A Counterexample to the Jacobian Conjecture ↗
Explicit three-variable counterexample · observed Oct 13, 2026
Source status date unavailable · observed Oct 13, 2026
The all-dimensions complex Jacobian conjecture.
Publication v8
Record details
Jacobian conjecture →Coverage & review
Coverage of this curation
The canonical statement, selected results and public proof evidence, with historical and contribution context.
Formal proofs: Formal polynomial determinant, fiber collision and no-inverse results have public pinned Lean sources. Analytic derivative correspondence is outside their stated scope; no local proof build is recorded.
AI-assisted work: The discovery report credits Levent Alpöge, Akhil Mathew for asking the question and Fable for work leading to the example. The July 20 ULAM verification manuscript prints no author name. Separate analysis: Shuhong Gao. Formalization: the alerad/alpoge-lean project’s reported AI-assisted workflow.
Search scope & review
Selected primary manuscripts, theorem declarations and project provenance reviewed on 13 October 2026; this is not an exhaustive literature survey; Gao’s abstract for counterexamples in every dimension n ≥ 3.
Checked Oct 13, 2026 · independent review pending.
- Independent mathematical review and local execution of the cited formal proofs are pending.
General status observations
Source identities & rights
A Counterexample to the Jacobian Conjecture ↗
ULAM research manuscript (no author named) · metadata-link · Original document rights remain with its authors.
OpenMath supplies original paraphrases and links; no proof text is republished. Rights / source terms
1 cited entry · exact source identity
- Version
- jacobian-conjecture:source:witness:v2
- Revision
- 2026-07-20
- Content SHA-256
- fcc91582150fa5ed7914bce443080c926771378b8e9bea7b9c2b866ddc6419b2
- Source update
- 2026-07-20
- Observed
- 2026-10-13
- Check cadence
- 90 days
- Explicit map and verification of determinant and fiber ↗
jacobian-conjecture:entry:witness:v2
Counterexamples to the Jacobian conjecture in dimensions greater than two ↗
Shuhong Gao · metadata-link · Original document rights remain with its authors.
OpenMath supplies original paraphrases and links; no proof text is republished. Rights / source terms
2 cited entries · exact source identity
- Version
- jacobian-conjecture:source:analysis:v2
- Revision
- 2026-07-31
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-07-31
- Observed
- 2026-10-13
- Check cadence
- 90 days
- Introduction and analysis of the counterexample ↗
jacobian-conjecture:entry:analysis:v2 - Abstract ↗
jacobian-conjecture:entry:all-dimensions:v1
Alpöge counterexample in Lean ↗
alerad/alpoge-lean project · metadata-link · Original document rights remain with its authors.
OpenMath supplies original paraphrases and links; no proof text is republished. Rights / source terms
1 cited entry · exact source identity
- Version
- jacobian-conjecture:source:lean:v1
- Revision
- b8eec94711a9b73151929b78cbd7cb999a008590
- Content SHA-256
- 2cdfa772891c7b7f815c23039de71b971e02beada5fa4fad63350d3b2887a372
- Source update
- 2026-07-23
- Observed
- 2026-10-13
- Check cadence
- 90 days
- README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean ↗
jacobian-conjecture:entry:lean:v1
Publication history
- Problem ID
- problem-jacobian-conjecture
- Publication checksum
- af21c4342c54f7cf755d0f90327d4555850fec5713b3d2e0b2b03ccde7aceb77
- Published
- 2026-10-20T21:33:54.610377+00:00
- Publication reason
- Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 1 · Oct 13, 2026
Initial checked pilot publication; source attribution, rights, wording and coverage reviewed by OpenMath staff.
- Version 2 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 3 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 4 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 5 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 6 · Oct 14, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 7 · Oct 15, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 8 · Oct 20, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.