Proved
Formal polynomial counterexample reported complete ↗
The pinned algebraic statements, not an analytic derivative identification.
Observed Oct 13, 2026
Theorem · version 2
A constant determinant with three inputs sharing an output
Earlier version 2. View current version 3 →
Statement
Set and . The map has . The distinct inputs (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) all map to (−1/4,0,0).
Assumptions
- Use the displayed map over a characteristic-zero field.
Scope
A concrete failure of injectivity and hence of a polynomial inverse. The coordinate degrees are 7, 6 and 4; a cubic fiber relation does not make this a cubic coordinate map.
Evidence
Source reports
Proved
Algebraic witness analyzed ↗
The explicit non-injective map.
Observed Oct 13, 2026
Replaces an earlier source assessment. Version history ↓
Formal proofs
Lean: the three-point fiber ↗
Complete formal proof reported externally
The polynomial witness is source-reported complete; OpenMath has not built it. The project also proves jacobian_det and not_injective.
Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026
Declaration & revision
Alpoge.three_point_fiberCommit b8eec94711a9b73151929b78cbd7cb999a008590
- An explicit Jacobian counterexample: Explicit map and verification of determinant and fiber →
- Counterexamples to the Jacobian conjecture in dimensions greater than two: Introduction and analysis of the counterexample →
- Alpöge counterexample in Lean: README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean →
Connections
In this problem · publication v2
Jacobian conjecture →
Alternative answer · Proved
Related statements
related toThe two-variable Jacobian question
The three-variable witness settles the universal formulation, while the two-variable restriction remains a different question.
A collision in ℂ³ supplies no counterexample in ℂ².
Details & sources
A constant determinant with three inputs sharing an output →
Proved · version 2
Set and . The map has . The distinct inputs (0,0,−1/4), (1,−3/2,13/2) and (−1,3/2,13/2) all map to (−1/4,0,0).
- Use the displayed map over a characteristic-zero field.
Scope: A concrete failure of injectivity and hence of a polynomial inverse. The coordinate degrees are 7, 6 and 4; a cubic fiber relation does not make this a cubic coordinate map.
The two-variable Jacobian question →
Open · version 2
Every polynomial map F: → with constant nonzero Jacobian determinant has a polynomial inverse.
- Exactly two complex variables.
Scope: The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.
Relationship not independently reviewed.
Connection in Jacobian conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Alpöge counterexample in Lean →
- README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean → · Original passage ↗
Recorded wording
The source proves constant formal polynomial Jacobian determinant, a three-point fiber, and failure of a polynomial inverse for the recorded Jacobian conjecture statement.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
alerad/alpoge-lean project · Project documentation · Publication date Jul 23, 2026
Public formal source; build and independent mathematical review are separate checks.
Alpöge counterexample in Lean — pinned source ↗No OpenMath execution recorded
Public source inspected for statement and assumptions; this proof project has not been executed by OpenMath.
Revision & attribution
Source revision: b8eec94711a9b73151929b78cbd7cb999a008590. Observed Oct 13, 2026.
The project reports an AI-assisted formalization and computer-algebra workflow; this is distinct from the original discovery.
Source updated Jul 23, 2026.
Alpöge counterexample in Lean — pinned source: b8eec94711a9b73151929b78cbd7cb999a008590.
An explicit Jacobian counterexample →
- Explicit map and verification of determinant and fiber → · Original passage ↗
Recorded wording
An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Levent Alpöge · Preprint · Publication date Jul 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-07. Observed Oct 13, 2026.
Discovery is attributed to Levent Alpöge working with Claude Fable 5; mathematical authorship and the model contribution are separate roles.
Counterexamples to the Jacobian conjecture in dimensions greater than two →
- Introduction and analysis of the counterexample → · Original passage ↗
Recorded wording
Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Shuhong Gao · Preprint · Publication date Jul 31, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-07-31. Observed Oct 13, 2026.
No AI contribution is asserted by this source record.
Source updated Jul 31, 2026.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Algebraic witness analyzed ↗
The explicit non-injective map.
Observed Oct 13, 2026 · Source passage →
Replaces an earlier source assessment; see the original source for its history.
Proved
Formal polynomial counterexample reported complete ↗
The pinned algebraic statements, not an analytic derivative identification.
Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-jacobian-three-point-fiber
- Statement version
- 2
- SHA-256
- d56816663e049c4917a024584f6e0351b7a25e434504783f994d36e64d719809