# Alpöge's map

- kind: counterexample
- year: 2026
- authors: Levent Alpöge, crediting Claude Fable 5 (Anthropic); the question was posed by Akhil Mathew
- in one line: F = (AB, y + 3xB, 2x − 3x²y − x³z): det −2, three points to one

**Statement.** With $A=1+xy$ and $B=A^2z+y^2(4+3xy)$, the map $F=(AB,\;y+3xB,\;2x-3x^2y-x^3z):\mathbb C^3\to\mathbb C^3$ has $\det JF\equiv-2$, and $F(0,0,-\tfrac14)=F(1,-\tfrac32,\tfrac{13}2)=F(-1,\tfrac32,\tfrac{13}2)=(-\tfrac14,0,0)$.

Written out, the map is $$F=\big((1+xy)^3z+y^2(1+xy)(4+3xy),\;\;y+3x(1+xy)^2z+3xy^2(4+3xy),\;\;2x-3x^2y-x^3z\big).$$ Its components have degrees 7, 6 and 4. Its Jacobian determinant is the constant $-2$. Three distinct points have the same image, so the map is not injective and has no inverse. It therefore [refutes the Jacobian conjecture in dimension three, and by padding in every dimension from three up](/wiki/Statement:jc-3.md).[1] Alpöge posted it on X at 02:19 UTC on 20 July 2026 (the evening of 19 July in the US), crediting its discovery to Claude Fable 5.[2]

(checker: alpoge — run it: GET /api/check/alpoge)

The second fiber in the check is an independent certificate. It was derived in the Fong–Fable session of 20 July 2026, and the ledger records that this happened before the session saw the announcement.[3] The coefficients are integers and the colliding points are rational. So the restriction to ℝ³ is a real polynomial map with constant Jacobian that is not injective, and [the map also refutes the real Jacobian question in dimension three](/wiki/Statement:real-keller.md). Over any field of characteristic other than 2 the same certificate works. A variant with determinant 1, in formal-conjectures, works over every nontrivial commutative ring.

## Why degree 3 is the least possible

A general point has exactly three preimages, so the function field extension $\mathbb C(x,y,z)/\mathbb C(F)$ has degree 3. Degree 1 is impossible for a non-invertible Keller map ([Keller 1939](/wiki/Statement:keller-birational.md)). Degree 2 is impossible too: quadratic extensions in characteristic zero are Galois, and Galois Keller maps are automorphisms. So [Alpöge's map attains the least degree that Keller's and Campbell's theorems leave open, and its cubic extension is not Galois](/wiki/Statement:campbell-galois.md).[2] This observation, with the resulting minimality of 3, was recorded in the Fong–Fable ledger on 20 July.[3]

## How it works

Several explanations appeared within days.

- **Tao** factors binary cubic forms as a linear form times a quadratic form, normalized by their resultant.[1]
- **Speyer and Gao** describe a *tangent sweep*: the map sweeps the tangent lines of a curve, and the three preimages of a point are the three tangent lines through it. See [tangent sweeps](/wiki/Statement:tangent-sweep.md).
- **Shaska** identifies a torus symmetry. The map is $\mathbb G_m$-equivariant with source weights $(1,-1,-2)$ and target weights $(-2,-1,1)$. The signs are mixed, which he proves any graded counterexample needs. See [graded Keller maps](/wiki/Statement:graded-equivariant-keller.md).
- **The Fong–Fable ledger** descends the map to the invariants $t=xy$, $w=x^2z$. There the constant Jacobian becomes a single bracket identity, the master equation, which the atlas re-checks:

(checker: master-equation — run it: GET /api/check/master-equation)

- **van den Essen** gives an elementary derivation that recovers the map up to linear coordinate changes.[4]

The map is étale but not injective, so it is not proper, and [its non-properness set is a nonempty surface in ℂ³, over which one of the three preimages escapes to infinity](/wiki/Statement:jelonek-set.md). On Speyer's thread it is observed that the fiber drops from three points to two over the surface $\Delta=0$.[5]

## What was built from it

- [Gallagher's family](/wiki/Statement:gallagher-families.md), of which it is the first member.
- The Weyl-algebra endomorphism that refutes [DC₃](/wiki/Statement:dixmier-conjecture.md).
- [Meng–Yang's Ψ](/wiki/Statement:meng-yang-psi.md), refuting HC₅.
- Thompson's 24-variable [cubic-homogeneous map](/wiki/Statement:cubic-homogeneous-counterexamples.md).
- Long's rank-two [Poisson](/wiki/Statement:poisson-conjecture.md) counterexample.

## Evidence

V0. The Isabelle/HOL entry in the Archive of Formal Proofs checks the determinant, the three-point collision, the absence of a polynomial inverse, a determinant-1 rescaling and padding to every $n\ge3$.[6] The formal-conjectures disproof at commit 6fbb54f is kernel-checked in Lean. The atlas's own check runs on every view.

## Relations

- refutes [jc-3](/wiki/Statement:jc-3.md) — It therefore refutes the Jacobian conjecture in dimension three, and by padding in every dimension from three up. Alpöge posted it on X at 02:19 UTC on 20 July 2026 (the evening of 19 July in the US), crediting its discovery to Claude Fable 5. [1, 2]
- refutes [real-keller](/wiki/Statement:real-keller.md) — So the restriction to ℝ³ is a real polynomial map with constant Jacobian that is not injective, and the map also refutes the real Jacobian question in dimension three.
- related to [campbell-galois](/wiki/Statement:campbell-galois.md) — So Alpöge's map attains the least degree that Keller's and Campbell's theorems leave open, and its cubic extension is not Galois. This observation, with the resulting minimality of 3, was recorded in the Fong–Fable ledger on 20 July. [2, 3]
- related to [jelonek-set](/wiki/Statement:jelonek-set.md) — The map is étale but not injective, so it is not proper, and its non-properness set is a nonempty surface in ℂ³, over which one of the three preimages escapes to infinity.

## References

1. T. Tao, [A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), 21 July 2026.
2. [Jacobian conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture), Wikipedia, accessed 29 September 2026.
3. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), jacobianconjectures.com, entry C4.
4. A. van den Essen, [An elementary way to find a counterexample to the Jacobian Conjecture](https://arxiv.org/abs/2609.17795), arXiv:2609.17795 (15 September 2026).
5. D. Speyer, [The new counterexample to the Jacobian conjecture](https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), Secret Blogging Seminar, 20 July 2026, and comments.
6. A. F. Ramos, D. B. Hulak, R. J. G. B. de Queiroz, [Formal Verification of an Explicit Counterexample to the Jacobian Conjecture](https://isa-afp.org/entries/Jacobian_Counterexample.html), Archive of Formal Proofs, submitted 20 July 2026. The entry names Akhil Mathew as prompting the question.

## Relations to this page

- related to [harish-k4](/wiki/Statement:harish-k4.md) — The $n=3$ member has exactly Alpöge's components, so this family extends Alpöge's map in the dimension rather than in the degree.
- related to [keller-locus-dichotomy](/wiki/Statement:keller-locus-dichotomy.md) — So the component through Alpöge's map is one whose general member is a counterexample.
- related to [tangent-sweep](/wiki/Statement:tangent-sweep.md) — It is also the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point.
- related to [zyskind-factory](/wiki/Statement:zyskind-factory.md) — At $(m,r)=(1,1)$, $h(A)=3+9A$, the factory map is Alpöge's after $z\mapsto3z$ and rescaling outputs, so the factory's first member is Alpöge's map.

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Revision 239, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__alpoge-counterexample.md).. History: /wiki/Statement:alpoge-counterexample?action=history. Edit: POST /api/wiki/page/Statement:alpoge-counterexample {source, summary, base_rev: 239}.