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Formal Verification of an Explicit Counterexample to the Jacobian Conjecture

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It also constructs identity-padded polynomial maps, proves their block-Jacobian determinant is unchanged, and thereby obtains counterexamples in every finite dimension at least three.

The Jacobian conjecture is false in every dimension n ≥ 3Verified

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As quoted

The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicted

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As quoted

The theory defines polynomial invertibility and proves that the map has no polynomial inverse.

The explicit three-dimensional polynomial map with constant nonzero Jacobian determinant has no polynomial inverse.Unassessed

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As quoted

The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicted

Not yet read by this claim’s Steward.

As quoted

The Jacobian conjecture asks whether a polynomial self-map of affine space over a field of characteristic zero has a polynomial inverse whenever its Jacobian determinant is a nonzero constant.

Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverseContradicted

Not yet read by this claim’s Steward.

As quoted

The theory defines polynomial invertibility and proves that the map has no polynomial inverse.

The explicit three-dimensional polynomial map with constant nonzero Jacobian determinant has no polynomial inverse.Unassessed

Not yet read by this claim’s Steward.

As quoted

It also constructs identity-padded polynomial maps, proves their block-Jacobian determinant is unchanged, and thereby obtains counterexamples in every finite dimension at least three.

The Jacobian conjecture is false in every dimension n ≥ 3Verified

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As quoted

Isabelle proves that the formal Jacobian entries are the corresponding complex analytic partial derivatives and that the determinant is identically \(-2\).

The Alpöge–Fable map F:ℂ³→ℂ³ has Jacobian determinant identically equal to −2.Unassessed

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The announcement credits Akhil Mathew with prompting the question and the AI system Claude Fable with work leading to the map

An AI system produced work leading to a counterexample to the Jacobian conjecture.Unassessed

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Formal Verification of an Explicit Counterexample to the Jacobian Conjecture

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Link: https://isa-afp.org/entries/Jacobian_Counterexample.html
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