{"schema_version":"claim-reader-v1","record":{"available_versions":[1],"current_version":1,"assessment_summary":"consistent","claim":{"statement":"For P,Q ∈ ℂ[x] and a standard real Gaussian X, if E[P(X)^m] = 0 for all positive integers m, then E[Q(X)P(X)^m] = 0 for all sufficiently large m.","id":"claim-gaussian-moments-one-variable","slug":"gaussian-moments-one-variable","version":1,"kind":"theorem","title":"The one-variable case","assumptions":["One real Gaussian coordinate; complex coefficients.","The eventual threshold may depend on Q."],"scope":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","source_entry_ids":["gaussian-moments:entry:original:v1"],"formal_declarations":[]},"sha256":"da8cfccf49575db21b4e274b58f44390fd72214238c787231284d3030b1af6ed","contexts":[{"assessment_history":[{"report_id":"gaussian-moments:report:one:v1","source_id":"gaussian-moments-original","source_version_id":"gaussian-moments:source:original:v1","source_title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","source_label":"One-variable case proved","state":"resolved","scope":"Complex polynomials in one standard real Gaussian variable.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","entry_id":"gaussian-moments:entry:original:v1","source_date":"2015-06-17"}],"problem_id":"problem-gaussian-moments","slug":"gaussian-moments","title":"Gaussian moments conjecture","publication_version":6,"relationship":{"claim":{"id":"claim-gaussian-moments-one-variable","version":1},"relation":"special_case","relationship_note":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","result":"proved","report_ids":["gaussian-moments:report:one:v1"],"independent_review":"pending"},"assessments":[{"report_id":"gaussian-moments:report:one:v1","source_id":"gaussian-moments-original","source_version_id":"gaussian-moments:source:original:v1","source_title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","source_label":"One-variable case proved","state":"resolved","scope":"Complex polynomials in one standard real Gaussian variable.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","entry_id":"gaussian-moments:entry:original:v1","source_date":"2015-06-17"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"For P,Q ∈ ℂ[x] and a standard real Gaussian X, if E[P(X)^m] = 0 for all positive integers m, then E[Q(X)P(X)^m] = 0 for all sufficiently large m.","id":"claim-gaussian-moments-one-variable","slug":"gaussian-moments-one-variable","version":1,"kind":"theorem","title":"The one-variable case","assumptions":["One real Gaussian coordinate; complex coefficients.","The eventual threshold may depend on Q."],"scope":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","source_entry_ids":["gaussian-moments:entry:original:v1"],"formal_declarations":[],"relation":"special_case","relationship_note":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","result":"proved","report_ids":["gaussian-moments:report:one:v1"],"independent_review":"pending"}],"entries":[{"id":"gaussian-moments:entry:original:v1","source_version_id":"gaussian-moments:source:original:v1","item_id":"original","url":"https://arxiv.org/abs/1506.05192v1","wording":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","wording_kind":"paraphrase","locator":"Conjecture 1.1, Theorem 1.6 and Proposition 4.2","supersedes":null}],"sources":[{"work":{"authors":["Harm Derksen","Arno van den Essen","Wenhua Zhao"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"2015-06-17"}},"id":"gaussian-moments:source:original:v1","source_id":"gaussian-moments-original","title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","owner":"Harm Derksen, Arno van den Essen, Wenhua Zhao","url":"https://arxiv.org/abs/1506.05192v1","subject_scope":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","revision":"2015-06-17","content_sha256":"182136b363b272c1d1221aa2ca5c6603fd52942efe423463f46997f473503660","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/1506.05192v1","attribution":"Harm Derksen, Arno van den Essen, Wenhua Zhao","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}]}],"assessments":[{"report_id":"gaussian-moments:report:one:v1","source_id":"gaussian-moments-original","source_version_id":"gaussian-moments:source:original:v1","source_title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","source_label":"One-variable case proved","state":"resolved","scope":"Complex polynomials in one standard real Gaussian variable.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","entry_id":"gaussian-moments:entry:original:v1","source_date":"2015-06-17"}],"sources":[{"work":{"authors":["Harm Derksen","Arno van den Essen","Wenhua Zhao"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"2015-06-17"}},"id":"gaussian-moments:source:original:v1","source_id":"gaussian-moments-original","title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","owner":"Harm Derksen, Arno van den Essen, Wenhua Zhao","url":"https://arxiv.org/abs/1506.05192v1","subject_scope":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","revision":"2015-06-17","content_sha256":"182136b363b272c1d1221aa2ca5c6603fd52942efe423463f46997f473503660","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/1506.05192v1","attribution":"Harm Derksen, Arno van den Essen, Wenhua Zhao","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"gaussian-moments:entry:original:v1","source_version_id":"gaussian-moments:source:original:v1","item_id":"original","url":"https://arxiv.org/abs/1506.05192v1","wording":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","wording_kind":"paraphrase","locator":"Conjecture 1.1, Theorem 1.6 and Proposition 4.2","supersedes":null}],"formalizations":[],"tasks":[],"assessment_state":"proved","publication":{"version":6,"sha256":"0bb8acc244117ea8e632eda4b971e702b2cabea186cf80866d55bcb871185dc8","published_at":"2026-10-21T11:59:54.972264+00:00","reason":"Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.","withdrawn":false,"draft":{"reader_guide":{"reading":[{"entry_id":"gaussian-moments:entry:original:v1","role":"original_question","note":"Consult the original question and its hypotheses."},{"entry_id":"gaussian-moments:entry:long:v1","role":"proof","note":"Read the mathematical argument; inspect its Claim for assessment and scope."}],"definitions":[{"id":"gaussian-moments-definition","term":"Pure and mixed moments","explanation":"A pure moment is E[P(X)^m]; a mixed moment here is E[Q(X)P(X)^m]. These are ordinary complex powers, not absolute-value moments.","source_entry_ids":["gaussian-moments:entry:original:v1"]}],"proofs":[],"bounds":[],"witnesses":[{"id":"gaussian-moments-three-variable-witness-witness-detail","claim":{"id":"claim-gaussian-moments-three-variable-witness","version":1},"kind":"polynomial","title":"The three-variable polynomial witness","fields":[{"label":"Polynomial and identities","value":"Set Z = (X₁ + iX₂)/√2, W = (X₁ − iX₂)/√2, P₃ = (1 + Z)(W − (2 + Z)T²/2) and Q₃ = Z. For every integer m ≥ 1, E[P₃^m] = 0 and E[Q₃P₃^m] = m!.","notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Set "},{"kind":"math","text":"Z = (X₁ + iX₂)/√2","latex":"Z=(X_1+iX_2)/\\sqrt2"},{"kind":"text","text":", "},{"kind":"math","text":"W = (X₁ − iX₂)/√2","latex":"W=(X_1-iX_2)/\\sqrt2"},{"kind":"text","text":", "},{"kind":"math","text":"P₃ = (1 + Z)(W − (2 + Z)T²/2)","latex":"P_3=(1+Z)\\left(W-\\frac{2+Z}{2}T^2\\right)"},{"kind":"text","text":" and "},{"kind":"math","text":"Q₃ = Z","latex":"Q_3=Z"},{"kind":"text","text":". For every integer m ≥ 1, "},{"kind":"math","text":"E[P₃^m] = 0","latex":"\\mathbb E[P_3^m]=0"},{"kind":"text","text":" and "},{"kind":"math","text":"E[Q₃P₃^m] = m!","latex":"\\mathbb E[Q_3P_3^m]=m!"},{"kind":"text","text":"."}]}}],"limitation":"The identities hold for every positive integer order in the cited proof. A finite table of moments illustrates them; it does not prove the all-orders claim.","source_entry_ids":["gaussian-moments:entry:long:v1"]},{"id":"gaussian-moments-four-variable-witness-witness-detail","claim":{"id":"claim-gaussian-moments-four-variable-witness","version":1},"kind":"polynomial","title":"The four-variable polynomial witness","fields":[{"label":"Polynomial and identities","value":"For j = 1,2 put Zⱼ = (Xⱼ + iYⱼ)/√2 and Wⱼ = (Xⱼ − iYⱼ)/√2. Set P₄ = (1 + Z₂)(W₁(1 − Z₁) + W₂) and Q₄ = Z₂. For every m ≥ 1, E[P₄^m] = 0 and E[Q₄P₄^m] = m!.","notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For j = 1,2 put "},{"kind":"math","text":"Zⱼ = (Xⱼ + iYⱼ)/√2","latex":"Z_j=(X_j+iY_j)/\\sqrt2"},{"kind":"text","text":" and "},{"kind":"math","text":"Wⱼ = (Xⱼ − iYⱼ)/√2","latex":"W_j=(X_j-iY_j)/\\sqrt2"},{"kind":"text","text":". Set "},{"kind":"math","text":"P₄ = (1 + Z₂)(W₁(1 − Z₁) + W₂)","latex":"P_4=(1+Z_2)(W_1(1-Z_1)+W_2)"},{"kind":"text","text":" and "},{"kind":"math","text":"Q₄ = Z₂","latex":"Q_4=Z_2"},{"kind":"text","text":". For every m ≥ 1, "},{"kind":"math","text":"E[P₄^m] = 0","latex":"\\mathbb E[P_4^m]=0"},{"kind":"text","text":" and "},{"kind":"math","text":"E[Q₄P₄^m] = m!","latex":"\\mathbb E[Q_4P_4^m]=m!"},{"kind":"text","text":"."}]}}],"limitation":"The identities hold for every positive integer order in the cited proof. A finite table of moments illustrates them; it does not prove the all-orders claim.","source_entry_ids":["gaussian-moments:entry:long:v1"]}],"contributors":{"summary":"Long is the human author. ChatGPT 5.6 Sol Pro and Claude Fable 5 receive distinct, self-reported witness-discovery credits; Derksen, van den Essen and Zhao supplied the original question and reduction.","source_entry_ids":["gaussian-moments:entry:original:v1","gaussian-moments:entry:long:v1"]},"scopes":[]},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"gaussian-moments-by-dimension","version":1,"subject":{"kind":"problem","id":"problem-gaussian-moments","sha256":"77c7475c07f8e3442d2786eeb3bae677f36b4fb2dacdd8ad2775c2e5112db371"},"purpose":"setup","title":"The answer by dimension","caption":"The conjecture asks the same question for polynomials in n Gaussian variables. It holds for one variable, fails from three up, and is open for two.","description":"Columns are the number n of independent standard Gaussian variables. With one variable the conjecture holds. With three, an explicit quartic P₃ has every pure moment zero while E[Z P₃^m] = m!. Adding independent variables does not change these expectations, so the same witness refutes every n ≥ 4. Two variables remain open.","assumptions":["Complex polynomials in n independent standard real Gaussian variables.","The n ≥ 4 cells use the same three-variable witness, unchanged by the extra coordinates."],"limitation":"Failure from three variables up says nothing about two. The four-variable cubic witness is a separate statement about degree, not shown here.","author":"OpenMath curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-21","independent_review":"pending","source_entry_ids":["gaussian-moments:entry:long:v1","gaussian-moments:entry:original:v1"],"payload":{"kind":"tracks","column_heading":"Variables n","columns":[{"id":"n1","label":"1"},{"id":"n2","label":"2"},{"id":"n3","label":"3"},{"id":"n4","label":"4+"}],"tracks":[{"id":"gmc","label":"Mixed moments eventually vanish?"}],"cells":[{"id":"one","track":"gmc","columns":["n1"],"label":"Holds","outcome":"holds","claim":{"id":"claim-gaussian-moments-one-variable","version":1}},{"id":"two","track":"gmc","columns":["n2"],"label":"Open","outcome":"open","claim":{"id":"claim-gaussian-moments-two-variable","version":1}},{"id":"three","track":"gmc","columns":["n3"],"label":"Fails: quartic witness","outcome":"fails","claim":{"id":"claim-gaussian-moments-three-variable-witness","version":1}},{"id":"higher","track":"gmc","columns":["n4"],"label":"Fails: same witness","outcome":"fails","claim":{"id":"claim-gaussian-moments-higher-dimensions","version":1}}]}}],"visuals":[{"representation":{"id":"gaussian-moments-by-dimension","version":1},"section":"question"}],"claim_relations":[{"claim":{"id":"claim-gaussian-moments-universal","version":1},"relation":"primary_formulation","relationship_note":"The all-dimensional assertion fails already at n = 3.","result":"refuted","report_ids":["gaussian-moments:report:refutation:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-three-variable-witness","version":1},"relation":"counterexample","relationship_note":"Total degree four in three real coordinates. Ignoring extra coordinates gives failure for every n ≥ 3.","result":"proved","report_ids":["gaussian-moments:report:identities:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-four-variable-witness","version":1},"relation":"supporting_result","relationship_note":"A different witness: degree three with four real coordinates. The lower-dimensional quartic does not replace this degree statement.","result":"proved","report_ids":["gaussian-moments:report:identities:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-one-variable","version":1},"relation":"special_case","relationship_note":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","result":"proved","report_ids":["gaussian-moments:report:one:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-two-variable","version":1},"relation":"related_variant","relationship_note":"This restriction is unresolved in Long’s dimension discussion.","result":"unproved","report_ids":["gaussian-moments:report:two:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-consequence","version":1},"relation":"conditional_implication","relationship_note":"This is a one-way implication. Refuting the Gaussian premise does not logically refute the Jacobian conclusion; the small Gaussian witnesses are not themselves Keller maps.","result":"proved","report_ids":["gaussian-moments:report:consequence:v1"],"independent_review":"pending"},{"claim":{"id":"claim-gaussian-moments-higher-dimensions","version":1},"relation":"conditional_implication","relationship_note":"Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture.","result":"proved","report_ids":["gaussian-moments:report:higher-dimensions:v1"],"independent_review":"pending","reading":{"placement":"consequence","note":"Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture."}}],"milestones":[{"id":"gaussian-moments-formulation","date":{"value":"2015-06-17"},"date_basis":"preprint","category":"foundation","title":"Gaussian vanishing and polynomial inversion","summary":"Derksen, van den Essen and Zhao formulate the conjecture and prove its one-variable case.","limitation":"Their implication to the Jacobian conjecture is global, rather than an equivalence in matching dimensions.","claims":[{"id":"claim-gaussian-moments-universal","version":1},{"id":"claim-gaussian-moments-one-variable","version":1}],"source_entry_ids":["gaussian-moments:entry:original:v1"],"independent_review":"pending"},{"id":"gaussian-moments-witnesses","date":{"value":"2026-07-20"},"date_basis":"preprint","category":"related_result","title":"Two explicit polynomial witnesses","summary":"Long publishes the cubic four-variable and quartic three-variable constructions.","limitation":"Publication date records the source; the paper describes discovery order without providing separate calendar dates.","claims":[{"id":"claim-gaussian-moments-three-variable-witness","version":1},{"id":"claim-gaussian-moments-four-variable-witness","version":1}],"source_entry_ids":["gaussian-moments:entry:long:v1"],"independent_review":"pending"}],"activities":[{"id":"gaussian-moments-dimension-two","title":"Explore the remaining dimension","kind":"research_direction","source_entry_ids":["gaussian-moments:entry:long:v1"],"summary":"The source leaves GMC(2) as the nearest unresolved restriction.","limitation":"Higher-dimensional counterexamples cannot be projected down while preserving their moment identities.","related_claim_ids":["claim-gaussian-moments-two-variable"]}],"curation":{"checked_on":"2026-10-13","selection":"Original question, distinct results, proof provenance, selected history and an explanatory schematic.","search_scope":"Selected primary papers, exact formal declarations, project provenance and official descriptions checked on 13 October 2026.","formal_status":"No formal proof project for these explicit Gaussian identities was located in the selected primary sources. Independent mathematical assessment is pending.","ai_status":"Long is the human author. ChatGPT 5.6 Sol Pro and Claude Fable 5 receive distinct, self-reported witness-discovery credits; Derksen, van den Essen and Zhao supplied the original question and reduction.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For every n ≥ 1 and every P ∈ ℂ[x₁,…,xₙ], if E[P(X)^m] = 0 for every m ≥ 1, then for each Q ∈ ℂ[x₁,…,xₙ] there is M such that E[Q(X)P(X)^m] = 0 for every m ≥ M.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1 and every "},{"kind":"math","text":"P ∈ ℂ[x₁,…,xₙ]","latex":"P\\in\\mathbb C[x_1,\\ldots,x_n]"},{"kind":"text","text":", if "},{"kind":"math","text":"E[P(X)^m] = 0","latex":"\\mathbb E[P(X)^m]=0"},{"kind":"text","text":" for every m ≥ 1, then for each "},{"kind":"math","text":"Q ∈ ℂ[x₁,…,xₙ]","latex":"Q\\in\\mathbb C[x_1,\\ldots,x_n]"},{"kind":"text","text":" there is M such that "},{"kind":"math","text":"E[Q(X)P(X)^m] = 0","latex":"\\mathbb E[Q(X)P(X)^m]=0"},{"kind":"text","text":" for every m ≥ M."}]},"id":"claim-gaussian-moments-universal","slug":"gaussian-moments-universal","version":1,"kind":"conjecture","title":"Eventual mixed-moment vanishing in every dimension","assumptions":["X consists of n independent standard real Gaussians.","Positive integer powers; the threshold may depend on P and Q."],"scope":"The all-dimensional assertion fails already at n = 3.","source_entry_ids":["gaussian-moments:entry:original:v1","gaussian-moments:entry:long:v1"],"formal_declarations":[]},{"statement":"Set Z = (X₁ + iX₂)/√2, W = (X₁ − iX₂)/√2, P₃ = (1 + Z)(W − (2 + Z)T²/2) and Q₃ = Z. For every integer m ≥ 1, E[P₃^m] = 0 and E[Q₃P₃^m] = m!.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Set "},{"kind":"math","text":"Z = (X₁ + iX₂)/√2","latex":"Z=(X_1+iX_2)/\\sqrt2"},{"kind":"text","text":", "},{"kind":"math","text":"W = (X₁ − iX₂)/√2","latex":"W=(X_1-iX_2)/\\sqrt2"},{"kind":"text","text":", "},{"kind":"math","text":"P₃ = (1 + Z)(W − (2 + Z)T²/2)","latex":"P_3=(1+Z)\\left(W-\\frac{2+Z}{2}T^2\\right)"},{"kind":"text","text":" and "},{"kind":"math","text":"Q₃ = Z","latex":"Q_3=Z"},{"kind":"text","text":". For every integer m ≥ 1, "},{"kind":"math","text":"E[P₃^m] = 0","latex":"\\mathbb E[P_3^m]=0"},{"kind":"text","text":" and "},{"kind":"math","text":"E[Q₃P₃^m] = m!","latex":"\\mathbb E[Q_3P_3^m]=m!"},{"kind":"text","text":"."}]},"id":"claim-gaussian-moments-three-variable-witness","slug":"gaussian-moments-three-variable-witness","version":1,"kind":"theorem","title":"A quartic witness in three real variables","assumptions":["X₁, X₂ and T are independent standard real Gaussians.","Z and W are conjugate coordinates, not independent complex Gaussian variables."],"scope":"Total degree four in three real coordinates. Ignoring extra coordinates gives failure for every n ≥ 3.","source_entry_ids":["gaussian-moments:entry:long:v1"],"formal_declarations":[]},{"statement":"For j = 1,2 put Zⱼ = (Xⱼ + iYⱼ)/√2 and Wⱼ = (Xⱼ − iYⱼ)/√2. Set P₄ = (1 + Z₂)(W₁(1 − Z₁) + W₂) and Q₄ = Z₂. For every m ≥ 1, E[P₄^m] = 0 and E[Q₄P₄^m] = m!.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For j = 1,2 put "},{"kind":"math","text":"Zⱼ = (Xⱼ + iYⱼ)/√2","latex":"Z_j=(X_j+iY_j)/\\sqrt2"},{"kind":"text","text":" and "},{"kind":"math","text":"Wⱼ = (Xⱼ − iYⱼ)/√2","latex":"W_j=(X_j-iY_j)/\\sqrt2"},{"kind":"text","text":". Set "},{"kind":"math","text":"P₄ = (1 + Z₂)(W₁(1 − Z₁) + W₂)","latex":"P_4=(1+Z_2)(W_1(1-Z_1)+W_2)"},{"kind":"text","text":" and "},{"kind":"math","text":"Q₄ = Z₂","latex":"Q_4=Z_2"},{"kind":"text","text":". For every m ≥ 1, "},{"kind":"math","text":"E[P₄^m] = 0","latex":"\\mathbb E[P_4^m]=0"},{"kind":"text","text":" and "},{"kind":"math","text":"E[Q₄P₄^m] = m!","latex":"\\mathbb E[Q_4P_4^m]=m!"},{"kind":"text","text":"."}]},"id":"claim-gaussian-moments-four-variable-witness","slug":"gaussian-moments-four-variable-witness","version":1,"kind":"theorem","title":"A cubic witness in four real variables","assumptions":["The four real coordinates X₁,Y₁,X₂,Y₂ are independent standard Gaussians.","Each Wⱼ is the conjugate of Zⱼ."],"scope":"A different witness: degree three with four real coordinates. The lower-dimensional quartic does not replace this degree statement.","source_entry_ids":["gaussian-moments:entry:long:v1"],"formal_declarations":[]},{"statement":"For P,Q ∈ ℂ[x] and a standard real Gaussian X, if E[P(X)^m] = 0 for all positive integers m, then E[Q(X)P(X)^m] = 0 for all sufficiently large m.","id":"claim-gaussian-moments-one-variable","slug":"gaussian-moments-one-variable","version":1,"kind":"theorem","title":"The one-variable case","assumptions":["One real Gaussian coordinate; complex coefficients.","The eventual threshold may depend on Q."],"scope":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","source_entry_ids":["gaussian-moments:entry:original:v1"],"formal_declarations":[]},{"statement":"For P,Q ∈ ℂ[x₁,x₂] and independent standard real Gaussians X₁,X₂, does E[P(X)^m] = 0 for every positive integer m imply E[Q(X)P(X)^m] = 0 for all sufficiently large m?","id":"claim-gaussian-moments-two-variable","slug":"gaussian-moments-two-variable","version":1,"kind":"conjecture","title":"The remaining two-variable question","assumptions":["Exactly two real Gaussian coordinates.","Complex coefficients and a Q-dependent threshold."],"scope":"This restriction is unresolved in Long’s dimension discussion.","source_entry_ids":["gaussian-moments:entry:original:v1","gaussian-moments:entry:long:v1"],"formal_declarations":[]},{"statement":"If the Gaussian Moments Conjecture holds in every positive dimension, then every polynomial map F: ℂⁿ → ℂⁿ in every n ≥ 1 with nonzero constant Jacobian determinant has a polynomial inverse.","id":"claim-gaussian-moments-consequence","slug":"gaussian-moments-consequence","version":1,"kind":"theorem","title":"The global implication to polynomial inversion","assumptions":["The premise is the all-dimensional Gaussian conjecture, not one fixed dimension.","Complex polynomial maps with nonzero constant Jacobian determinant."],"scope":"This is a one-way implication. Refuting the Gaussian premise does not logically refute the Jacobian conclusion; the small Gaussian witnesses are not themselves Keller maps.","source_entry_ids":["gaussian-moments:entry:original:v1"],"formal_declarations":[]},{"statement":"If complex polynomials P,Q in three independent standard real Gaussian variables satisfy E[P^m] = 0 and E[QP^m] = m! for every positive integer m, then for every n ≥ 3 these polynomials, independent of the extra coordinates, refute the Gaussian Moments Conjecture in n variables.","id":"claim-gaussian-moments-higher-dimensions","slug":"gaussian-moments-higher-dimensions","version":1,"kind":"theorem","title":"The same witness works in every higher dimension","assumptions":["The exact three-variable identities hold for every positive integer m.","Additional coordinates are independent standard real Gaussians."],"scope":"Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture.","source_entry_ids":["gaussian-moments:entry:long:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"counterexample","extent":"full_statement","summary":"Long’s quartic witness refutes the universal statement in three variables, hence in every higher dimension. Dimension one is proved in the original source; dimension two remains a related open question.","report_ids":["gaussian-moments:report:refutation:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-gaussian-moments","slug":"gaussian-moments","title":"Gaussian moments conjecture","question":"If every positive power of a complex polynomial has zero Gaussian expectation, must its products with any fixed polynomial eventually have zero expectation?","scope":"For n independent standard real Gaussian variables X, let P and Q belong to ℂ[x₁,…,xₙ]. The hypothesis is E[P(X)^m] = 0 for every integer m ≥ 1. The proposed conclusion is: for each Q there is a threshold M, allowed to depend on Q, such that E[Q(X)P(X)^m] = 0 for every m ≥ M. Complex coefficients are essential; these are ordinary powers, not absolute moments.","aliases":["Gaussian Moments Conjecture","GMC","Gaussian moment vanishing"],"fields":["Probability","Algebra"],"msc":["60E15","14R15"],"topics":[],"background":"undergraduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["gaussian-moments:entry:original:v1","gaussian-moments:entry:long:v1"],"status_report_ids":["gaussian-moments:report:refutation:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"known","scope":"Source comparison and reader curation; independent mathematical assessment remains pending.","rationale":"Long’s quartic witness refutes the universal statement in three variables, hence in every higher dimension. Dimension one is proved in the original source; dimension two remains a related open question.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["gaussian-moments:report:refutation:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","url":"https://arxiv.org/abs/1506.05192v1","note":"Conjecture 1.1, Theorem 1.6 and Proposition 4.2"},{"title":"Small Counterexamples to the Gaussian Moments Conjecture","url":"https://arxiv.org/abs/2607.18186v1","note":"Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure"}],"formalizations":[],"related":[{"slug":"jacobian-conjecture","title":"Jacobian conjecture","relation":"related_result","note":"The original all-dimensional Gaussian conjecture would imply the Jacobian conjecture. Failure of the former alone does not refute the latter, and this reduction is not dimension preserving."}],"tasks":[],"insights":[{"id":"gaussian-moments-coefficient","title":"Read the coefficient identity, not just sample moments","summary":"For either witness and its distinguished coordinate Z, the source derives E[A(Z)P^m] = m! [z^m] A(z)(1+z)^(m−1). Taking A = 1 or A = z gives the two headline identities.","review":"pending","kind":"approach","source_entry_ids":["gaussian-moments:entry:long:v1"],"claims":[{"id":"claim-gaussian-moments-three-variable-witness","version":1},{"id":"claim-gaussian-moments-four-variable-witness","version":1}],"limitation":"A few moment values illustrate the mechanism; the coefficient identity establishes all positive orders."},{"id":"gaussian-moments-impact","title":"The same witness works in every higher dimension","summary":"If complex polynomials P,Q in three independent standard real Gaussian variables satisfy E[P^m] = 0 and E[QP^m] = m! for every positive integer m, then for every n ≥ 3 these polynomials, independent of the extra coordinates, refute the Gaussian Moments Conjecture in n variables.","review":"pending","kind":"impact","source_entry_ids":["gaussian-moments:entry:long:v1"],"claims":[{"id":"claim-gaussian-moments-higher-dimensions","version":1}],"scenario":{"id":"claim-gaussian-moments-three-variable-witness","version":1},"limitation":"Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture."}],"coverage_gaps":["No linked formal proof of the moment identities was found in the selected sources.","The two-variable case and comprehensive later literature require continued review."],"checked_on":"2026-10-13"},"sources":[{"work":{"authors":["Harm Derksen","Arno van den Essen","Wenhua Zhao"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"2015-06-17"}},"id":"gaussian-moments:source:original:v1","source_id":"gaussian-moments-original","title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","owner":"Harm Derksen, Arno van den Essen, Wenhua Zhao","url":"https://arxiv.org/abs/1506.05192v1","subject_scope":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","revision":"2015-06-17","content_sha256":"182136b363b272c1d1221aa2ca5c6603fd52942efe423463f46997f473503660","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/1506.05192v1","attribution":"Harm Derksen, Arno van den Essen, Wenhua Zhao","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Christopher D. Long"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"Long credits ChatGPT 5.6 Sol Pro with the four-variable polynomial, Claude Fable 5 with the three-variable reduction and checks, and ChatGPT with editing. Long is the paper’s author and responsible for its mathematical claims.","artifacts":[],"published":{"value":"2026-07-20"}},"id":"gaussian-moments:source:long:v1","source_id":"gaussian-moments-long","title":"Small Counterexamples to the Gaussian Moments Conjecture","owner":"Christopher D. Long","url":"https://arxiv.org/abs/2607.18186v1","subject_scope":"Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.","revision":"2026-07-20","content_sha256":"d568de4b8568f627548eaacb7c65b1b852a52f1017aa5402150250e3d5cf6738","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2607.18186v1","attribution":"Christopher D. Long","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"gaussian-moments:entry:original:v1","source_version_id":"gaussian-moments:source:original:v1","item_id":"original","url":"https://arxiv.org/abs/1506.05192v1","wording":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","wording_kind":"paraphrase","locator":"Conjecture 1.1, Theorem 1.6 and Proposition 4.2","supersedes":null},{"id":"gaussian-moments:entry:long:v1","source_version_id":"gaussian-moments:source:long:v1","item_id":"long","url":"https://arxiv.org/abs/2607.18186v1","wording":"Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.","wording_kind":"paraphrase","locator":"Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure","supersedes":null}],"reports":[{"id":"gaussian-moments:report:refutation:v1","entry_id":"gaussian-moments:entry:long:v1","state":"refuted","source_label":"Universal conjecture refuted by an explicit polynomial","scope":"Three independent real Gaussian coordinates suffice; the construction extends to every higher dimension.","source_date":"2026-07-20","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.18186v1","supersedes":null},{"id":"gaussian-moments:report:identities:v1","entry_id":"gaussian-moments:entry:long:v1","state":"resolved","source_label":"Moment identities proved in the cited paper","scope":"Exact coefficient identities for the three- and four-variable polynomials, not a finite numerical moment test.","source_date":"2026-07-20","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.18186v1","supersedes":null},{"id":"gaussian-moments:report:one:v1","entry_id":"gaussian-moments:entry:original:v1","state":"resolved","source_label":"One-variable case proved","scope":"Complex polynomials in one standard real Gaussian variable.","source_date":"2015-06-17","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","supersedes":null},{"id":"gaussian-moments:report:two:v1","entry_id":"gaussian-moments:entry:long:v1","state":"reported_open","source_label":"Two-variable case remains open in the cited survey","scope":"The three-variable counterexample does not settle dimension two.","source_date":"2026-07-20","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.18186v1","supersedes":null},{"id":"gaussian-moments:report:consequence:v1","entry_id":"gaussian-moments:entry:original:v1","state":"resolved","source_label":"Conditional mathematical implication in the cited source","scope":"This is a one-way implication. Refuting the Gaussian premise does not logically refute the Jacobian conclusion; the small Gaussian witnesses are not themselves Keller maps.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","supersedes":null},{"id":"gaussian-moments:report:higher-dimensions:v1","entry_id":"gaussian-moments:entry:long:v1","state":"resolved","source_label":"Witness extends to every higher dimension","scope":"Use the same polynomial in its first three coordinates, independent of all extra Gaussian coordinates.","source_date":"2026-07-20","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.18186v1","supersedes":null}]},"status":{"state":"refuted","label":"Resolved · counterexample","extent":"full_statement","summary":"Long’s quartic witness refutes the universal statement in three variables, hence in every higher dimension. Dimension one is proved in the original source; dimension two remains a related open question.","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":"gaussian-moments:report:one:v1","source_id":"gaussian-moments-original","source_version_id":"gaussian-moments:source:original:v1","source_title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","source_label":"One-variable case proved","state":"resolved","scope":"Complex polynomials in one standard real Gaussian variable.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","entry_id":"gaussian-moments:entry:original:v1","source_date":"2015-06-17"}],"problem_id":"problem-gaussian-moments","slug":"gaussian-moments","title":"Gaussian moments conjecture","publication_version":6,"relationship":{"claim":{"id":"claim-gaussian-moments-one-variable","version":1},"relation":"special_case","relationship_note":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","result":"proved","report_ids":["gaussian-moments:report:one:v1"],"independent_review":"pending"},"assessments":[{"report_id":"gaussian-moments:report:one:v1","source_id":"gaussian-moments-original","source_version_id":"gaussian-moments:source:original:v1","source_title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","source_label":"One-variable case proved","state":"resolved","scope":"Complex polynomials in one standard real Gaussian variable.","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/1506.05192v1","entry_id":"gaussian-moments:entry:original:v1","source_date":"2015-06-17"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"For P,Q ∈ ℂ[x] and a standard real Gaussian X, if E[P(X)^m] = 0 for all positive integers m, then E[Q(X)P(X)^m] = 0 for all sufficiently large m.","id":"claim-gaussian-moments-one-variable","slug":"gaussian-moments-one-variable","version":1,"kind":"theorem","title":"The one-variable case","assumptions":["One real Gaussian coordinate; complex coefficients.","The eventual threshold may depend on Q."],"scope":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","source_entry_ids":["gaussian-moments:entry:original:v1"],"formal_declarations":[],"relation":"special_case","relationship_note":"The original paper proves GMC(1); it is compatible with higher-dimensional counterexamples.","result":"proved","report_ids":["gaussian-moments:report:one:v1"],"independent_review":"pending"}],"entries":[{"id":"gaussian-moments:entry:original:v1","source_version_id":"gaussian-moments:source:original:v1","item_id":"original","url":"https://arxiv.org/abs/1506.05192v1","wording":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","wording_kind":"paraphrase","locator":"Conjecture 1.1, Theorem 1.6 and Proposition 4.2","supersedes":null}],"sources":[{"work":{"authors":["Harm Derksen","Arno van den Essen","Wenhua Zhao"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"2015-06-17"}},"id":"gaussian-moments:source:original:v1","source_id":"gaussian-moments-original","title":"The Gaussian Moments Conjecture and the Jacobian Conjecture","owner":"Harm Derksen, Arno van den Essen, Wenhua Zhao","url":"https://arxiv.org/abs/1506.05192v1","subject_scope":"Introduces Gaussian moment vanishing, proves the one-variable case and gives a global implication to the Jacobian conjecture.","revision":"2015-06-17","content_sha256":"182136b363b272c1d1221aa2ca5c6603fd52942efe423463f46997f473503660","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/1506.05192v1","attribution":"Harm Derksen, Arno van den Essen, Wenhua Zhao","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}]}}}