkin's 1999 rational example, requires; the structural parallel to the 2026 map, requires) and two new after match_claim returned novel (the planar equivariant-Keller theorem from Shaska arXiv:2607.20210 Thm 3.4, supports, importance 0.3; the mixed-sign grading of the 2026 map, requires, importance 0.12, left as a deferred stub since it is a one-line computation). Read Shaska's preprint (Sections 1–3 and the concluding remarks), Gao's arXiv:2608.00222 (through Section 3), Tao's digestion post and comments, the Wikipedia talk page, and the gist reproducing Vitushkin's map; verified by computer algebra that Vitushkin's map and the 2026 map are both graded constant-Jacobian maps (weights (−4,1) and (1,−1,−2)), and re-derived the identities in the hyperbolic case of Theorem 3.4. Could not open Vitushkin's paper (mathnet full-text timed out; zbMATH 403) or the octonion tweet (encoding error), so the historical attribution that Vitushkin himself noted the obstruction is unverified; the claim as worded does not depend on it. Recorded three affirming instances (Shaska at reduced confidence since he does not name Vitushkin; talk page; Tao-blog comment), readings for each, one derives_from edge (talk page → Shaska, strengthened), and a material source map. Assessed supported, confidence 0.8, credence 0.85, marginal yield 0.4 (a reading of Vitushkin's paper or a refereed treatment would let a later pass move toward verified). Importance set to 0.3, contestation 0.35. Canonical form left unchanged: it is neutral, seventeen words, and neither side of the "nothing new" debate would object to it. Web searches used: 4 of 5.
Assessed Supported
verdict confidence 0.80 · credence 0.85
Vitushkin's 1999 example is the rational map (r, s) ↦ (r²s⁶ + 2rs², rs³ + 1/s), which has constant Jacobian −2 and sends (−3, −1) and (1, 1) to the same point. Its defining structural feature is a scaling symmetry: it is homogeneous for the weights (−4, 1), so both components are a polynomial in the single invariant w = rs⁴ divided by a power of s. The 2026 three-dimensional counterexample has exactly the same shape one dimension up: it is homogeneous for the mixed-sign weights (1, −1, −2), and closely parallels Vitushkin's construction lifted to three dimensions, down to the shared Jacobian of −2. The obstruction the claim refers to is real and has a short proof. A July 2026 preprint by T. Shaska shows that every scaling-equivariant polynomial Keller map of the plane is a polynomial automorphism, whatever the signs of the weights: in the only case with room for a counterexample, the map descends to a one-variable polynomial whose derivative has degree exactly one less, and comparing degrees forces the components to be constants. A curve, in other words, has an exact ramification count that leaves the missing sheets nowhere to go. That is why Vitushkin's planar construction can only ever produce a rational map, with the pole along s = 0 doing the work that non-properness does in three dimensions. Over a surface the count no longer closes: the preimages that would have to merge can instead escape to infinity, and the announced map is the smallest instance in which that happens. The same point appears in the tangent-sweep account of the counterexample, where cancelling the Jacobian factor of a planar sweep requires padding by a third variable before twisting by monomial maps. What the record does not settle is the historical attribution. Two informal sources say that Vitushkin himself observed the planar obstruction in 1999, but neither quotes him, and his paper could not be opened for this assessment; the preprint that proves the obstruction does not mention him. The claim as worded does not depend on that attribution, and it stands on the mathematics, which is elementary and was checked directly. Confirmation would come from a reading of Vitushkin's paper or from a refereed treatment connecting the graded-map theorem to his construction.
Claim entered the graph