{"id":"200077c5-966e-4692-9d4d-3901681ff1cd","arxiv_id":"2608.01579","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":9.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A computer-assisted proof constructs a bounded simply connected non-circular planar domain with a nonconstant Neumann eigenfunction equal to 1 on the boundary, disproving Schiffer's and Pompeiu's conjectures.","lead":"Two long-standing conjectures in geometry and analysis say only a disk can have a special vibration mode that is constant on its boundary. The paper constructs a non-circular domain with such a mode, using a computer-assisted proof, disproving both conjectures in the plane.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the analytic reduction and a posteriori contraction proof appear sound; the remaining risk is implementation-level certificate correctness.","rationale":"The reader's weakest assumption—correctness of the exhaustive coefficient and tail estimates and of the interval enclosures—is also the only substantive risk I see. I examined the main mathematical steps and found them coherent: the conformal reduction is standard; the disk-polynomial basis and positive linearisation are used correctly; the inverse K satisfies the stated Laplacian and trace identities (I verified the (0,1) case by hand); the weighted Banach-algebra constants are correct; and the radii-polynomial bounds (66)–(67) leave a comfortable margin. The tail analysis in §3.5 is intricate but appears to cover all omitted rows and columns: finite residual support is bounded, enumerated boundary columns are handled by interval arithmetic, and the far g- and shape-tail columns are controlled by monotone estimates. The final reconstructed domain, univalence, noncircularity, and analytic regularity arguments are standard and consistent. I found no internal inconsistency or obviously false estimate. The only way the central claim could fail is an unnoticed bug in the computational certificate, which is precisely the standard caveat for this genre. Since the paper ships a detailed verification archive and the reader already assigned moderate confidence, I see no reason to change the verdict.","tokens_in":21394,"tokens_out":37734,"duration_ms":461738,"concrete_test":"Run the supplied reproduction on a clean machine from the Zenodo archive (./reproduce.sh; python3 verify_certificate.py) and independently recompute the largest shape-tail column (j=36) and the j=151 far-tail monotone bound using an independent interval library (e.g., Arb) with the recurrences (40)–(42). If either recomputed value exceeds the certified bounds Zp,∂≤0.6202382261408258 or Zp,∞≤0.5997353899953235, the global bound (59) would fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify a load-bearing mathematical flaw. The reduction to the cubic equation, the inverse K in Lemma 2.4, the weighted-algebra estimates, and the radii-polynomial argument are internally consistent. I spot-checked the K inverse identity for the (0,1) mode and the weighted column-sum argument; both work. The tail partition in §3.5 covers the stated support classes, and the monotone shape-tail argument for j≥151 is plausible and internally documented. The genuinely load-bearing assumption is that the shipped 2471×2471 frozen binary64 inverse R, the directed-MPFR interval enclosures, and the exact-rational checker are free of implementation or enumeration errors. This is the standard residual risk for computer-assisted proofs; it is mitigated by byte-identical 256-bit audits, a widened 192-bit audit, and a standalone exact-rational verifier, but I have not independently executed the certificate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a computer-assisted proof of the existence of a bounded, simply connected, noncircular domain Ω⊂R^2 with real-analytic D10-symmetric Jordan boundary and a nonconstant real-analytic function u satisfying (Δ+k^2)u=0 in Ω, u=1, ∂_ν u=0 on ∂Ω, for some k∈(31.967007261,31.967007293). Green's identity then shows that the Fourier transform of the indicator of Ω vanishes on the circle |ξ|=k, so Ω fails the Pompeiu property. The authors conclude that Ω is simultaneously a counterexample to Schiffer's conjecture and to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. The proof transfers the problem to the unit disc via a ten-fold symmetric conformal map and reduces it to the cubic operator equation F(g,p)=g+|p|^2(1+Kg)=0 on weighted coefficient spaces of disk polynomials, where K is an explicit inverse of the Laplacian on the trace-compatible range. The existence of a zero is established by a Newton–Kantorovich/radii-polynomial argument with rigorous interval arithmetic, an exact dyadic numerical centre, a frozen 2471×2471 binary64 approximate inverse, directed MPFR enclosures, and a standalone exact-rational certificate checker.","tokens_in":21630,"tokens_out":19078,"duration_ms":212234,"significance":"If the computational certificate is correct, this is a landmark negative resolution of two long-standing rigidity conjectures. The analytic reduction is a substantial contribution in its own right: the construction of the explicit inverse K with sharp norm bound, the positive disk-polynomial linearisation algebra, and the careful partition of finite and tail contributions are elegant and well matched to the problem. The computational part is unusually thorough: exact dyadic data, frozen binary64 inverse, directed MPFR interval enclosures, byte-identical 256-bit audits, a widened 192-bit audit, and a standalone exact-rational verifier are all provided. The main residual risk is implementation-level: correctness of the frozen inverse, the interval enclosures, and the exhaustive tail enumeration. The paper itself identifies this assumption, and the supplied checks mitigate it to the standard of current computer-assisted proofs. I did not independently execute the certificate, but I found no mathematical gap in the analytic reduction or in the a posteriori contraction argument.","major_comments":[{"comment":"The validity of Theorem 1.1 rests on the absence of implementation and enumeration errors in the frozen 2471×2471 binary64 approximate inverse, the directed-MPFR interval enclosures, and the exhaustive tail partition. The manuscript explicitly identifies this as load-bearing. The certificate provides strong mitigation—byte-identical 256-bit audits, a widened 192-bit audit, a standalone exact-rational checker, and a reproduction script—but I did not execute the certificate. This is a standard residual risk for computer-assisted proofs; I do not regard it as a mathematical flaw, but it is the point on which the existence theorem depends.","section":"Appendix A.1/A.5"}],"minor_comments":[{"comment":"The sentence 'real-valuedness gives f_{-ℓ,s}=f_{ℓ,s}' should read 'real-valuedness gives f_{-ℓ,s} = \\overline{f_{ℓ,s}}' (or 'conjugation gives'), since as printed the two stated symmetry conditions are identical and the conclusion that the coefficients are real is obscured.","section":"§2.2, after Eq. (11)"},{"comment":"The monotone shape-tail bound for j≥151 is stated in one sentence: 'On positive disk-polynomial indices, each recurrence column has nonnegative coefficients with sum one.' For signed coefficient sequences one uses the triangle inequality through the convex recurrence (40). Please add a short display or lemma making the factor-ρ cancellation fully explicit, since this bound contributes a large part of Z (0.5997 of 0.6202).","section":"§3.5.3, Eqs. (55)–(58)"},{"comment":"The first row, labelled 'Principal tail', describes the identification map Jtail rather than an estimate. Consider renaming the row to 'Tail identification' to avoid confusing it with the numerical bounds in the other rows.","section":"Table 1"},{"comment":"The phrase 'passing resemblance to a shortcake biscuit' is informal; the quantitative error bound is clear, but a more neutral wording may be preferable for a journal caption.","section":"Figure 1 caption"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is not a numerical experiment: the paper sets up a rigorous a posteriori contraction in a weighted coefficient algebra, and the existence proof is anchored by interval enclosures, an exact rational verifier, and a frozen binary64 inverse with controlled error. Second, the geometry is genuinely new: it's the first bounded simply connected noncircular planar domain with a constant-boundary Neumann eigenfunction. If the certificate is sound, that settles Schiffer's conjecture and the planar Pompeiu conjecture.\n\nWhat the paper does well: the reduction to the cubic operator equation F = g + |p|^2(1+Kg) is clean, the disk-polynomial framework with K as an explicit inverse of the Laplacian on the compatible range is worked out in detail, and the norm bounds (like ||K||=1/8) are proven. The computational side is unusually careful: exact dyadic centre coefficients, two byte-identical 256-bit audits, an exact leaf-to-theorem checker, and a reproduction script. The authors are honest about the residual risk: the entire validation rests on the correctness of the enumerate-and-enclose step for the 2471×2471 matrix and the tail classifications. That's a standard caveat for computer-assisted proofs, and they've done a lot to mitigate it.\n\nThe soft spots are proportionate. I didn't independently execute the certificate, so my confidence is moderate, not high. The tail bounds in Section 3.5 are plausible but intricate; a referee should spot-check the support classification and the monotone argument for j≥151. Nothing in the analytic reduction looks circular to me: the numerical centre is a computational input, not a fitted parameter, and the contraction theorem is a genuine existence proof around it.\n\nBottom line: this is a paper for specialists in geometric analysis and computer-assisted proofs. If the certificate holds, it's a major result. I'd accept it for peer review without hesitation, with the caveat that the referee should run the verifier or at least independently check key enclosures.\n\nRecommendation: send it to review, and make reproducibility a condition.","headline":"A serious computer-assisted counterexample to two long-standing conjectures—worth reading closely, and worth sending to a referee who will actually run the certificate.","tokens_in":22082,"tokens_out":2262,"would_cite":true,"duration_ms":25713,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J05","35N25","35P05","42B10","47J05","65G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the existence of a bounded simply connected noncircular domain with real-analytic boundary carrying a nonconstant Helmholtz–Neumann eigenfunction that is constant on the boundary, a counterexample to both Schiffer's and Po","keywords":["Pompeiu problem","Schiffer conjecture","overdetermined Neumann eigenvalue problem","conformal mapping","disk polynomials","computer-assisted proof","interval arithmetic","Fourier zero set"],"falsifier":"Regenerate the certificate from source and recompute the 24,001-row support enumeration and the complete g-tail and shape-tail column sums in exact rational arithmetic. If any certified bound in Table 2 is exceeded, or if the radii polynomial at r = 10^-6 is not negative, the central existence claim fails.","tokens_in":21302,"feed_emoji":"🧮","tokens_out":8239,"duration_ms":98433,"temperature":0.7,"pith_summary":"Two long-standing rigidity questions in planar analysis are answered in the negative. The paper constructs a bounded, simply connected, noncircular domain with real-analytic boundary which carries a nonconstant eigenfunction of the Helmholtz operator with $u$ constant and $\\partial_\\nu u = 0$ on the boundary; therefore it is a counterexample both to Schiffer's conjecture and, by Green's identity, to the planar Pompeiu conjecture. The proof is computer-assisted but fully certified: the domain is obtained as the conformal image of the unit disc under a map close to an explicitly listed degree-301 polynomial, and the analytic problem is reduced to a cubic operator equation whose zero is verified by a contraction argument in a weighted coefficient algebra. The upshot is that the two classical conjectures fail in the plane at a specific high frequency, with all numerical estimates enclosed by exact arithmetic.","feed_headline":"Ten-lobed domain breaks Pompeiu and Schiffer conjectures","feed_subtitle":"Certified interval arithmetic proves a noncircular real-analytic domain with a boundary-constant Neumann eigenfunction.","key_machinery":"The load-bearing identity is the cubic operator equation $F(g,p) = g + |p|^2(1 + Kg) = 0$ on real coefficient spaces of ten-fold symmetric disk polynomials, the orthogonal polynomials in radius and angle on the unit disc. $K$ is a three-term inverse of the Laplacian on modes orthogonal to harmonic polynomials, with zero Dirichlet and Neumann traces. Because the disk-polynomial linearisation coefficients are nonnegative and sum to one, the coefficient space is a Banach algebra with norm-one multiplication, so the quadratic and cubic estimates reduce to weighted $\\ell^1$ sums; the same positivity makes the infinite tails monotone, permitting finite enumeration plus rigorous 'everything beyond is smaller' bounds that close the contraction.","core_discovery":"The central claim is that for a specific ten-fold symmetric domain $\\Omega = \\varphi(\\mathbb{D})$, where $\\varphi$ is a conformal map whose coefficients are within $10^{-6}$ of the listed polynomial centre, and for $k \\in (31.967007261, 31.967007293)$, there is a nonconstant real-analytic $u$ satisfying $(\\Delta + k^2)u = 0$ in $\\Omega$, $u = 1$ and $\\partial_\\nu u = 0$ on the boundary. On the fixed unit disc the problem becomes the cubic operator equation $F(g,p) = g + |p|^2(1 + Kg) = 0$, with $g = \\Delta(U - 1)$, $p = k\\varphi'$, and $K$ an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces. A posteriori radii-polynomial estimates, assembled from interval arithmetic and monotone tail bounds, certify a unique zero of $F$ in a small ball around an approximate solution, which pulls back to the claimed eigenfunction on $\\Omega$.","manual_tex_pass":"2026-08-05-schiffer","pith_inferences":["The construction begins from a bifurcation in the ten-fold angular sector near a higher zero of a Wronskian; if that mechanism is generic, Schiffer counterexamples may exist in other symmetry sectors and at many frequencies, not just this isolated one.","Because the proof only needs certified control of a cubic equation in a coefficient algebra, the same pipeline should adapt to other m-fold symmetric domains or to overdetermined problems with nonzero constant Neumann data, provided the corresponding tail bounds are re-certified.","The Fourier-circle property ties this counterexample to the regularity theory of nonscattering inhomogeneities: at frequency k the domain is formally invisible to a constant incident field, and the real-analytic boundary is consistent with known regularity results.","Future rigidity statements will likely need hypotheses that exclude high-frequency, high-symmetry modes, since the present proof shows such modes can break the classical conclusions."],"forward_implications":["Both conjectures are false as stated: no rigidity theorem of the Schiffer or Pompeiu type survives for bounded simply connected Lipschitz domains without extra hypotheses.","At the certified frequency, a plane wave of direction k e1 has zero integral over every rigid motion of Omega, giving an explicit continuous witness of Pompeiu failure.","The boundary of Omega is a real-analytic Jordan curve of critical points of u, with Hessian equal to -k^2 nu tensor nu; the paper supplies the global analytic extension that local Cauchy data alone cannot guarantee.","The validated domain is quantitatively close to a printed finite curve: boundary parametrisation error below 7.13e-11 and noncircularity certified through a first nonzero shape coefficient of magnitude greater than 12.16."],"supporting_citations":[{"why":"supplies the disk-polynomial inverse recurrence and the positive linearisation coefficients used to build K and the Banach algebra estimates.","marker":"[3]"},{"why":"provides the positivity of the linearisation coefficients that underlies the coefficient-space algebra estimate.","marker":"[25]"},{"why":"supplies the radii-polynomial contraction criterion that converts interval bounds into an existence and uniqueness theorem.","marker":"[21]"},{"why":"gives the Fourier–Laplace characterisation connecting Pompeiu failure to Fourier zero sets.","marker":"[7]"},{"why":"establishes the equivalence between Fourier cancellation and the overdetermined Neumann boundary-value problem used to link the two conjectures.","marker":"[37]"},{"why":"shows that Pompeiu counterexamples must have real-analytic boundary, the regularity class needed by the constructed domain.","marker":"[38]"},{"why":"provides the boundary regularity estimates used to pass from W^{2,q} solutions to smooth solutions.","marker":"[1]"},{"why":"provides the analytic elliptic regularity used to extend the solution real-analytically across the boundary.","marker":"[29]"}],"fun_headline_variants":["Computer-assisted proof topples two planar conjectures","Lobed disk refutes Pompeiu and Schiffer conjectures","Exact counterexample to Pompeiu and Schiffer found","Noncircular domain breaks Schiffer's conjecture"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof stands on the correctness of the exhaustive coefficient and tail estimates: every omitted row and column must really be covered by the monotone bounds, and the frozen 2471-by-2471 approximate inverse with its interval enclosures must have no enumeration or implementation error.","fun_headline_variants_meta":{"raw":{"variants":["Computer-assisted proof topples two planar conjectures","Lobed disk refutes Pompeiu and Schiffer conjectures","Exact counterexample to Pompeiu and Schiffer found","Noncircular domain breaks Schiffer's conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1588,"prompt_tokens":917,"completion_tokens":671,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":606}},"tokens_in":661,"tokens_out":671,"duration_ms":8247,"temperature":1.0,"reasoning_tokens":606,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:43:20.402997+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Regenerate the certificate from source and recompute the 24,001-row support enumeration and the complete g-tail and shape-tail column sums in exact rational arithmetic. If any certified bound in Table 2 is exceeded, or if the radii polynomial at r = 10^-6 is not negative, the central existence claim fails.","supporting_citations":[{"cited_title":"Arioli and H","cited_arxiv_id":null,"evidence_quote":"supplies the disk-polynomial inverse recurrence and the positive linearisation coefficients used to build K and the Banach algebra estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the positivity of the linearisation coefficients that underlies the coefficient-space algebra estimate."},{"cited_title":"Hungria, J.-P","cited_arxiv_id":null,"evidence_quote":"supplies the radii-polynomial contraction criterion that converts interval bounds into an existence and uniqueness theorem."},{"cited_title":"Brown, B","cited_arxiv_id":null,"evidence_quote":"gives the Fourier–Laplace characterisation connecting Pompeiu failure to Fourier zero sets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the equivalence between Fourier cancellation and the overdetermined Neumann boundary-value problem used to link the two conjectures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"shows that Pompeiu counterexamples must have real-analytic boundary, the regularity class needed by the constructed domain."},{"cited_title":"Agmon, A","cited_arxiv_id":null,"evidence_quote":"provides the boundary regularity estimates used to pass from W^{2,q} solutions to smooth solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the analytic elliptic regularity used to extend the solution real-analytically across the boundary."}],"review_version":1}