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The map F(x,y,z) = (u³z + y²u(4 + 3s), y + 3xu²z + 3xy²(4 + 3s), 2x − 3x²y − x³z) has det JF = −2. 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).","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Set "},{"kind":"math","text":"s = xy","latex":"s=xy"},{"kind":"text","text":" and "},{"kind":"math","text":"u = 1 + s","latex":"u=1+s"},{"kind":"text","text":". The map "},{"kind":"math","text":"F(x,y,z) = (u³z + y²u(4 + 3s), y + 3xu²z + 3xy²(4 + 3s), 2x − 3x²y − x³z)","latex":"F(x,y,z)=\\begin{pmatrix}u^3z+y^2u(4+3s)\\\\ y+3xu^2z+3xy^2(4+3s)\\\\2x-3x^2y-x^3z\\end{pmatrix}","display":true},{"kind":"text","text":" has "},{"kind":"math","text":"det JF = −2","latex":"\\det JF=-2"},{"kind":"text","text":". 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)."}]},"id":"claim-jacobian-three-point-fiber","slug":"jacobian-three-point-fiber","version":3,"kind":"theorem","title":"A constant determinant with three inputs sharing an output","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.","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"formal_declarations":["Alpoge.three_point_fiber"]},{"statement":"Every polynomial map F: ℂ² → ℂ² with constant nonzero Jacobian determinant has a polynomial inverse.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Every polynomial map F: "},{"kind":"math","text":"ℂ²","latex":"\\mathbb C^2"},{"kind":"text","text":" → "},{"kind":"math","text":"ℂ²","latex":"\\mathbb C^2"},{"kind":"text","text":" with constant nonzero Jacobian determinant has a polynomial inverse."}]},"id":"claim-jacobian-dimension-two","slug":"jacobian-dimension-two","version":2,"kind":"conjecture","title":"The two-variable Jacobian question","assumptions":["Exactly two complex variables."],"scope":"The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.","source_entry_ids":["jacobian-conjecture:entry:analysis:v2"],"formal_declarations":[]},{"statement":"There exists a polynomial map F: ℚ³ → ℚ³ with constant Jacobian determinant −2 and no two-sided polynomial compositional inverse.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"There exists a polynomial map F: "},{"kind":"math","text":"ℚ³","latex":"\\mathbb Q^3"},{"kind":"text","text":" → "},{"kind":"math","text":"ℚ³","latex":"\\mathbb Q^3"},{"kind":"text","text":" with constant Jacobian determinant −2 and no two-sided polynomial compositional inverse."}]},"id":"claim-jacobian-no-polynomial-inverse","slug":"jacobian-no-polynomial-inverse","version":2,"kind":"theorem","title":"No polynomial inverse for the three-variable witness","assumptions":["Polynomial maps and composition are taken over ℚ."],"scope":"The source witnesses this existence statement using the three-variable counterexample. Analytic derivative correspondence and further geometric interpretations are outside this formal result.","source_entry_ids":["jacobian-conjecture:entry:lean:v1"],"formal_declarations":["Alpoge.FormalConjecturesJC.jacobian_conjecture_false"]}],"problem":{"resolution_scope":{"resolution":"counterexample","extent":"full_statement","summary":"The cited explicit three-variable witness refutes the universal complex formulation. Its determinant and colliding rational inputs are exposed as separate, inspectable evidence.","report_ids":["jacobian-conjecture:report:counterexample:v2"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-jacobian-conjecture","slug":"jacobian-conjecture","title":"Jacobian conjecture","question":"Must every polynomial map F: ℂⁿ → ℂⁿ with a constant nonzero Jacobian determinant have a polynomial inverse?","question_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Must every polynomial map F: "},{"kind":"math","text":"ℂⁿ","latex":"\\mathbb C^n"},{"kind":"text","text":" → ℂⁿ with a constant nonzero Jacobian determinant have a polynomial inverse?"}]},"scope":"Complex polynomial maps in finite dimension n ≥ 1. A local derivative condition is being compared with global polynomial invertibility; the two-variable restriction is a separate question.","aliases":["Keller conjecture","Jacobian problem"],"fields":["Algebraic geometry"],"msc":["14R15","14R10"],"topics":[],"background":"graduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"status_report_ids":["jacobian-conjecture:report:counterexample:v2"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"known","scope":"Source comparison and page curation only; independent mathematical review is pending.","rationale":"The cited explicit three-variable witness refutes the universal complex formulation. Its determinant and colliding rational inputs are exposed as separate, inspectable evidence.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["jacobian-conjecture:report:counterexample:v2"],"independent_mathematical_review":"pending"},"literature":[{"title":"A Counterexample to the Jacobian Conjecture","url":"https://www.ulam.ai/research/jacobian.pdf","note":"Explicit map and verification of determinant and fiber"},{"title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","url":"https://arxiv.org/abs/2608.00222v1","note":"Introduction and analysis of the counterexample"},{"title":"Alpöge counterexample in Lean","url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","note":"README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean"}],"formalizations":[{"title":"Lean: the three-point fiber","url":"https://github.com/alerad/alpoge-lean/blob/b8eec94711a9b73151929b78cbd7cb999a008590/Alpoge/Basic.lean","note":"The polynomial witness is source-reported complete; OpenMath has not built it. The project also proves jacobian_det and not_injective.","target":{"id":"claim-jacobian-three-point-fiber","version":3},"proof_status":"reported_complete","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"declaration":"Alpoge.three_point_fiber","source_commit":"b8eec94711a9b73151929b78cbd7cb999a008590","alignment":"staff_linked","checked_on":"2026-10-13"},{"title":"Lean: no polynomial inverse for the rational witness","url":"https://github.com/alerad/alpoge-lean/blob/b8eec94711a9b73151929b78cbd7cb999a008590/Alpoge/FormalConjectures.lean","note":"The endpoint witnesses the statement over ℚ with three variables. The explicit rational points also give the complex non-injectivity; analytic correspondence was not formalized by this project.","target":{"id":"claim-jacobian-no-polynomial-inverse","version":2},"proof_status":"reported_complete","source_entry_ids":["jacobian-conjecture:entry:lean:v1"],"declaration":"Alpoge.FormalConjecturesJC.jacobian_conjecture_false","source_commit":"b8eec94711a9b73151929b78cbd7cb999a008590","alignment":"staff_linked","checked_on":"2026-10-13"}],"related":[],"tasks":[],"insights":[{"id":"jacobian-conjecture-approach","title":"Check a witness instead of a universal obstruction","summary":"The counterexample can be inspected through exact polynomial differentiation and substitution of three rational inputs.","review":"pending","kind":"approach","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:lean:v1"],"claims":[{"id":"claim-jacobian-three-point-fiber","version":3}],"limitation":"A determinant calculation alone would not demonstrate failure of global inversion."},{"id":"jacobian-conjecture-impact","title":"Local invertibility does not guarantee a global inverse","summary":"A constant nonzero Jacobian can coexist with several inputs having the same output, invalidating the universal polynomial-inverse conclusion.","review":"pending","kind":"impact","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"claims":[{"id":"claim-jacobian-three-point-fiber","version":3},{"id":"claim-jacobian-no-polynomial-inverse","version":2}],"scenario":{"id":"claim-jacobian-three-point-fiber","version":3},"limitation":"The remaining two-variable question requires its own argument."}],"coverage_gaps":["Independent mathematical review and local execution of the cited formal proofs are pending."],"checked_on":"2026-10-13"},"sources":[{"work":{"authors":["ULAM research manuscript (no author named)"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"self_reported","ai_note":"The manuscript cites Levent Alpöge’s announcement, credits Akhil Mathew for asking the question and Fable for the work leading to the example. No named author is printed on this verification manuscript; these discovery credits are not its bibliographic authorship.","artifacts":[],"published":{"value":"2026-07-20"}},"id":"jacobian-conjecture:source:witness:v2","source_id":"jacobian-conjecture-witness","title":"A Counterexample to the Jacobian Conjecture","owner":"ULAM research manuscript (no author named)","url":"https://www.ulam.ai/research/jacobian.pdf","subject_scope":"An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.","revision":"2026-07-20","content_sha256":"fcc91582150fa5ed7914bce443080c926771378b8e9bea7b9c2b866ddc6419b2","update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-07-20","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://www.ulam.ai/research/jacobian.pdf","attribution":"ULAM research manuscript (no author named)","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90},{"work":{"authors":["Shuhong Gao"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"not_reported","ai_note":"No AI contribution is asserted by this source record.","artifacts":[],"published":{"value":"2026-07-31"}},"id":"jacobian-conjecture:source:analysis:v2","source_id":"jacobian-conjecture-analysis","title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","owner":"Shuhong Gao","url":"https://arxiv.org/abs/2608.00222v1","subject_scope":"Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.","revision":"2026-07-31","content_sha256":null,"update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-07-31","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://arxiv.org/abs/2608.00222v1","attribution":"Shuhong Gao","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90},{"work":{"authors":["alerad/alpoge-lean project"],"kind":"research_project","publication":"project","publication_note":"Public formal source; build and independent mathematical review are separate checks.","ai_role":"self_reported","ai_note":"The project reports an AI-assisted formalization and computer-algebra workflow; this is distinct from the original discovery.","artifacts":[{"kind":"formal_code","title":"Alpöge counterexample in Lean — pinned source","url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","version":"b8eec94711a9b73151929b78cbd7cb999a008590","assessment":"not_executed","note":"Public source inspected for statement and assumptions; this proof project has not been executed by OpenMath."}],"published":{"value":"2026-07-23"}},"id":"jacobian-conjecture:source:lean:v1","source_id":"jacobian-conjecture-lean","title":"Alpöge counterexample in Lean","owner":"alerad/alpoge-lean project","url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","subject_scope":"The source proves constant formal polynomial Jacobian determinant, a three-point fiber, and failure of a polynomial inverse for the recorded Jacobian conjecture statement.","revision":"b8eec94711a9b73151929b78cbd7cb999a008590","content_sha256":"2cdfa772891c7b7f815c23039de71b971e02beada5fa4fad63350d3b2887a372","update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-07-23","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","attribution":"alerad/alpoge-lean project","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90}],"entries":[{"id":"jacobian-conjecture:entry:witness:v2","source_version_id":"jacobian-conjecture:source:witness:v2","item_id":"witness","url":"https://www.ulam.ai/research/jacobian.pdf","wording":"An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.","wording_kind":"paraphrase","locator":"Explicit map and verification of determinant and fiber","supersedes":"jacobian-conjecture:entry:witness:v1"},{"id":"jacobian-conjecture:entry:analysis:v2","source_version_id":"jacobian-conjecture:source:analysis:v2","item_id":"analysis","url":"https://arxiv.org/abs/2608.00222v1","wording":"Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.","wording_kind":"paraphrase","locator":"Introduction and analysis of the counterexample","supersedes":"jacobian-conjecture:entry:analysis:v1"},{"id":"jacobian-conjecture:entry:lean:v1","source_version_id":"jacobian-conjecture:source:lean:v1","item_id":"lean","url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","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.","wording_kind":"paraphrase","locator":"README trust boundary; Alpoge/Basic.lean and Alpoge/FormalConjectures.lean","supersedes":null}],"reports":[{"id":"jacobian-conjecture:report:counterexample:v2","entry_id":"jacobian-conjecture:entry:witness:v2","state":"refuted","source_label":"Explicit three-variable counterexample","scope":"The all-dimensions complex Jacobian conjecture.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://www.ulam.ai/research/jacobian.pdf","supersedes":"jacobian-conjecture:report:counterexample:v1"},{"id":"jacobian-conjecture:report:analysis:v2","entry_id":"jacobian-conjecture:entry:analysis:v2","state":"resolved","source_label":"Algebraic witness analyzed","scope":"The explicit non-injective map.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","supersedes":"jacobian-conjecture:report:analysis:v1"},{"id":"jacobian-conjecture:report:two-open:v2","entry_id":"jacobian-conjecture:entry:analysis:v2","state":"reported_open","source_label":"Dimension two remains open","scope":"Complex dimension two.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","supersedes":"jacobian-conjecture:report:two-open:v1"},{"id":"jacobian-conjecture:report:formal:v1","entry_id":"jacobian-conjecture:entry:lean:v1","state":"resolved","source_label":"Formal polynomial counterexample reported complete","scope":"The pinned algebraic statements, not an analytic derivative identification.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://github.com/alerad/alpoge-lean/tree/b8eec94711a9b73151929b78cbd7cb999a008590","supersedes":null}]},"status":{"state":"refuted","label":"Resolved · counterexample","extent":"full_statement","summary":"The cited explicit three-variable witness refutes the universal complex formulation. Its determinant and colliding rational inputs are exposed as separate, inspectable evidence.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-13","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"]}},"selected_context":{"assessment_history":[{"report_id":"jacobian-conjecture:report:counterexample:v2","source_id":"jacobian-conjecture-witness","source_version_id":"jacobian-conjecture:source:witness:v2","source_title":"A Counterexample to the Jacobian Conjecture","source_label":"Explicit three-variable counterexample","state":"refuted","scope":"The all-dimensions complex Jacobian conjecture.","observed_on":"2026-10-13","evidence_url":"https://www.ulam.ai/research/jacobian.pdf","entry_id":"jacobian-conjecture:entry:witness:v2","supersedes":"jacobian-conjecture:report:counterexample:v1"}],"problem_id":"problem-jacobian-conjecture","slug":"jacobian-conjecture","title":"Jacobian conjecture","publication_version":5,"relationship":{"claim":{"id":"claim-jacobian-polynomial-inverse","version":3},"relation":"primary_formulation","relationship_note":"Universal conjecture; one non-injective map in dimension three suffices to refute it.","result":"refuted","report_ids":["jacobian-conjecture:report:counterexample:v2"],"independent_review":"pending"},"assessments":[{"report_id":"jacobian-conjecture:report:counterexample:v2","source_id":"jacobian-conjecture-witness","source_version_id":"jacobian-conjecture:source:witness:v2","source_title":"A Counterexample to the Jacobian Conjecture","source_label":"Explicit three-variable counterexample","state":"refuted","scope":"The all-dimensions complex Jacobian conjecture.","observed_on":"2026-10-13","evidence_url":"https://www.ulam.ai/research/jacobian.pdf","entry_id":"jacobian-conjecture:entry:witness:v2","supersedes":"jacobian-conjecture:report:counterexample:v1"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"For every integer n ≥ 1, every polynomial map F: ℂⁿ → ℂⁿ whose Jacobian determinant is a nonzero constant has a polynomial inverse.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every integer n ≥ 1, every polynomial map F: "},{"kind":"math","text":"ℂⁿ","latex":"\\mathbb C^n"},{"kind":"text","text":" → "},{"kind":"math","text":"ℂⁿ","latex":"\\mathbb C^n"},{"kind":"text","text":" whose Jacobian determinant is a nonzero constant has a polynomial inverse."}]},"id":"claim-jacobian-polynomial-inverse","slug":"jacobian-polynomial-inverse","version":3,"kind":"conjecture","title":"Constant Jacobian implies a polynomial inverse","assumptions":["The ground field is ℂ.","The determinant is a nonzero constant polynomial."],"scope":"Universal conjecture; one non-injective map in dimension three suffices to refute it.","source_entry_ids":["jacobian-conjecture:entry:witness:v2","jacobian-conjecture:entry:analysis:v2"],"formal_declarations":[],"relation":"primary_formulation","relationship_note":"Universal conjecture; one non-injective map in dimension three suffices to refute it.","result":"refuted","report_ids":["jacobian-conjecture:report:counterexample:v2"],"independent_review":"pending"}],"entries":[{"id":"jacobian-conjecture:entry:witness:v2","source_version_id":"jacobian-conjecture:source:witness:v2","item_id":"witness","url":"https://www.ulam.ai/research/jacobian.pdf","wording":"An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.","wording_kind":"paraphrase","locator":"Explicit map and verification of determinant and fiber","supersedes":"jacobian-conjecture:entry:witness:v1"},{"id":"jacobian-conjecture:entry:analysis:v2","source_version_id":"jacobian-conjecture:source:analysis:v2","item_id":"analysis","url":"https://arxiv.org/abs/2608.00222v1","wording":"Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.","wording_kind":"paraphrase","locator":"Introduction and analysis of the counterexample","supersedes":"jacobian-conjecture:entry:analysis:v1"}],"sources":[{"work":{"authors":["ULAM research manuscript (no author named)"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"self_reported","ai_note":"The manuscript cites Levent Alpöge’s announcement, credits Akhil Mathew for asking the question and Fable for the work leading to the example. No named author is printed on this verification manuscript; these discovery credits are not its bibliographic authorship.","artifacts":[],"published":{"value":"2026-07-20"}},"id":"jacobian-conjecture:source:witness:v2","source_id":"jacobian-conjecture-witness","title":"A Counterexample to the Jacobian Conjecture","owner":"ULAM research manuscript (no author named)","url":"https://www.ulam.ai/research/jacobian.pdf","subject_scope":"An explicit three-variable polynomial map has constant determinant −2 and a fiber containing three distinct rational points.","revision":"2026-07-20","content_sha256":"fcc91582150fa5ed7914bce443080c926771378b8e9bea7b9c2b866ddc6419b2","update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-07-20","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://www.ulam.ai/research/jacobian.pdf","attribution":"ULAM research manuscript (no author named)","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90},{"work":{"authors":["Shuhong Gao"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"not_reported","ai_note":"No AI contribution is asserted by this source record.","artifacts":[],"published":{"value":"2026-07-31"}},"id":"jacobian-conjecture:source:analysis:v2","source_id":"jacobian-conjecture-analysis","title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","owner":"Shuhong Gao","url":"https://arxiv.org/abs/2608.00222v1","subject_scope":"Algebraic analysis of the three-variable counterexample, including its non-injectivity; the two-dimensional complex case remains open.","revision":"2026-07-31","content_sha256":null,"update_method":"Source comparison followed by a new immutable curation revision.","identifier_policy":"Stable source and entry identities; changed wording receives a new version.","observed_on":"2026-10-13","source_updated_on":"2026-07-31","rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://arxiv.org/abs/2608.00222v1","attribution":"Shuhong Gao","rights_note":"OpenMath supplies original paraphrases and links; no proof text is republished.","check_interval_days":90}]}}}