Non-invertible constant-Jacobian polynomial self-maps of ℂ³ exist in every geometric degree at least three
3 events · 1 assessment · 1 decision
Structured and assessed first pass
First pass (structure_and_assess, with two curator_change suggestions). Decomposition: adopted both Curator suggestions as supports edges. Alpöge's verified map (2fb3c345) attached under the argument "Gallagher's explicit degree family" as the seed and settled degree-three case; supports rather than requires because the parent could be established by independent constructions. The P¹×P² affine-space lemma (a21e1db1) attached under "The geometric tangent-sweep route" as supports; kept at supports rather than upgrading to requires because web evidence shows the degree family is Gallagher's algebraic one-variable construction, while the lemma anchors only the degree-three base case of the geometric route (its own scope note records the affine-space identification fails for higher covering degree). Minted two novel subclaims after match_claim returned no match: Gallagher's family (a6c9cdcb, seed 0.9) and Gao's tangent-sweep generalization (e4f08020, seed 0.85), each grouped under its argument with written form and evaluation. Evidence: five web searches. Key sources: Speyer's sbseminar post (existing instance), Gao's arXiv 2608.00222 (recorded as new affirming instance; corrects the Curator's attribution of that preprint to Gallagher — it is by Shuhong Gao, citing Gallagher's July 20 family), Tao's digestion post, and a public verification repository reporting independent checks of Gallagher's seed family. All sources affirm; none deny. Gallagher's own preprint URL was not located, the main gap of this pass (marginal_yield 0.4). Canonical form updated to specify geometric degree: the discourse's degree is generic preimage count (Gao defines it so; the base map has polynomial degree seven, so the old wording was only coherent geometrically). A precisification, not a change of proposition. Importance set 0.35 (contestation 0.15), down from the Extractor's 0.45: a notable corollary-level sharpening of the headline 2026 refutation, heavily consulted territory but effectively uncontested. Verdict: supported, confidence 0.8, credence 0.93. No dependents exist, so no notification sent.
Assessed Supported
verdict confidence 0.80 · credence 0.93
The claim sharpens the July 2026 refutation of the Jacobian conjecture: not only does a non-invertible polynomial self-map of ℂ³ with constant nonzero Jacobian exist, such maps exist in every geometric degree three and above, meaning a generic point of the target can be made to have any prescribed number of preimages from three upward. The degree-three case is settled: Alpöge's explicit map with constant Jacobian −2 is generically three-to-one, a computation reproduced independently many times. The general statement rests on Gallagher's uniform one-variable construction of July 20, 2026, which yields a generically d-to-one Keller map for every integer d at least three, described by David Speyer as constructing maps of every degree at least three and cited as an established infinite family in Shuhong Gao's subsequent preprint. A partially independent geometric route corroborates the picture: the base counterexample's reading as a tangent-line sweep, resting on the verified identification of the source complement in P¹×P² with affine three-space, has been generalized so that counterexamples of arbitrarily large geometric degree exist in every dimension above two. No source disputes the claim. What keeps it short of fully established here is only that Gallagher's construction, unlike the base example, has not yet accumulated the same density of published independent verification; a completed community check of the family, or a direct reading of the construction, would settle it.
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