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The two linked formalization projects self-report automated assistance; their 2026 work formalizes an established theorem.","source_entry_ids":["poincare-conjecture:entry:0:v1","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:extinction:v1","poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:smooth-four:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:wanted-statements:v1"]},"scopes":[{"id":"poincare-three-topological-scope-detail","claim":{"id":"claim-poincare-three-topological","version":2},"label":"The topological three-dimensional Poincaré theorem","source_entry_ids":["poincare-conjecture:entry:0:v1","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:wanted-statements:v1"],"facts":[{"label":"Dimension","value":"3"},{"label":"Category","value":"Topological · homeomorphism"}]},{"id":"poincare-three-smooth-scope-detail","claim":{"id":"claim-poincare-three-smooth","version":2},"label":"The smooth three-dimensional theorem","source_entry_ids":["poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:wanted-statements:v1"],"facts":[{"label":"Dimension","value":"3"},{"label":"Category","value":"Smooth · diffeomorphism"}]},{"id":"poincare-smooth-four-scope-detail","claim":{"id":"claim-poincare-smooth-four","version":1},"label":"The smooth four-dimensional Poincaré question","source_entry_ids":["poincare-conjecture:entry:smooth-four:v1"],"facts":[{"label":"Dimension","value":"4"},{"label":"Category","value":"Smooth · diffeomorphism"}]},{"id":"poincare-topological-four-scope-detail","claim":{"id":"claim-poincare-topological-four","version":1},"label":"The topological four-dimensional theorem","source_entry_ids":["poincare-conjecture:entry:smooth-four:v1"],"facts":[{"label":"Dimension","value":"4"},{"label":"Category","value":"Topological · homeomorphism"}]}],"connections":{"statements":[{"claim":{"id":"claim-poincare-topological-four","version":1},"label":"Topological Poincaré in dimension 4"},{"claim":{"id":"claim-poincare-smooth-four","version":1},"label":"Smooth Poincaré in dimension 4"},{"claim":{"id":"claim-poincare-three-topological","version":2},"label":"Topological Poincaré in dimension 3"}],"relationships":[{"connection_id":"poincare-smooth-versus-topological-four","title":"Topological and smooth structures","purpose":"comparison"},{"connection_id":"poincare-three-four-scope","title":"Dimensions 3 and 4"}]}},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"poincare-spheres-by-dimension","version":1,"subject":{"kind":"problem","id":"problem-poincare-conjecture","sha256":"1dc06c82cd7afd11c1a3dda373801ff070d6c517263e1f92f60528b89d3131e0"},"purpose":"setup","title":"Spheres in every dimension","caption":"The same question asked up to homeomorphism and up to diffeomorphism. 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Beyond them the cited source reports a standard smooth sphere also in dimension 61 and exotic spheres in every other dimension up to 63. Piecewise linear structures are not shown.","author":"OpenMath curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-21","independent_review":"pending","source_entry_ids":["poincare-conjecture:entry:0:v1","poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:milnor-smale:v1","poincare-conjecture:entry:milnor-spheres:v1","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:smooth-four:v1","poincare-conjecture:entry:wanted-statements:v1"],"payload":{"kind":"tracks","column_heading":"Dimension n","columns":[{"id":"n3","label":"3"},{"id":"n4","label":"4"},{"id":"n5","label":"5"},{"id":"n6","label":"6"},{"id":"n7","label":"7"},{"id":"n8","label":"8"},{"id":"n9","label":"9"},{"id":"n10","label":"10"},{"id":"n11","label":"11"},{"id":"n12","label":"12"},{"id":"n13","label":"13"}],"tracks":[{"id":"topological","label":"Homeomorphic to Sⁿ"},{"id":"smooth","label":"Diffeomorphic to Sⁿ"}],"cells":[{"id":"top-3","track":"topological","columns":["n3"],"label":"Perelman","outcome":"holds","claim":{"id":"claim-poincare-three-topological","version":2}},{"id":"top-4","track":"topological","columns":["n4"],"label":"Freedman","outcome":"holds","claim":{"id":"claim-poincare-topological-four","version":1}},{"id":"top-high","track":"topological","columns":["n5","n6","n7","n8","n9","n10","n11","n12","n13"],"label":"Smale: smooth homotopy spheres","outcome":"holds","claim":{"id":"claim-poincare-high-dimensions","version":1}},{"id":"smooth-3","track":"smooth","columns":["n3"],"label":"Standard","outcome":"holds","claim":{"id":"claim-poincare-three-smooth","version":2}},{"id":"smooth-4","track":"smooth","columns":["n4"],"label":"Open","outcome":"open","claim":{"id":"claim-poincare-smooth-four","version":1}},{"id":"smooth-5-6","track":"smooth","columns":["n5","n6"],"label":"Standard","outcome":"holds","claim":{"id":"claim-poincare-smooth-spheres-standard","version":1}},{"id":"smooth-7-11","track":"smooth","columns":["n7","n8","n9","n10","n11"],"label":"Exotic spheres exist","outcome":"fails","claim":{"id":"claim-poincare-exotic-spheres","version":1}},{"id":"smooth-12","track":"smooth","columns":["n12"],"label":"Standard","outcome":"holds","claim":{"id":"claim-poincare-smooth-spheres-standard","version":1}},{"id":"smooth-13","track":"smooth","columns":["n13"],"label":"Exotic","outcome":"fails","claim":{"id":"claim-poincare-exotic-spheres","version":1}}]}},{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"poincare-three-dimensional-proof","version":1,"subject":{"kind":"problem","id":"problem-poincare-conjecture","sha256":"1dc06c82cd7afd11c1a3dda373801ff070d6c517263e1f92f60528b89d3131e0"},"purpose":"setup","title":"How the three-dimensional case was proved","caption":"Perelman’s Ricci flow with surgery proves the smooth statement; Moise’s smoothing theorem transfers it to topological manifolds. Select a step to read the exact statement.","description":"Start with a smooth simply connected closed three-manifold and any Riemannian metric. Ricci flow with surgery becomes extinct in finite time, which identifies the manifold with the smooth three-sphere. Every topological three-manifold admits a smooth structure (Moise), so the topological statement follows.","assumptions":["Closed, simply connected three-manifolds.","Each step is a cited statement; the arrows show how the proof uses them, not a formal derivation."],"limitation":"An outline only. The detailed proofs are in Perelman’s papers and the Morgan–Tian account, and the step from finite extinction to the smooth sphere needs further arguments not drawn here.","author":"OpenMath curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-21","independent_review":"pending","source_entry_ids":["poincare-conjecture:entry:0:v1","poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:extinction:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:wanted-statements:v1"],"payload":{"kind":"pipeline","steps":[{"id":"extinction","label":"Finite extinction","note":"Ricci flow with surgery on a simply connected closed smooth three-manifold becomes extinct in finite time.","claim":{"id":"claim-poincare-finite-extinction","version":1}},{"id":"smooth","label":"Smooth three-sphere","note":"Hence every closed, simply connected smooth three-manifold is diffeomorphic to the standard three-sphere.","claim":{"id":"claim-poincare-three-smooth","version":2}},{"id":"moise","label":"Moise smoothing","note":"Every closed topological three-manifold admits a compatible smooth structure.","claim":{"id":"claim-poincare-moise-smoothing","version":1}},{"id":"topological","label":"Topological three-sphere","note":"So every closed, simply connected topological three-manifold is homeomorphic to the three-sphere.","claim":{"id":"claim-poincare-three-topological","version":2}}]}}],"visuals":[{"representation":{"id":"poincare-spheres-by-dimension","version":1},"section":"question"},{"representation":{"id":"poincare-three-dimensional-proof","version":1},"section":"question"}],"claim_relations":[{"claim":{"id":"claim-poincare-three-topological","version":2},"relation":"primary_formulation","relationship_note":"Compact Hausdorff three-manifolds without boundary; simply connected includes connectedness.","result":"proved","report_ids":["poincare-conjecture:report:0:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-three-smooth","version":2},"relation":"equivalent_formulation","relationship_note":"Smoothness is additional structure; in dimension three the classification agrees with the topological theorem via smoothing and uniqueness of smooth structures.","result":"proved","report_ids":["poincare-conjecture:report:poincare-three-smooth:v1"],"independent_review":"pending","reading":{"placement":"formulation","note":"Smoothness is additional structure; in dimension three the classification agrees with the topological theorem via smoothing and uniqueness of smooth structures."}},{"claim":{"id":"claim-poincare-moise-smoothing","version":1},"relation":"supporting_result","relationship_note":"This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold.","result":"proved","report_ids":["poincare-conjecture:report:poincare-moise-smoothing:v1"],"independent_review":"pending","reading":{"placement":"background","note":"This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold."}},{"claim":{"id":"claim-poincare-finite-extinction","version":1},"relation":"supporting_result","relationship_note":"Extinction concerns the flow with surgery, including discarded components; it does not mean that the original manifold vanishes under a homeomorphism.","result":"proved","report_ids":["poincare-conjecture:report:poincare-finite-extinction:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-smooth-four","version":1},"relation":"related_variant","relationship_note":"This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.","result":"unproved","report_ids":["poincare-conjecture:report:poincare-smooth-four:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-topological-four","version":1},"relation":"related_variant","relationship_note":"Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.","result":"proved","report_ids":["poincare-conjecture:report:poincare-topological-four:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-high-dimensions","version":1},"relation":"related_variant","relationship_note":"A different dimension: dimension five and above were settled before dimensions four and three.","result":"proved","report_ids":["poincare-conjecture:report:high-dimensions:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-smooth-spheres-standard","version":1},"relation":"related_variant","relationship_note":"A different question: smooth spheres in other dimensions, where diffeomorphism and homeomorphism can differ.","result":"proved","report_ids":["poincare-conjecture:report:smooth-spheres-standard:v1"],"independent_review":"pending"},{"claim":{"id":"claim-poincare-exotic-spheres","version":1},"relation":"related_variant","relationship_note":"A different question: smooth spheres in other dimensions, where diffeomorphism and homeomorphism can differ.","result":"proved","report_ids":["poincare-conjecture:report:exotic-spheres:v1"],"independent_review":"pending"}],"connections":[{"id":"poincare-smooth-versus-topological-four","source":{"id":"claim-poincare-topological-four","version":1},"target":{"id":"claim-poincare-smooth-four","version":1},"relation":"related_to","assertion":null,"explanation":"Homeomorphism and diffeomorphism are different requirements in dimension four.","limitation":"The proved topological theorem does not settle the smooth conjecture.","source_entry_ids":["poincare-conjecture:entry:smooth-four:v1"],"independent_review":"pending"},{"id":"poincare-three-four-scope","source":{"id":"claim-poincare-three-topological","version":2},"target":{"id":"claim-poincare-smooth-four","version":1},"relation":"analogue_of","assertion":null,"explanation":"The four-dimensional smooth question concerns a different dimension and equivalence notion.","limitation":"The three-dimensional theorem is fully proved; this analogue is not an unresolved part of it.","source_entry_ids":["poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:smooth-four:v1"],"independent_review":"pending"}],"milestones":[{"id":"poincare-conjecture:milestone:question","date":{"value":"1904"},"date_basis":"publication","category":"foundation","title":"Poincaré poses the question","summary":"The loop-contraction question becomes the three-dimensional conjecture.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-three-topological","version":2}],"source_entry_ids":["poincare-conjecture:entry:dimension-scope:v1"],"independent_review":"pending"},{"id":"poincare-conjecture:milestone:extinction","date":{"value":"2003-07-17"},"date_basis":"preprint","category":"related_result","title":"Perelman’s finite-extinction argument","summary":"The final preprint supplies extinction for the relevant Ricci flows with surgery.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-finite-extinction","version":1}],"source_entry_ids":["poincare-conjecture:entry:extinction:v1"],"independent_review":"pending"},{"id":"poincare-conjecture:milestone:account","date":{"value":"2007"},"date_basis":"publication","category":"related_result","title":"A detailed proof account","summary":"Morgan and Tian publish the proof and three-sphere conclusion.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-three-topological","version":2},{"id":"claim-poincare-three-smooth","version":2}],"source_entry_ids":["poincare-conjecture:entry:proof:v1"],"independent_review":"pending"},{"id":"poincare-conjecture:milestone:qin","date":{"value":"2026-09-27"},"date_basis":"preprint","category":"formalization","title":"A source-pinned Lean formalization report","summary":"The authors report smooth and topological proofs with release-specific endpoint audits.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-three-topological","version":2},{"id":"claim-poincare-three-smooth","version":2}],"source_entry_ids":["poincare-conjecture:entry:qin-formal:v1"],"independent_review":"pending"},{"id":"poincare-conjecture:milestone:frenzy","date":{"value":"2026-09-28"},"date_basis":"formalization","category":"formalization","title":"External comparator verification is reported","summary":"FrenzyMath records statement and kernel checks at its named proof commit.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-three-topological","version":2},{"id":"claim-poincare-three-smooth","version":2}],"source_entry_ids":["poincare-conjecture:entry:frenzy-verification:v2"],"independent_review":"pending"},{"id":"poincare-conjecture:milestone:frenzypaper","date":{"value":"2026-10-06"},"date_basis":"preprint","category":"formalization","title":"An AI-assisted formalization account","summary":"The preprint describes the project and public evidence.","limitation":"Historical mathematics, external formal reports and OpenMath verification are separate evidence.","claims":[{"id":"claim-poincare-three-topological","version":2},{"id":"claim-poincare-three-smooth","version":2}],"source_entry_ids":["poincare-conjecture:entry:frenzy-formal:v1"],"independent_review":"pending"}],"activities":[{"id":"poincare-external-formalization","title":"Two public formalization projects","kind":"formalization","source_entry_ids":["poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2"],"summary":"The selected 2026 projects report complete smooth and topological three-dimensional proofs, with exact release or verification revisions.","limitation":"These source reports do not constitute OpenMath executions or independently reviewed mathematical correspondence.","related_claim_ids":["claim-poincare-three-topological","claim-poincare-three-smooth"]},{"id":"poincare-verification-date-correction","title":"Verification date distinguished from publication date","kind":"source_correction","source_entry_ids":["poincare-conjecture:entry:frenzy-verification:v2"],"summary":"The project report records checks on 28 September 2026. Its publication date is not established, and that verification event is not used as a bibliographic date.","limitation":"This metadata correction changes no mathematical statement, tested proof commit or externally reported check outcome.","related_claim_ids":["claim-poincare-three-smooth","claim-poincare-three-topological"]}],"curation":{"checked_on":"2026-10-12","selection":"The accepted topological theorem, smooth formulation, proof ingredients, dimension-four scope and recent formalization evidence.","search_scope":"Clay’s official description and Morgan–Tian account; Perelman’s finite-extinction paper; Manolescu’s 2026 survey; two 2026 formalization papers, exact public source revisions and the FrenzyMath verification report; Mathlib wanted declarations; Milnor’s 2014 ICM lecture slides for smooth spheres by dimension.","formal_status":"Complete Lean proofs of the topological and smooth three-dimensional statements are reported by two external projects. Links below pin the relevant source revisions. Mathlib’s wanted declarations remain statement-only at the inspected revision. No OpenMath proof build or independent correspondence review is recorded.","ai_status":"Both formalization projects self-report automated assistance; this attribution is separate from theorem truth and verification evidence.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"Every closed, simply connected topological three-manifold is homeomorphic to the standard three-sphere.","id":"claim-poincare-three-topological","slug":"poincare-three-topological","version":2,"kind":"theorem","title":"The topological three-dimensional Poincaré theorem","assumptions":["Compact Hausdorff three-manifolds without boundary; simply connected includes connectedness."],"scope":"Compact Hausdorff three-manifolds without boundary; simply connected includes connectedness.","source_entry_ids":["poincare-conjecture:entry:0:v1","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:wanted-statements:v1"],"formal_declarations":["DifferentialGeometry.Topology.poincare_conjecture","PoincareConjecture.ComparatorTargets.topologicalPoincareSkeleton","SimplyConnectedSpace.nonempty_homeomorph_sphere_three"]},{"statement":"Every closed, simply connected smooth three-manifold is diffeomorphic to the standard smooth three-sphere.","id":"claim-poincare-three-smooth","slug":"poincare-three-smooth","version":2,"kind":"theorem","title":"The smooth three-dimensional theorem","assumptions":["Smoothness is additional structure; in dimension three the classification agrees with the topological theorem via smoothing and uniqueness of smooth structures."],"scope":"Smoothness is additional structure; in dimension three the classification agrees with the topological theorem via smoothing and uniqueness of smooth structures.","source_entry_ids":["poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:dimension-scope:v1","poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:frenzy-formal:v1","poincare-conjecture:entry:frenzy-verification:v2","poincare-conjecture:entry:wanted-statements:v1"],"formal_declarations":["DifferentialGeometry.PDE.RicciFlow.Surgery.Topology.smoothPoincareConjecture_holds","PoincareConjecture.ComparatorTargets.smoothPoincareSkeleton","SimplyConnectedSpace.nonempty_sdiffeomorph_sphere_three"]},{"statement":"Every closed topological three-manifold admits a smooth structure compatible with its topology.","id":"claim-poincare-moise-smoothing","slug":"poincare-moise-smoothing","version":1,"kind":"theorem","title":"Smoothing a topological three-manifold","assumptions":["This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold."],"scope":"This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold.","source_entry_ids":["poincare-conjecture:entry:qin-formal:v1","poincare-conjecture:entry:dimension-scope:v1"],"formal_declarations":[]},{"statement":"A closed, oriented, simply connected smooth three-manifold with an initial smooth Riemannian metric admits Ricci flow with surgery that becomes extinct in finite time.","id":"claim-poincare-finite-extinction","slug":"poincare-finite-extinction","version":1,"kind":"theorem","title":"Finite extinction in the simply connected case","assumptions":["Extinction concerns the flow with surgery, including discarded components; it does not mean that the original manifold vanishes under a homeomorphism."],"scope":"Extinction concerns the flow with surgery, including discarded components; it does not mean that the original manifold vanishes under a homeomorphism.","source_entry_ids":["poincare-conjecture:entry:extinction:v1","poincare-conjecture:entry:proof:v1","poincare-conjecture:entry:qin-formal:v1"],"formal_declarations":[]},{"statement":"Every closed smooth four-manifold homotopy equivalent to the four-sphere is diffeomorphic to the standard smooth four-sphere.","id":"claim-poincare-smooth-four","slug":"poincare-smooth-four","version":1,"kind":"conjecture","title":"The smooth four-dimensional Poincaré question","assumptions":["This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis."],"scope":"This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.","source_entry_ids":["poincare-conjecture:entry:smooth-four:v1"],"formal_declarations":[]},{"statement":"Every closed topological four-manifold homotopy equivalent to the four-sphere is homeomorphic to the four-sphere.","id":"claim-poincare-topological-four","slug":"poincare-topological-four","version":1,"kind":"theorem","title":"The topological four-dimensional theorem","assumptions":["Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture."],"scope":"Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.","source_entry_ids":["poincare-conjecture:entry:smooth-four:v1"],"formal_declarations":[]},{"statement":"If Mⁿ is a differentiable homotopy sphere of dimension n ≥ 5, then Mⁿ is homeomorphic to Sⁿ. In fact, Mⁿ is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed n-balls under a suitable diffeomorphism.","id":"claim-poincare-high-dimensions","slug":"poincare-high-dimensions","version":1,"kind":"theorem","title":"The Poincaré theorem in dimensions five and higher (Smale)","assumptions":["Differentiable homotopy spheres of dimension n ≥ 5.","Proved by Smale (1960), with related proofs by Stallings–Zeeman and Wallace."],"scope":"The conclusion is a homeomorphism to Sⁿ. The smooth conclusion is a diffeomorphism to two balls glued along their boundaries, not necessarily to Sⁿ.","source_entry_ids":["poincare-conjecture:entry:milnor-smale:v1"],"formal_declarations":[]},{"statement":"For n = 1, 2, 3, 5, 6, 12 and 61, every smooth manifold homeomorphic to Sⁿ is diffeomorphic to the standard smooth Sⁿ.","id":"claim-poincare-smooth-spheres-standard","slug":"poincare-smooth-spheres-standard","version":1,"kind":"theorem","title":"Dimensions where every smooth sphere is standard","assumptions":["Closed smooth manifolds homeomorphic to the n-sphere.","Reported by Milnor from Kervaire–Milnor, with Perelman for n = 3."],"scope":"The source states that the group of oriented diffeomorphism classes of manifolds homeomorphic to Sⁿ is trivial for these n. Dimension four is excluded: there the question is open.","source_entry_ids":["poincare-conjecture:entry:milnor-spheres:v1"],"formal_declarations":[]},{"statement":"For every n ≤ 63 other than 1, 2, 3, 4, 5, 6, 12 and 61, some smooth manifold homeomorphic to Sⁿ is not diffeomorphic to the standard smooth Sⁿ.","id":"claim-poincare-exotic-spheres","slug":"poincare-exotic-spheres","version":1,"kind":"theorem","title":"Exotic spheres in every other dimension up to 63","assumptions":["Closed smooth manifolds homeomorphic to the n-sphere.","Reported by Milnor from Kervaire–Milnor; the first examples are Milnor’s exotic 7-spheres."],"scope":"The source states that the group of oriented diffeomorphism classes is non-trivial for these n. Since Sⁿ has an orientation-reversing diffeomorphism, a non-trivial class is not diffeomorphic to Sⁿ at all. The smooth Poincaré statement therefore fails in these dimensions.","source_entry_ids":["poincare-conjecture:entry:milnor-spheres:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"proved","extent":"full_statement","summary":"The three-dimensional theorem is proved by Perelman’s work, with detailed proof accounts. Recent sources also report complete Lean formalizations; their exact statements and externally checked revisions are listed separately.","report_ids":["poincare-conjecture:report:0:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"theorem","problem_id":"problem-poincare-conjecture","slug":"poincare-conjecture","title":"Poincaré conjecture","question":"Is every closed, simply connected three-dimensional manifold homeomorphic to the three-sphere?","scope":"Closed means compact without boundary. Simply connected includes path connectedness and says every loop can be contracted to a point. 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introduction, Theorem 0.1 and Corollary 0.2","url":"https://www.claymath.org/resource/ricci-flow-and-the-poincare-conjecture/","wording":"Morgan and Tian give a detailed proof of Perelman’s result, using Ricci flow with surgery and finite-time extinction to obtain the three-sphere conclusion.","wording_kind":"paraphrase","locator":"Clay monograph 3; introduction, Theorem 0.1 and Corollary 0.2","supersedes":null},{"id":"poincare-conjecture:entry:extinction:v1","source_version_id":"perelman-extinction@arxiv-math-0307245v1","item_id":"Abstract; finite-extinction argument","url":"https://arxiv.org/abs/math/0307245v1","wording":"Perelman establishes finite extinction for the relevant three-manifold class under Ricci flow with surgery; the simply connected closed case is used in the Poincaré proof.","wording_kind":"paraphrase","locator":"Abstract; finite-extinction argument","supersedes":null},{"id":"poincare-conjecture:entry:dimension-scope:v1","source_version_id":"milnor-poincare-description@clay-description-observed-2026-10-12","item_id":"§§1–3; dimension-three comparison on page 5","url":"https://www.claymath.org/wp-content/uploads/2022/06/poincare.pdf","wording":"Milnor distinguishes topological from smooth classification and explains why the distinctions disappear in dimension three. A homology sphere alone does not satisfy the Poincaré hypothesis.","wording_kind":"paraphrase","locator":"§§1–3; dimension-three comparison on page 5","supersedes":null},{"id":"poincare-conjecture:entry:smooth-four:v1","source_version_id":"manolescu-four-manifolds@icm-2026-doi-25M1806703","item_id":"§3, Conjecture 3.1 and preceding topological classification","url":"https://epubs.siam.org/doi/10.1137/25M1806703","wording":"The survey states the smooth four-dimensional Poincaré conjecture as an open question, distinct from Freedman’s topological four-dimensional result.","wording_kind":"paraphrase","locator":"§3, Conjecture 3.1 and preceding topological classification","supersedes":null},{"id":"poincare-conjecture:entry:qin-formal:v1","source_version_id":"qin-liao-khaitan-chow-poincare@arxiv-2609.33842v1","item_id":"§1, Theorems 1–3; §3, release and verification","url":"https://arxiv.org/abs/2609.33842v1","wording":"The authors report Lean proofs of the smooth theorem and Moise smoothing, yielding the topological theorem. Their release v0.1.3 report names commit 7a48598d35109aa99d1cc678e2724c213cdf4ff3 and audits the endpoint closures for the three standard Lean axioms, without sorryAx or project axioms.","wording_kind":"paraphrase","locator":"§1, Theorems 1–3; §3, release and verification","supersedes":null},{"id":"poincare-conjecture:entry:frenzy-formal:v1","source_version_id":"frenzymath-poincare@arxiv-2610.08329v1","item_id":"Formal theorem endpoints and verification discussion","url":"https://arxiv.org/abs/2610.08329v1","wording":"The team reports an AI-assisted Lean formalization of both the smooth and topological three-dimensional theorem, with public verification resources.","wording_kind":"paraphrase","locator":"Formal theorem endpoints and verification discussion","supersedes":null},{"id":"poincare-conjecture:entry:frenzy-verification:v1","source_version_id":"frenzymath-poincare-verification@432c38f2aa5a30efb13871292d17b4a3309a496a","item_id":"Local integration verification; verification.json beside README","url":"https://github.com/frenzymath/Poincare-Conjecture/blob/432c38f2aa5a30efb13871292d17b4a3309a496a/archive/PoincareConjecture/references/ricci-flow/mapher/integration-verification/README.md","wording":"The project’s report records successful statement comparison and Lean/Nanoda checks on 28 September 2026 for proof commit 1876d7dc2c85325a3f62ce9776af98f910c5db04, permitting only the three standard axioms. Later documentation and import commits are not the tested revision; the report is project-supplied evidence.","wording_kind":"paraphrase","locator":"Local integration verification; verification.json beside README","supersedes":null},{"id":"poincare-conjecture:entry:frenzy-verification:v2","source_version_id":"frenzymath-poincare-verification@432c38f2aa5a30efb13871292d17b4a3309a496a-metadata-v2","item_id":"Local integration verification; verification.json beside README","url":"https://github.com/frenzymath/Poincare-Conjecture/blob/432c38f2aa5a30efb13871292d17b4a3309a496a/archive/PoincareConjecture/references/ricci-flow/mapher/integration-verification/README.md","wording":"The project’s report records successful statement comparison and Lean/Nanoda checks on 28 September 2026 for proof commit 1876d7dc2c85325a3f62ce9776af98f910c5db04, permitting only the three standard axioms. Later documentation and import commits are not the tested revision; the report is project-supplied evidence.","wording_kind":"paraphrase","locator":"Local integration verification; verification.json beside README","supersedes":"poincare-conjecture:entry:frenzy-verification:v1"},{"id":"poincare-conjecture:entry:wanted-statements:v1","source_version_id":"mathlib-poincare-wanted@9454a4c25a2b923eb55dba3098f2d6ae0484d09b","item_id":"SimplyConnectedSpace.nonempty_homeomorph_sphere_three and nonempty_sdiffeomorph_sphere_three","url":"https://github.com/leanprover-community/mathlib4/blob/9454a4c25a2b923eb55dba3098f2d6ae0484d09b/Wanted/Geometry/Manifold/PoincareConjecture.lean","wording":"This Mathlib revision declares the three-dimensional topological and smooth Poincaré statements with proof_wanted. These are formal statements seeking library proofs, not evidence of a completed proof in Mathlib.","wording_kind":"paraphrase","locator":"SimplyConnectedSpace.nonempty_homeomorph_sphere_three and nonempty_sdiffeomorph_sphere_three","supersedes":null},{"id":"poincare-conjecture:entry:milnor-smale:v1","source_version_id":"milnor-poincare-description@clay-description-observed-2026-10-12","item_id":"Smale Theorem, page 4","url":"https://www.claymath.org/wp-content/uploads/2022/06/poincare.pdf","wording":"Smale Theorem: a differentiable homotopy sphere of dimension n ≥ 5 is homeomorphic to the n-sphere, and diffeomorphic to a manifold obtained by gluing two closed n-balls along their boundaries.","wording_kind":"paraphrase","locator":"§3 Higher Dimensions, p. 4, Smale Theorem","supersedes":null},{"id":"poincare-conjecture:entry:milnor-spheres:v1","source_version_id":"milnor-four-centuries@icm-2014-slides-sha256-89cf5c46","item_id":"Slide 33, “To Conclude: An Open Problem”","url":"https://www.math.stonybrook.edu/~jack/abelPrint.pdf","wording":"For each n, the oriented diffeomorphism classes of manifolds homeomorphic to Sⁿ form a semigroup under connected sum. For n ≠ 4 it is a finite abelian group (Kervaire–Milnor, with Perelman); it is trivial for n = 1, 2, 3, 5, 6, 12, 61 and non-trivial for every other n ≤ 63 except 4. The smooth analogue in dimension four remains open.","wording_kind":"paraphrase","locator":"Slide 33, “To Conclude: An Open Problem”","supersedes":null}],"reports":[{"id":"poincare-conjecture:report:0:v1","entry_id":"poincare-conjecture:entry:0:v1","state":"resolved","source_label":"Solved","scope":"The cited source item and version only; this is not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-01","evidence_url":"https://www.claymath.org/millennium/poincare-conjecture/","supersedes":null},{"id":"poincare-conjecture:report:poincare-three-smooth:v1","entry_id":"poincare-conjecture:entry:proof:v1","state":"resolved","source_label":"Established result in cited source","scope":"Smoothness is additional structure; in dimension three the classification agrees with the topological theorem via smoothing and uniqueness of smooth structures.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://www.claymath.org/resource/ricci-flow-and-the-poincare-conjecture/","supersedes":null},{"id":"poincare-conjecture:report:poincare-moise-smoothing:v1","entry_id":"poincare-conjecture:entry:qin-formal:v1","state":"resolved","source_label":"Established result in cited source","scope":"This existence statement supplies the smooth input for the topological theorem; it alone does not classify the manifold.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/2609.33842v1","supersedes":null},{"id":"poincare-conjecture:report:poincare-finite-extinction:v1","entry_id":"poincare-conjecture:entry:extinction:v1","state":"resolved","source_label":"Established result in cited source","scope":"Extinction concerns the flow with surgery, including discarded components; it does not mean that the original manifold vanishes under a homeomorphism.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://arxiv.org/abs/math/0307245v1","supersedes":null},{"id":"poincare-conjecture:report:poincare-smooth-four:v1","entry_id":"poincare-conjecture:entry:smooth-four:v1","state":"reported_open","source_label":"Open question in cited source","scope":"This is a separate conjecture in dimension four; simple connectivity alone is insufficient for its hypothesis.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://epubs.siam.org/doi/10.1137/25M1806703","supersedes":null},{"id":"poincare-conjecture:report:poincare-topological-four:v1","entry_id":"poincare-conjecture:entry:smooth-four:v1","state":"resolved","source_label":"Established result in cited source","scope":"Freedman’s topological conclusion does not identify a smooth structure or imply the smooth four-dimensional conjecture.","source_date":null,"observed_on":"2026-10-12","evidence_url":"https://epubs.siam.org/doi/10.1137/25M1806703","supersedes":null},{"id":"poincare-conjecture:report:high-dimensions:v1","entry_id":"poincare-conjecture:entry:milnor-smale:v1","state":"resolved","source_label":"Established result reported in the official description","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://www.claymath.org/wp-content/uploads/2022/06/poincare.pdf","supersedes":null},{"id":"poincare-conjecture:report:smooth-spheres-standard:v1","entry_id":"poincare-conjecture:entry:milnor-spheres:v1","state":"resolved","source_label":"Established result reported in the author’s lecture","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":"2014-08-15","observed_on":"2026-10-21","evidence_url":"https://www.math.stonybrook.edu/~jack/abelPrint.pdf","supersedes":null},{"id":"poincare-conjecture:report:exotic-spheres:v1","entry_id":"poincare-conjecture:entry:milnor-spheres:v1","state":"resolved","source_label":"Established result reported in the author’s lecture","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":"2014-08-15","observed_on":"2026-10-21","evidence_url":"https://www.math.stonybrook.edu/~jack/abelPrint.pdf","supersedes":null}]},"status":{"state":"proved","label":"Proved","extent":"full_statement","summary":"The three-dimensional theorem is proved by Perelman’s work, with detailed proof accounts. Recent sources also report complete Lean formalizations; their exact statements and externally checked revisions are listed separately.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-12","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"]}},"selected_context":null}}