{"schema_version":"claim-reader-v1","record":{"available_versions":[2,1],"current_version":2,"assessment_summary":"consistent","claim":{"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":[]},"sha256":"68010ca5e2dcb4581b3c3fc43162e4da4eac23e453f7e3c1a72661fd7a7eb7cf","contexts":[{"assessment_history":[{"report_id":"jacobian-conjecture:report:two-open:v2","source_id":"jacobian-conjecture-analysis","source_version_id":"jacobian-conjecture:source:analysis:v2","source_title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","source_label":"Dimension two remains open","state":"reported_open","scope":"Complex dimension two.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","entry_id":"jacobian-conjecture:entry:analysis:v2","supersedes":"jacobian-conjecture:report:two-open:v1"}],"problem_id":"problem-jacobian-conjecture","slug":"jacobian-conjecture","title":"Jacobian conjecture","publication_version":8,"relationship":{"claim":{"id":"claim-jacobian-dimension-two","version":2},"relation":"related_variant","relationship_note":"The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.","result":"unproved","report_ids":["jacobian-conjecture:report:two-open:v2"],"independent_review":"pending"},"assessments":[{"report_id":"jacobian-conjecture:report:two-open:v2","source_id":"jacobian-conjecture-analysis","source_version_id":"jacobian-conjecture:source:analysis:v2","source_title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","source_label":"Dimension two remains open","state":"reported_open","scope":"Complex dimension two.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","entry_id":"jacobian-conjecture:entry:analysis:v2","supersedes":"jacobian-conjecture:report:two-open:v1"}],"connections":[{"id":"jacobian-three-versus-two","source":{"id":"claim-jacobian-three-point-fiber","version":3},"target":{"id":"claim-jacobian-dimension-two","version":2},"relation":"related_to","assertion":null,"explanation":"The three-variable witness settles the universal formulation, while the two-variable restriction remains a different question.","limitation":"A collision in ℂ³ supplies no counterexample in ℂ².","source_entry_ids":["jacobian-conjecture:entry:analysis:v2"],"independent_review":"pending"}],"related_claims":[{"id":"claim-jacobian-three-point-fiber","version":3,"slug":"jacobian-three-point-fiber","title":"A constant determinant with three inputs sharing an output"}],"progress":[],"tasks":[],"statements":[{"statement":"Set s = xy and u = 1 + s. 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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":2,"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:v1","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":"For the displayed three-variable map F over ℚ, there are no polynomial tuples defining a two-sided compositional inverse; the mirrored universal Jacobian conjecture statement is false.","id":"claim-jacobian-no-polynomial-inverse","slug":"jacobian-no-polynomial-inverse","version":1,"kind":"theorem","title":"The explicit map has no polynomial inverse","assumptions":["Polynomial Jacobian and composition over ℚ."],"scope":"A formal algebraic counterexample. Identification with analytic Fréchet derivatives and further geometric interpretations are outside the claimed formalization.","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:v1"],"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:v1","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"status_report_ids":["jacobian-conjecture:report:counterexample:v1"],"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:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"An explicit Jacobian counterexample","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":2},"proof_status":"reported_complete","source_entry_ids":["jacobian-conjecture:entry:witness:v1","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: failure of the recorded polynomial-inverse assertion","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":1},"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:v1","jacobian-conjecture:entry:lean:v1"],"claims":[{"id":"claim-jacobian-three-point-fiber","version":2}],"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:v1","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"claims":[{"id":"claim-jacobian-three-point-fiber","version":2},{"id":"claim-jacobian-no-polynomial-inverse","version":1}],"scenario":{"id":"claim-jacobian-three-point-fiber","version":2},"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":["Levent Alpöge"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"self_reported","ai_note":"Discovery is attributed to Levent Alpöge working with Claude Fable 5; mathematical authorship and the model contribution are separate roles.","artifacts":[],"published":{"value":"2026-07"}},"id":"jacobian-conjecture:source:witness:v1","source_id":"jacobian-conjecture-witness","title":"An explicit Jacobian counterexample","owner":"Levent Alpöge","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","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":null,"rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://www.ulam.ai/research/jacobian.pdf","attribution":"Levent Alpöge","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:v1","source_version_id":"jacobian-conjecture:source:witness:v1","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":null},{"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:v1","entry_id":"jacobian-conjecture:entry:witness:v1","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":null},{"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:two-open:v2","source_id":"jacobian-conjecture-analysis","source_version_id":"jacobian-conjecture:source:analysis:v2","source_title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","source_label":"Dimension two remains open","state":"reported_open","scope":"Complex dimension two.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","entry_id":"jacobian-conjecture:entry:analysis:v2","supersedes":"jacobian-conjecture:report:two-open:v1"}],"problem_id":"problem-jacobian-conjecture","slug":"jacobian-conjecture","title":"Jacobian conjecture","publication_version":2,"relationship":{"claim":{"id":"claim-jacobian-dimension-two","version":2},"relation":"related_variant","relationship_note":"The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.","result":"unproved","report_ids":["jacobian-conjecture:report:two-open:v2"],"independent_review":"pending"},"assessments":[{"report_id":"jacobian-conjecture:report:two-open:v2","source_id":"jacobian-conjecture-analysis","source_version_id":"jacobian-conjecture:source:analysis:v2","source_title":"Counterexamples to the Jacobian conjecture in dimensions greater than two","source_label":"Dimension two remains open","state":"reported_open","scope":"Complex dimension two.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.00222v1","entry_id":"jacobian-conjecture:entry:analysis:v2","supersedes":"jacobian-conjecture:report:two-open:v1"}],"connections":[{"id":"jacobian-three-versus-two","source":{"id":"claim-jacobian-three-point-fiber","version":2},"target":{"id":"claim-jacobian-dimension-two","version":2},"relation":"related_to","assertion":null,"explanation":"The three-variable witness settles the universal formulation, while the two-variable restriction remains a different question.","limitation":"A collision in ℂ³ supplies no counterexample in ℂ².","source_entry_ids":["jacobian-conjecture:entry:analysis:v2"],"independent_review":"pending"}],"related_claims":[{"id":"claim-jacobian-three-point-fiber","version":2,"slug":"jacobian-three-point-fiber","title":"A constant determinant with three inputs sharing an output"}],"progress":[],"tasks":[],"statements":[{"statement":"Set s = xy and u = 1 + s. 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":2,"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:v1","jacobian-conjecture:entry:analysis:v2","jacobian-conjecture:entry:lean:v1"],"formal_declarations":["Alpoge.three_point_fiber"],"relation":"alternative_answer","relationship_note":"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.","result":"proved","report_ids":["jacobian-conjecture:report:analysis:v2","jacobian-conjecture:report:formal:v1"],"independent_review":"pending","reading":{"placement":"answer","note":"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."}},{"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":[],"relation":"related_variant","relationship_note":"The source leaves this dimension open; the three-variable witness cannot be specialized to refute it.","result":"unproved","report_ids":["jacobian-conjecture:report:two-open:v2"],"independent_review":"pending"}],"entries":[{"id":"jacobian-conjecture:entry:witness:v1","source_version_id":"jacobian-conjecture:source:witness:v1","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":null},{"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}],"sources":[{"work":{"authors":["Levent Alpöge"],"kind":"research_paper","publication":"preprint","publication_note":"Public manuscript or exposition; publication does not imply independent OpenMath review.","ai_role":"self_reported","ai_note":"Discovery is attributed to Levent Alpöge working with Claude Fable 5; mathematical authorship and the model contribution are separate roles.","artifacts":[],"published":{"value":"2026-07"}},"id":"jacobian-conjecture:source:witness:v1","source_id":"jacobian-conjecture-witness","title":"An explicit Jacobian counterexample","owner":"Levent Alpöge","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","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":null,"rights":"metadata-link","license":"Original document rights remain with its authors.","terms_url":"https://www.ulam.ai/research/jacobian.pdf","attribution":"Levent Alpöge","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}]}}}