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The corresponding eigenvalue is k², not k.","source_entry_ids":["schiffer-pompeiu:entry:computer:v1","schiffer-pompeiu:entry:certificate:v1","schiffer-pompeiu:entry:computer-lean:v1"],"formal_declarations":["PompeiuSchiffer.Numerics.UnconditionalCounterexamples.exists_common_domain"]},{"statement":"For infinitely many integers N ≥ 2 there are a bounded simply connected star-shaped non-disk Ω with C∞ boundary and N-fold symmetry, λ > 0 and a nonconstant u ∈ C∞(Ω̄) satisfying Δu + λu = 0 in Ω, u = 1 and ∂νu = 0 on ∂Ω.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For infinitely many integers N ≥ 2 there are a bounded simply connected star-shaped non-disk Ω with C∞ boundary and N-fold symmetry, λ > 0 and a nonconstant "},{"kind":"math","text":"u ∈ C∞(Ω̄)","latex":"u\\in C^\\infty(\\overline\\Omega)"},{"kind":"text","text":" satisfying "},{"kind":"math","text":"Δu + λu = 0","latex":"\\Delta u+\\lambda u=0"},{"kind":"text","text":" in Ω, u = 1 and "},{"kind":"math","text":"∂νu = 0","latex":"\\partial_\\nu u=0"},{"kind":"text","text":" on ∂Ω."}]},"id":"claim-schiffer-smooth-symmetry-family","slug":"schiffer-smooth-symmetry-family","version":1,"kind":"theorem","title":"Infinitely many smooth symmetry orders","assumptions":["Planar star-shaped domains with smooth boundary.","Existence for infinitely many integer symmetry orders, not a prescribed N."],"scope":"Cao-Labora–de Dios Pont’s uniform bifurcation theorem supplies an infinite family without a numerical certificate.","source_entry_ids":["schiffer-pompeiu:entry:bifurcation:v1"],"formal_declarations":[]},{"statement":"There exist a nonempty bounded star-shaped planar non-disk Ω with C² boundary and a global real C² field u such that −Δu = u in Ω, u = 1 and ∇u = 0 on ∂Ω.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"There exist a nonempty bounded star-shaped planar non-disk Ω with C² boundary and a global real C² field u such that "},{"kind":"math","text":"−Δu = u","latex":"-\\Delta u=u"},{"kind":"text","text":" in Ω, u = 1 and "},{"kind":"math","text":"∇u = 0","latex":"\\nabla u=0"},{"kind":"text","text":" on ∂Ω."}]},"id":"claim-schiffer-normalized-c2-witness","slug":"schiffer-normalized-c2-witness","version":2,"kind":"theorem","title":"The normalized C² formal statement","assumptions":["C² boundary is specified by a global C² defining function with nonzero gradient on its zero set.","Star-shaped about the origin; hence the domain is contractible and simply connected."],"scope":"The linked formal statement uses eigenvalue one after dilation and a vanishing full boundary gradient. It does not claim the analytic frequency interval or the manuscript’s C∞ regularity.","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"formal_declarations":["Schiffer.Comparator.Schiffer_Star_Shaped"]},{"statement":"Every bounded simply connected planar non-disk Ω with smooth boundary has the Pompeiu property: if a continuous real f has integral zero over every rigid-motion image of Ω, then f is identically zero.","id":"claim-pompeiu-planar-rigidity","slug":"pompeiu-planar-rigidity","version":1,"kind":"conjecture","title":"The planar Pompeiu property question","assumptions":["Rigid motions include translations and orthogonal transformations.","Integration is with respect to planar Lebesgue measure."],"scope":"A related rigidity formulation through domain integrals. Its counterexample evidence is linked to the specific constructions.","source_entry_ids":["schiffer-pompeiu:entry:computer:v1","schiffer-pompeiu:entry:bifurcation:v1"],"formal_declarations":[]},{"statement":"There exist a nonempty bounded star-shaped planar non-disk Ω with C² boundary and a continuous real f, not identically zero, such that the integral of f over every rigid-motion image of Ω equals zero.","id":"claim-pompeiu-star-shaped-c2-witness","slug":"pompeiu-star-shaped-c2-witness","version":1,"kind":"theorem","title":"A formal non-disk failing the Pompeiu property","assumptions":["Planar Lebesgue measure and all affine isometries.","The star-shaped C² witness provided by the cited formal construction."],"scope":"The theorem derives Pompeiu failure from its own Schiffer witness through Green’s identity. It is a related conclusion, not a third geometric construction.","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"formal_declarations":["Schiffer.Comparator.Pompeiu_Star_Shaped"]},{"statement":"For any bounded planar domain Ω with C² boundary, if a global C² real field u satisfies −Δu = u in Ω, u = 1 and ∇u = 0 on ∂Ω, then there exists a continuous real f, not identically zero, with integral zero over every rigid-motion image of Ω.","id":"claim-schiffer-pompeiu-consequence","slug":"schiffer-pompeiu-consequence","version":1,"kind":"theorem","title":"From boundary data to vanishing domain averages","assumptions":["C² boundary in the defining-function sense of the cited formal source.","Planar Lebesgue integration and all affine isometries; the global C² field satisfies the normalized equation."],"scope":"The implication needs its stated boundary regularity and field assumptions. It does not classify every domain failing the Pompeiu property or assert an equivalence for rough domains.","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"counterexample","extent":"full_statement","summary":"The cited August papers give two distinct planar non-disk constructions: a certified analytic tenfold witness and an infinite smooth bifurcation family. Their formal and computational evidence have different exact scopes.","report_ids":["schiffer-pompeiu:report:rigidity:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-schiffer-pompeiu","slug":"schiffer-pompeiu","title":"Schiffer and Pompeiu conjectures","question":"Must a bounded simply connected planar domain be a disk if it supports a nonconstant Helmholtz solution with constant boundary value and zero normal derivative?","scope":"Let Ω be a bounded simply connected planar domain with smooth boundary. Schiffer’s rigidity question assumes a positive eigenvalue λ and a nonconstant real solution of Δu + λu = 0 in Ω, with u = 1 and ∂νu = 0 on the boundary. The value one is a normalization. The related Pompeiu question asks whether every non-disk such domain has the Pompeiu property: a continuous function whose integral over every rigid copy of Ω vanishes must be identically zero.","aliases":["Schiffer conjecture","Pompeiu problem","Pompeiu conjecture","Overdetermined Helmholtz problem"],"fields":["Analysis","Partial differential equations","Spectral geometry"],"msc":["35P05","35N25"],"topics":[],"background":"graduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["schiffer-pompeiu:entry:computer:v1","schiffer-pompeiu:entry:bifurcation:v1","schiffer-pompeiu:entry:lean:v1","schiffer-pompeiu:entry:certificate:v1","schiffer-pompeiu:entry:computer-lean:v1"],"status_report_ids":["schiffer-pompeiu:report:rigidity: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":"The cited August papers give two distinct planar non-disk constructions: a certified analytic tenfold witness and an infinite smooth bifurcation family. Their formal and computational evidence have different exact scopes.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["schiffer-pompeiu:report:rigidity:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures","url":"https://arxiv.org/abs/2608.01579v1","note":"Main theorem, computational certificate appendix and AI declaration"},{"title":"Counterexamples to Schiffer’s Conjecture","url":"https://arxiv.org/abs/2608.05114v1","note":"Theorem 1.1, Corollary 1.2 and AI/Lean acknowledgement"},{"title":"Lean proof of the star-shaped Schiffer/Pompeiu construction","url":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","note":"Schiffer.Comparator.Schiffer_Star_Shaped and Pompeiu_Star_Shaped; simply-connected corollaries"},{"title":"Numerical certificate and reproduction report","url":"https://github.com/MColbrook/Pompeiu_Schiffer/tree/2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d/pompeiu_validation_certificate","note":"REPRODUCTION_REPORT.md; validation archive; Zenodo DOI 10.5281/zenodo.21765287"},{"title":"Lean proof of the analytic tenfold construction","url":"https://github.com/MColbrook/Pompeiu_Schiffer/blob/2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d/lean-proof/SchifferCert/Numerics/UnconditionalCounterexamples.lean","note":"PompeiuSchiffer.Numerics.UnconditionalCounterexamples.exists_common_domain; Comparator definitions and verification record"}],"formalizations":[{"title":"Star-shaped C² Schiffer counterexample","url":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","note":"Exact global C² normalized statement. Smoothness, the tenfold analytic frequency interval and every symmetry order are not attributed to this declaration.","target":{"id":"claim-schiffer-normalized-c2-witness","version":2},"proof_status":"reported_complete","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"declaration":"Schiffer.Comparator.Schiffer_Star_Shaped","source_commit":"2938e277969c329caf154e48a3d8823f3635c7f1","alignment":"staff_linked","checked_on":"2026-10-13"},{"title":"Star-shaped C² Pompeiu counterexample","url":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","note":"A continuous nonzero real test function has zero integral over every rigid copy of the formal domain. The simply-connected corollary follows from contractibility.","target":{"id":"claim-pompeiu-star-shaped-c2-witness","version":1},"proof_status":"reported_complete","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"declaration":"Schiffer.Comparator.Pompeiu_Star_Shaped","source_commit":"2938e277969c329caf154e48a3d8823f3635c7f1","alignment":"staff_linked","checked_on":"2026-10-13"},{"title":"Analytic tenfold Schiffer and Pompeiu counterexample","url":"https://github.com/MColbrook/Pompeiu_Schiffer/blob/2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d/lean-proof/SchifferCert/Numerics/UnconditionalCounterexamples.lean","note":"The full statement includes real-analytic Jordan boundary, tenfold dihedral symmetry, the published frequency interval, a real analytic classical Schiffer field and Pompeiu failure. The formal proof checks its own coarser exact numerical bounds using the same dyadic centre as the interval certificate.","target":{"id":"claim-schiffer-analytic-tenfold-witness","version":2},"proof_status":"reported_complete","source_entry_ids":["schiffer-pompeiu:entry:computer-lean:v1"],"declaration":"PompeiuSchiffer.Numerics.UnconditionalCounterexamples.exists_common_domain","source_commit":"2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d","alignment":"staff_linked","checked_on":"2026-10-13"}],"related":[],"tasks":[],"insights":[{"id":"schiffer-pompeiu-geometry","title":"One boundary with two conditions","summary":"Constant boundary value and zero normal derivative overdetermine the Helmholtz equation. The constructions arrange noncircular geometry while satisfying both.","review":"pending","kind":"approach","source_entry_ids":["schiffer-pompeiu:entry:computer:v1","schiffer-pompeiu:entry:bifurcation:v1"],"claims":[{"id":"claim-schiffer-planar-disk-rigidity","version":1},{"id":"claim-schiffer-analytic-tenfold-witness","version":2},{"id":"claim-schiffer-smooth-symmetry-family","version":1}],"limitation":"The domain must remain simply connected; examples on annuli would not refute this question."},{"id":"schiffer-pompeiu-impact","title":"From boundary data to vanishing domain averages","summary":"For any bounded planar domain Ω with C² boundary, if a global C² real field u satisfies −Δu = u in Ω, u = 1 and ∇u = 0 on ∂Ω, then there exists a continuous real f, not identically zero, with integral zero over every rigid-motion image of Ω.","review":"pending","kind":"impact","source_entry_ids":["schiffer-pompeiu:entry:lean:v1"],"claims":[{"id":"claim-schiffer-pompeiu-consequence","version":1}],"scenario":{"id":"claim-schiffer-normalized-c2-witness","version":2},"limitation":"The implication needs its stated boundary regularity and field assumptions. It does not classify every domain failing the Pompeiu property or assert an equivalence for rough domains."}],"coverage_gaps":["Independent mathematical assessment of the cited constructions remains pending.","No OpenMath execution of the certificate or either Lean project is recorded."],"checked_on":"2026-10-13"},"sources":[{"work":{"authors":["Matthew J. Colbrook","George Stepaniants"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The paper credits GPT-5.5 Codex for debugging early computational code, checked line by line by the authors. The authors own the mathematical argument and certificate.","artifacts":[],"published":{"value":"2026-08-03"}},"id":"schiffer-pompeiu:source:computer:v1","source_id":"schiffer-pompeiu-computer","title":"A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures","owner":"Matthew J. Colbrook, George Stepaniants","url":"https://arxiv.org/abs/2608.01579v1","subject_scope":"Constructs a non-disk simply connected domain with real-analytic Jordan boundary and tenfold symmetry, using certified numerical estimates.","revision":"2026-08-03","content_sha256":"0a228277208a58f7a4a918a57c0db911c2b4354cee05587f47a4cfa11345b1b7","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/2608.01579v1","attribution":"Matthew J. Colbrook, George Stepaniants","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Gonzalo Cao-Labora","Jaume de Dios Pont"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The authors credit GPT-5.5/5.6 and Claude Opus 4.8/Fable 5 for exploration, checks and exposition; GPT-5.6 assisted the early-draft Lean verification. The authors retain responsibility.","artifacts":[],"published":{"value":"2026-08-05"}},"id":"schiffer-pompeiu:source:bifurcation:v1","source_id":"schiffer-pompeiu-bifurcation","title":"Counterexamples to Schiffer’s Conjecture","owner":"Gonzalo Cao-Labora, Jaume de Dios Pont","url":"https://arxiv.org/abs/2608.05114v1","subject_scope":"Constructs infinitely many symmetry orders with smooth star-shaped non-disk counterexamples through a uniform bifurcation argument.","revision":"2026-08-05","content_sha256":"857b2f4975bfc4b2ccc113020bf6e9fc9e1ef36c4caa09c09ff73e08f7840026","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/2608.05114v1","attribution":"Gonzalo Cao-Labora, Jaume de Dios Pont","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Jaume de Dios Pont","Gonzalo Cao-Labora"],"kind":"research_project","publication":"project","publication_note":"Public formal source; execution and statement correspondence are separate checks.","ai_role":"self_reported","ai_note":"The repository attributes formalization to OpenAI Codex. Its metadata reports agent review rather than independent human peer review. The formal path and C² scope differ from the later smooth manuscript.","artifacts":[{"kind":"formal_code","title":"Lean proof of the star-shaped Schiffer/Pompeiu construction — pinned source","url":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","version":"2938e277969c329caf154e48a3d8823f3635c7f1","assessment":"not_executed","note":"Exact source inspected; OpenMath has not executed this formal proof project.","sha256":"02cdbc8f521968df2c2da3c6caad9fc1e93245230efff82ae4509615dc390e96"}]},"id":"schiffer-pompeiu:source:lean:v1","source_id":"schiffer-pompeiu-lean","title":"Lean proof of the star-shaped Schiffer/Pompeiu construction","owner":"Jaume de Dios Pont, Gonzalo Cao-Labora","url":"https://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","subject_scope":"Public theorem chain constructs a bounded star-shaped non-disk with C² boundary and a global C² field satisfying normalized Helmholtz and zero full boundary gradient, then derives failure of the Pompeiu property.","revision":"2938e277969c329caf154e48a3d8823f3635c7f1","content_sha256":"02cdbc8f521968df2c2da3c6caad9fc1e93245230efff82ae4509615dc390e96","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://github.com/jaumededios/Schiffer/blob/2938e277969c329caf154e48a3d8823f3635c7f1/Schiffer/Solution.lean","attribution":"Jaume de Dios Pont, Gonzalo Cao-Labora","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Matthew J. 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Colbrook, George Stepaniants","url":"https://github.com/MColbrook/Pompeiu_Schiffer/tree/2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d/pompeiu_validation_certificate","subject_scope":"The exact dyadic input centre, interval audits, tail bounds and standalone checker support the computer-assisted witness.","revision":"2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d","content_sha256":"66b8c4fcf6bf61f9a529ad70af05de7cdfc8970d3d92bfbf2bfa911033f5f0be","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://github.com/MColbrook/Pompeiu_Schiffer/tree/2fb3f2ecafd3233dbca390d73e764fe6e04a9c7d/pompeiu_validation_certificate","attribution":"Matthew J. 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