[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"vDtRFpkMWr":3},"# The Dixmier conjecture implies the Jacobian conjecture, and is false in dimension 3 and above\n\nA Lean 4 proof of four statements from Google DeepMind's [Formal Conjectures](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean) (pinned commit `7bdc4a42`), all in `FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean`:\n\n| Formal Conjectures' name | What it says |\n|---|---|\n| [`dixmier_conjecture.variants.jacobian_implication`](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L132) | The Dixmier conjecture in dimension n implies the Jacobian conjecture in dimension n. |\n| [`dixmier_conjecture.variants.dimension_ge_three`](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L123) | The Dixmier conjecture is false in every dimension n ≥ 3. |\n| [`dixmier_conjecture.variants.dimension_three`](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L114) | The Dixmier conjecture is false in dimension 3. |\n| [`dixmier_conjecture`](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L89) | The Dixmier conjecture, generalised to all dimensions, is false: it fails for some n. |\n\nThe Dixmier conjecture says that, over a field of characteristic zero, every endomorphism of the n-th Weyl algebra is an automorphism. The Weyl algebra is the algebra generated by x₁, …, xₙ and y₁, …, yₙ with x_i y_i − y_i x_i = 1 and all other pairs commuting: the canonical commutation relations of quantum mechanics. Dixmier posed it in 1968 for the first Weyl algebra (Bull. Soc. Math. France 96, Problem 1).\n\n**The implication.** Formal Conjectures credits it to H. Bass, E. Connell and D. Wright, \"The Jacobian conjecture: reduction of degree and formal expansion of the inverse\", Bull. Amer. Math. Soc. 7 (1982), and K. Adjamagbo and A. van den Essen include it in the chain of implications of their Theorem 7 (arXiv:math/0608009). The converse holds stably, by three independent proofs (Tsuchimoto 2005, Belov-Kanel and Kontsevich 2007, Bavula), which this repository does not use.\n\n**The other three statements** follow from the implication, because the Jacobian conjecture fails in dimension 3 and above, and that is proved here as a Lean theorem rather than cited. Formal Conjectures' [`JacobianConjecture.G`](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Wikipedia/JacobianConjecture.lean#L48) is a polynomial map of K³, a variant of what Formal Conjectures calls Alpöge/Fable's counterexample (L. Alpöge, 2026). Formal Conjectures proves that it has [Jacobian determinant 1](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Wikipedia/JacobianConjecture.lean#L68) and [sends (1, 0, 1) and (0, 3, −71) to the same point](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Wikipedia/JacobianConjecture.lean#L85), and from that [proves the Jacobian conjecture false](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Wikipedia/JacobianConjecture.lean#L105). This repository restates the map as `DixJac.cG`, proves both facts again, and pads it with identity coordinates to a counterexample in every dimension n ≥ 3 (`DixJac.not_jacobianConjectureProp`, in `DixJac/Pad.lean`).\n\n**Earlier work.** The failure of the Dixmier conjecture in dimension 3 was found within days of Alpöge's counterexample, in July 2026, in [An Explicit Counterexample to the Dixmier Conjecture in A3](https://paratelligent.com/research/papers/an-explicit-counterexample-to-the-dixmier-conjecture-in-a3-H8jVaRUn), which Teller and Naughton credit to Jeff Pickhardt and the Omniscience Research Agent (19 July), and independently by [Marcus Teller](https://github.com/frozaken/dixmier3), whose Lean proofs cover the determinant, the collision and the matrix identities, and [Brian Naughton](https://github.com/brian-naughton/dixmier-a3). [William Mayner](https://github.com/wmayner/dixmier-counterexample) circulated a note covering every n ≥ 3. This repository proves Formal Conjectures' statements in Lean, including the implication for every n and every field of characteristic zero.\n\n**What is not claimed.** Formal Conjectures marks the Dixmier conjecture in [dimension 1](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L97) and [dimension 2](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Arxiv/math.0608009/DixmierConjecture.lean#L106) as open, and it marks the [Jacobian conjecture in dimension 2](https://github.com/google-deepmind/formal-conjectures/blob/7bdc4a42f24c0316f81c73b26121794b2bed96a3/FormalConjectures/Wikipedia/JacobianConjecture.lean#L118) open too. Nothing here touches them, or the converse implication.\n\n## The proof\n\nThe Weyl algebra in Formal Conjectures is a bare quotient of a free algebra, so most of the work is a small library for it.\n\n1. **Polynomials inside the Weyl algebra** (`DixJac/Basic.lean`). Polynomials in the x's embed as a commutative subalgebra, and so do polynomials in the y's. Moving y_j past a polynomial p in the x's costs a derivative: y_j p = p y_j − ∂p/∂x_j.\n2. **The polynomial representation is faithful** (`DixJac/Faithful.lean`). The Weyl algebra acts on polynomials: x_i multiplies by X_i and y_i acts as −∂/∂X_i. Every element can be written in normal order, with the x's on the left. Commuting with x_i differentiates the y-part, so by induction on degree an element acting as zero has every derivative zero; in characteristic zero it is then a constant, and that constant is zero. So the representation is faithful, and anything that commutes with every x_i is a polynomial in the x's.\n3. **The endomorphism** (`DixJac/Construction.lean`). Given a polynomial map F with invertible Jacobian determinant, let A be the inverse of its Jacobian matrix. The vector fields D_i = Σ_j A_ij ∂_j satisfy D_i(F_k) = δ_ik and commute with each other: their commutator kills every F_k, and a derivation that kills every F_k is zero because the Jacobian is invertible. So x_k ↦ F_k, y_i ↦ Σ_j A_ij y_j respects the relations and defines an endomorphism of the Weyl algebra.\n4. **The inverse** (`DixJac/Main.lean`). If the Dixmier conjecture holds, that endomorphism has an inverse ψ. Each ψ(x_i) commutes with every x_j, so by step 2 it is a polynomial G_i in the x's. G is then a polynomial inverse of F on both sides.\n5. **Dimension 3 and above** (`DixJac/Pad.lean`, `DixJac/FormalTarget.lean`). The padded counterexample has a block-diagonal Jacobian with determinant 1 and still sends two points to one, so the Jacobian conjecture fails in every dimension n ≥ 3, and with step 4 so does the Dixmier conjecture.\n\nThe statements themselves are Formal Conjectures' own. `Challenge.lean` copies the definitions and the four statements verbatim, with only their attributes removed, and `scripts/check-fc-match.py` regenerates that text from Formal Conjectures at the pinned commit and requires an exact match. That compares text only: Formal Conjectures builds with Lean and Mathlib v4.33.1, this repository with Lean v4.35.0-rc3, and the script does not read what Formal Conjectures' files import. Palomar does not accept escaped name components such as `«math.0608009»` in theorem names, so `Challenge.lean` states the four theorems in the namespace `DixmierConjecture` with Formal Conjectures' leaf names. `Solution.lean` proves them under both names, and an `example` checks that each pair has the same statement at every universe level.\n\n## Building\n\nLean `v4.35.0-rc3` and Mathlib `5e0c4e5239cb0a2d86d68a884bf52cfd963fce22`. Formal Conjectures is not a dependency.\n\n```\nlake exe cache get\nlake build\nlake env lean audit/Axioms.lean\nlake env lean audit/Control.lean\npython3 scripts/check-lean-sources.py\npython3 scripts/check-fc-match.py --self-test\nruby scripts/validate-formalization.rb\n```\n\n`audit/Axioms.lean` checks that the four theorems and the key lemmas use only propext, Classical.choice and Quot.sound. `audit/Control.lean` checks that the axiom audit catches a `sorry`, specialises the implication to the rationals in dimension 2, checks the commutation relation and the derivative in the representation, and checks the counterexample. The self-test runs 28 altered copies of the inputs, each of which must fail. `scripts/verify-comparator.sh` runs Palomar's Comparator check; it needs Linux, so it runs in CI.\n\n## Licence\n\nApache 2.0. The files copied from Formal Conjectures keep its copyright notice.\n",1794686191343]