Non-invertible constant-Jacobian polynomial self-maps of ℂ³ exist in every geometric degree at least three
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim decomposes into a settled base case and a general construction, with a second, partially independent line of corroboration.
Base case. Alpöge's map on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one stands verified at high confidence: both properties are finite computations, reproduced many times since the July 19, 2026 announcement. This establishes geometric degree three outright.
General degrees. Gallagher's July 2026 construction yields a generically d-to-one Keller map of ℂ³ for every integer d at least three is the load-bearing step for degrees above three. Three sources read during this pass bear on it. David Speyer's Secret Blogging Seminar post of July 20, 2026 (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/, the claim's original recorded instance) asserts that the preprint constructs maps of ℂ³ with constant Jacobian of every degree at least three. Shuhong Gao's preprint (arxiv.org/abs/2608.00222, July 31, 2026) recounts the refutation "with an infinite family by Gallagher (July 20)" and describes a uniform one-variable construction producing a counterexample for every integer degree, recorded here as an affirming instance. A public verification repository (github.com/dasjoms/jacobian-conjecture-counterexample-exploration) reports independent verification of the core claims of the Gallagher seed family. Gallagher's preprint itself was not opened whole in this pass, which is the principal gap.
Independent corroboration. Gao's tangent-sweep construction produces counterexamples of arbitrarily large geometric degree in every dimension greater than two gives a second construction reaching infinitely many degrees in ℂ³, resting for its base case on the verified lemma that the source complement in P¹×P² is affine three-space. "Arbitrarily large" covers infinitely many degrees but not by itself every degree, so this route corroborates without independently establishing the full statement.
Interpretation. "Degree" is read as geometric degree (generic preimage count), the reading the discourse uses: Gao defines it explicitly, and the base map's polynomial degree is seven while the family is indexed from three, which is only coherent geometrically.
Stances and verdict. Every instance and every source read affirms the claim or a stronger version; none deny it. The verdict is supported rather than verified because the general construction was not examined directly and its published independent verification is thinner than the base example's. What would change the conclusion: a flaw found in Gallagher's uniform construction would drop the claim to the degrees reachable by Alpöge's example and Gao's construction (an infinite but possibly incomplete set); a direct reading or further independent verification of the family would raise it to verified.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because Alpöge's map has constant Jacobian −2 yet is generically three-to-one, a non-invertible constant-Jacobian self-map of ℂ³ exists in geometric degree three, and because Gallagher's uniform one-variable construction yields a generically d-to-one Keller map for every integer d at least three, such maps exist in every geometric degree at least three.
The inference is immediate: an explicit family with one generically d-to-one Keller map per degree yields existence in every degree at least three. The base case is settled by Alpöge's verified three-to-one map, so the argument stands or falls with Gallagher's uniform construction covering every degree, which multiple expert sources assert and none dispute but which has not yet been examined directly here.
The base counterexample has a geometric reading as a sweep of the tangent lines of a plane curve, which becomes a polynomial self-map of affine three-space because the source complement in P¹×P² is isomorphic to affine three-space; and because Gao's generalization of this tangent-sweep mechanism produces counterexamples of arbitrarily large geometric degree in every dimension greater than two, a second construction independently yields non-invertible constant-Jacobian self-maps of ℂ³ in infinitely many degrees at least three.
The route is sound as far as it reaches: the verified identification of the source complement with affine three-space makes the tangent-sweep reading a genuine polynomial counterexample, and Gao's generalization extends it to arbitrarily large geometric degree in dimension three. The caveat is one of scope: arbitrarily large degree gives infinitely many degrees, not every degree at least three, so this argument corroborates the claim strongly without establishing it on its own.
Provenance
Where this claim has been said, linked to its canonical form.
which constructs maps C^3 –> C^3 with constant Jacobian of every degree >= 3.
David Speyer describing Alexis Gallagher's new preprint: 'I imagine many of us are reading Alexis Gallagher's new preprint ... which constructs maps C^3 --> C^3 with constant Jacobian of every degree >= 3.'
The next day, Gallagher [12] gave a uniform one-variable construction producing, for every integer … d
Gao's preprint on counterexamples to the Jacobian conjecture in dimensions greater than two, recounting the July 2026 refutation in dimension three: it cites Gallagher's July 20 preprint as an established infinite family produced by a uniform one-variable construction, one counterexample for each integer degree. The passage as read was truncated by the search rendering at the ellipsis.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.