{"id":"f8e2f757-d74f-4966-b51f-05d5d1298742","arxiv_id":"2608.08953","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A certified computer proof constructs a non-circular, 26-fold-symmetric planar domain carrying a sign-changing Helmholtz eigenfunction with zero Dirichlet and constant nonzero Neumann boundary data, disproving the unrestricted planar Berenstein conjecture.","lead":"This paper gives a computer-checked counterexample to the long-standing Berenstein conjecture: a shaped, non-circular planar domain that carries an eigenfunction whose boundary values are zero everywhere and whose boundary pull (normal derivative) is constant everywhere. If correct, it shows that the disc cannot be recognized from this special boundary fingerprint.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The existence theorem rests on infinite-dimensional tail bounds (41), (45)-(47) whose monotonicity and weighted-shift steps are asserted or imported from [7]; a small error there could move Z above the certified value and invalidate the contraction, and the certificate does not independently check…","rationale":"The paper is a serious computer-assisted proof with a strong artefact: pinned SHA-256 digests, byte-identical 256-bit runs, a standalone verifier, exact rational re-aggregation, and a well-specified finite block (882 unknowns). My reading of the contraction estimates, the support lemmas, and the reconstruction argument found no concrete error. The most load-bearing concern is the control of the infinite-dimensional tail. The theorem's validity depends on the source-tail and shape-tail bounds (41) and (45)-(47); these are the only steps that are not independently checked by the verifier and whose proofs are partly asserted or imported from [7]. This matches the reader's weakest assumption. The correct response is to keep CONDITIONAL: the claimed refutation of the Berenstein conjecture is plausible and well-supported, but should be treated as settled only after an independent re-derivation/re-execution. If the independent test passes, ACCEPT would be warranted.","tokens_in":19603,"tokens_out":21484,"duration_ms":211808,"concrete_test":"Run an independent re-implementation of the certificate using a different interval-arithmetic library (e.g., Arb/arblib) or a formal proof assistant, with a fresh derivation of (41), (45)-(47) from the exact recurrences; verify that the recomputed Y, Z, ||A||, p(10^-6), and contraction bound agree with (49)-(54) to the stated precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the a posteriori step, Theorem 4.6 hinges on Z <= 0.4732202748897342. The finite part of Z is interval-certified, but the control of infinitely many omitted directions is analytic. After the support cutoffs (42), the source-tail bound (41) is declared 'monotone' without a displayed proof; for shape directions, (47) asserts for all j >= 61 that Z_a(j) <= (k°)^2 sigma^{-1} rho^{-41} ||H_41||_rho + A_-, where A_- in (46) is only described as the 'exact normalised norm' after cancellation. The deduction uses the weighted-shift bound (45), inherited from [7, Sec. 3.5], whose assumption 'every angular index in H_j is positive' is not re-verified for the present H_j = z^{mj} a° U° and the conjugate term a° z^{-mj} U° appearing in (23). Since Table 1 shows the shape-tail column 21 <= j <= 60 already gives the largest contribution to Z, a small error in the extremal-column or monotonicity argument could change the certified contraction. The verifier script recomputes the radii inequalities in exact arithmetic, but it only re-aggregates the same interval enclosures; it does not check the analytic tail majorants. Thus no independent check currently covers the step on which the infinite-dimensional existence claim rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a computer-assisted counterexample to the planar Berenstein conjecture in its unrestricted, sign-changing form. Theorem 1.1 asserts the existence of a bounded simply connected domain Ω in R^2 with real-analytic Jordan boundary, a frequency k in (27.4381178838, 27.4381198839), and a nonzero real-analytic function u such that (Δ + k^2)u = 0 in Ω, u = 0 and ∂_ν u = 1 on ∂Ω. The domain is D13-invariant, neither a disc nor centrally symmetric, the eigenfunction changes sign, and the boundary arclength Fourier transform vanishes on the circle |ξ| = k. The proof proceeds through a Fourier-circle equivalence (Proposition 2.1), a conformal transfer to the unit disc giving the coupled interior-boundary system (11), coefficient spaces built from D13-adapted disk polynomials, an explicit zero-Dirichlet inverse K_D and its Neumann trace N, a radii-polynomial contraction theorem (Theorem 4.1), and a computer-assisted verification of the finitely many inequalities with directed-rounding interval arithmetic. Appendix A supplies exact dyadic inputs, hashes, and a verifier script. The reconstruction part proves univalence, recovers the unsquared boundary condition by a positivity argument, extends U analytically across the boundary, and obtains the sign change via Serrin's theorem.","tokens_in":19782,"tokens_out":16083,"duration_ms":172898,"significance":"If the proof is correct, this is a major result: it disproves the planar Berenstein conjecture in the sign-changing case, complementing the recent Pompeiu-Schiffer counterexamples of Colbrook and Stepaniants, and it demonstrates that the fixed-disc/disk-polynomial/validated-tail framework extends to the Dirichlet endpoint with a coupled interior-boundary system. The paper is careful with normalizations, works with exact dyadic centers, and provides a detailed certificate with cryptographic hashes and a reproduction script. I found no circularity: the a posteriori contraction argument certifies existence around a fixed dyadic center rather than assuming the numerical solution. The analytic reductions, including Proposition 2.1 and the coefficient identities (18)-(20), are transparent and pass manual spot checks. The main risks are the partially unproved infinite-dimensional tail bounds and the fact that the entire numerical certificate is produced and checked by a single software pipeline.","major_comments":[{"comment":"The shape-tail bound is load-bearing: Table 1 shows that the 'remaining shape tail, j≥61' contributes 0.4506521964853019 to Z, and Z ≤ 0.4732202748897342 is one of the two quantitative hypotheses of Theorem 4.6. The manuscript states without proof that for j≥41 the weighted-shift argument of [7, Sec. 3.5] applies because 'every angular index in H_j is positive'. For H_j = z^{mj} a° U° with U° = K_D g° expanded in the real symmetric basis Φ_{ℓ,s} + Φ_{-ℓ,s}, this positivity is not immediate and should be proved or verified from the actual support of g°. Likewise, A_- in (46) is only described as the 'exact normalised norm' after cancellation; the cancellation of the principal term of DF2 and the resulting norm should be displayed. Without these details, the infinite tail of shape directions is not independently certified.","section":"§4.4, Eqs. (45)–(47)"},{"comment":"The source-tail bound is asserted to be monotone beyond the support cutoffs, but no proof is given. The extremal-column reduction to (41,0), (41,1), and (0,302) is plausible from the monotonicity of κ_{ℓ,s} = 1/[(m|ℓ|+2s)(m|ℓ|+2s+2)] and ζ_ℓ = 1/[2(mℓ+1)], but the case s = 0 with the additional trace term must be handled separately, and the present text does not do so. Since Table 1 lists the largest source-tail column meeting the retained block as 0.4593799524396485, a mistake in this step would directly affect the certified value of Z. Please include the short monotonicity argument.","section":"§4.4, Eq. (41) and following paragraph"},{"comment":"The theorem is certified by a single software pipeline: src/interval_assemble.cpp produces the interval enclosures, and verify_certificate.py aggregates exactly those enclosures. The verifier does not perform an independent computation of the interval bounds or of the analytic tail majorants (41), (45)–(47). Given that the main theorem rests on the certified bounds Y ≤ 3.621873700919759e-9 and Z ≤ 0.4732202748897342, I ask for a higher standard of evidence: a second independent implementation in a different language, a formal proof assistant check of the finite computations, or at minimum a complete public trace of all interval endpoints that enter Y and Z. This is a request for proportionate verification, not a demand for formal verification of the whole proof.","section":"§4.5, Appendix A"},{"comment":"Several essential ingredients are imported from the companion preprint [7] rather than proved in this paper: the positive disk-polynomial linearization (Corollary 3.1), the conformal transformation identities behind (10), the weighted-shift shape-tail bound (45), and the analytic-continuation argument used in Section 5.3. The paper states that some of these are independent of the symmetry order, but the manuscript should make the dependence explicit and, since [7] is a preprint, should either reproduce the arguments or give precise statements of the hypotheses verified for the present D13 data. The real-analytic regularity of u, the sign-recovery step, and the tail control all rely on these imports.","section":"§3.1, §4.4, §5.3"}],"minor_comments":[{"comment":"The line 'Φℓ,s = Φ−ℓ,s' should read 'overline{Φ_{ℓ,s}} = Φ_{-ℓ,s}', and the sentence should be split so that the conjugation identity and the bound sup_D |Φ_{ℓ,s}| ≤ 1 are stated separately.","section":"§3.1"},{"comment":"The expression 'DF1(x°)[0,0,z^{mj}] = (k°)^2(H_j + H_j)' is ambiguous: one of the two summands should be the conjugate term \\(\\overline{H_j}\\) arising from the second term in (23).","section":"§4.4, Eq. (43) and the displayed derivative"},{"comment":"The phrase 'exact normalised norm' is not defined; please state explicitly the norm and the normalization that produce A_-.","section":"§4.4, Eq. (46)"},{"comment":"The sentence containing '≤10−6. which together with (35)' has a typographical period before 'which'; it should read '≤10−6, which together with (35)'.","section":"§5.2"},{"comment":"The abstract writes \\(u \\in C^\\omega(\\overline{\\Omega})\\) while the body defines \\(C^\\omega(\\Omega)\\) as restrictions of real-analytic functions defined on a neighbourhood of Ω; please unify the notation.","section":"Abstract and Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"This is an impressive and significant result, and I found no internal contradiction or circularity in the analytic framework. My recommendation of major revision is driven by the burden of proof appropriate for a claimed disproof of a long-standing conjecture: the infinite-dimensional tail estimates are not fully proved in the text, and the numerical certificate is verified only by the authors' own pipeline. I would ask the editor to require either an independent re-execution of the finite computations or a detailed proof of the tail bounds and the imported lemmas from [7] before accepting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper deserves a serious referee and likely acceptance after minor revision. It constructs the first bounded simply connected counterexample to the unrestricted planar Berenstein conjecture, with real-analytic boundary and D13 symmetry, and the proof is a rigorous a posteriori contraction argument with interval arithmetic. The result is new and significant: it closes a four-decade-old problem and completes the pair of bounded planar counterexamples at the Dirichlet and Neumann endpoints.\n\nWhat's good: the Fourier-circle equivalence (Prop 2.1) is clean; the zero-Dirichlet inverse formulas (18)-(19) and Neumann trace (20) are exact and checkable; the dimension count 882 is right; the certificate is exemplary—pinned SHA-256 digests, exact dyadic inputs, two byte-identical 256-bit runs, and a verifier script that recomputes the radii inequalities in exact rational arithmetic. The authors are transparent about what is computed and what is analytic.\n\nThe soft spots are two, and both are addressable. First, the source-tail bound (41) is declared 'monotone' beyond the cutoff, but the monotonicity is not displayed. It's easy to prove—κ_{ℓ,s} and ζ_ℓ are decreasing in ℓ and s—but it should be a lemma, not an assertion. Second, the shape-tail bound (45) is imported from [7, Sec. 3.5]. The stress-test concern about whether every angular index in H_j is positive is actually resolved by reading the definitions: H_j is supported on ℓ = j + j0 ± ℓ' with 0≤j0≤20 and |ℓ'|≤20, so for j≥41 all indices are positive, and then (45) is an equality, not an inequality. The paper could say this in two lines. The larger caveat is that the proof inherits several lemmas from the authors' unreplicated 2026 preprint [7]. That is not circular—the references are explicit—but it does mean the full chain of confidence passes through a second document that hasn't been independently replicated. A referee who accepts [7]'s results will find this paper sound.\n\nI don't think the stress-test's worry about a small tail error invalidating the contraction is credible on the merits; the tail bounds are straightforward once the notation is unpacked. But the bookkeeping is exactly where computer-assisted proofs most often slip, so the authors should be asked to expand the proof of (41) and to state explicitly the support and positivity of H_j. An independent re-execution of the certificate would also be cheap insurance. None of this undermines the central argument.\n\nWho is this for? Anyone working on overdetermined eigenvalue problems, the Pompeiu-Schiffer circle, or computer-assisted proofs. It deserves peer review—not desk rejection—and I'd be happy to see it in an archive.\n\nRecommendation: send to a serious journal, but require the missing tail details before final acceptance.","headline":"A credible computer-assisted refutation of the planar Berenstein conjecture; the certificate is strong, the tail estimates are mostly standard but one monotonicity claim is underexplained.","tokens_in":20455,"tokens_out":7243,"would_cite":true,"duration_ms":66369,"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":"A certified construction produces a non-circular real-analytic domain whose sign-changing Dirichlet eigenfunction has constant normal derivative, disproving the unrestricted planar Berenstein conjecture.","keywords":["Berenstein conjecture","overdetermined Dirichlet eigenvalue problem","constant normal derivative","conformal mapping","disk polynomials","validated numerics","computer-assisted proof","Fourier zero sets"],"falsifier":"Run an independent implementation of the certificate checks: if it returns $Y>3.6219\\times10^{-9}$ or $Z>0.4732202748897342$ at the stated centre, or if two reproductions of the directed-rounding runs disagree, the existence theorem fails.","tokens_in":19218,"feed_emoji":"📐","tokens_out":9585,"duration_ms":91960,"temperature":0.7,"pith_summary":"The paper claims to construct a concrete bounded, simply connected planar domain $\\Omega$ with real-analytic boundary that is not a disc, together with a frequency $k\\approx 27.438$ and a nonzero real-analytic solution $u$ satisfying $(\\Delta+k^2)u=0$ in $\\Omega$, $u=0$, and $\\partial_\\nu u=1$ on $\\partial\\Omega$. If correct, this settles the unrestricted planar Berenstein conjecture in the negative: overdetermined Dirichlet--Neumann data alone do not force a disc. The proof pulls the problem back to the unit disc through a $D_{13}$-symmetric conformal map, rewrites it as a coupled interior--boundary polynomial system, and reduces existence to finitely many certified inequalities checked by interval arithmetic. A separate sign-recovery step converts the squared boundary condition $(Ng)^2=|a|^2$ back to $Ng=|a|$, and because the eigenfunction changes sign, the classical positivity symmetry theorem does not apply.","feed_headline":"Non-circular domain defeats planar Berenstein conjecture","feed_subtitle":"A verified 13-fold symmetric region satisfies u=0 and ∂νu=1 on its boundary without being a disc.","key_machinery":"The load-bearing object is the $D_{13}$-adapted disk-polynomial basis $\\Phi_{\\ell,s}$, with $\\Phi_{\\ell,s}(r,\\theta)=r^{13|\\ell|}P_s^{(0,13|\\ell|)}(2r^2-1)e^{i13\\ell\\theta}$. The positive product formula for these polynomials makes the weighted $\\ell^1$ coefficient space a Banach algebra with multiplication constant one. On this algebra the paper writes down the explicit zero-Dirichlet inverse $K_D$, with separate actions on harmonic modes $s=0$ and on modes $s\\ge 1$, and its Neumann trace $N$, with $N\\Phi_{\\ell,0}=e^{i13\\ell\\theta}/(2(13|\\ell|+1))$ and $N\\Phi_{\\ell,s}=0$ for $s\\ge1$. These give the coupled fixed-disc equations $F_1(g,k,a)=g+k^2|a|^2K_Dg=0$ and $F_2(g,k,a)=(Ng)^2-|a|^2=0$ on the boundary. A contraction-mapping radii-polynomial theorem reduces a zero of $F$ to finitely many inequalities, which are certified in interval arithmetic with directed rounding at 256-bit and guarded 192-bit precision.","core_discovery":"On its own terms, the central discovery is Theorem 1.1: there is a bounded simply connected domain $\\Omega\\subset\\mathbb{R}^2$ with real-analytic Jordan boundary, invariant under the dihedral group $D_{13}$ of order $26$ but neither a disc nor centrally symmetric, and a frequency $k$ in $(27.4381178838,27.4381198839)$ for which a nonzero real-valued $u\\in C^\\omega(\\overline{\\Omega})$ solves $(\\Delta+k^2)u=0$ in $\\Omega$, $u=0$, $\\partial_\\nu u=1$ on $\\partial\\Omega$. Equivalently, by the paper's Proposition 2.1, the arclength measure $\\sigma_{\\partial\\Omega}$ has Fourier transform $\\widehat{\\sigma_{\\partial\\Omega}}(k\\omega)=0$ for every direction $\\omega\\in\\mathbb{S}^1$. The function $u$ changes sign. The construction is not a formal adaptation of the companion Neumann-endpoint framework: the nonzero Neumann datum keeps the harmonic source modes present, so the problem becomes the coupled system $g+k^2|a|^2K_Dg=0$ and $(Ng)^2=|a|^2$ on the boundary, together with a separate sign-recovery argument.","pith_inferences":["The construction is likely parametric: the same symmetry-adapted branch could be continued to further sign-changing examples at other frequencies and symmetry orders (editorial inference).","The Fourier-circle equivalence turns the problem into a search for analytic Jordan curves whose arclength transform vanishes on a whole circle, which suggests geometric searches at frequencies tied to zeros of Bessel-function combinations (editorial inference).","If the certified arithmetic were replaced by a formally verified implementation, the result could become a fully machine-checked theorem; the current proof depends on the authors' own directed-rounding verifier (editorial inference).","The sign-recovery step indicates that the distinction between the squared and unsquared Neumann condition is the true obstruction separating the Berenstein conjecture from its sign-definite versions (editorial inference)."],"forward_implications":["The unrestricted planar Berenstein conjecture is false: a non-disc can carry a sign-changing Dirichlet eigenfunction with constant nonzero normal derivative.","There exists a non-circular real-analytic Jordan curve whose boundary arclength measure vanishes on the full Fourier circle $\\{\\xi:|\\xi|=k\\}$.","The overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on the eigenfunction.","For this counterexample the boundary obstruction is global: $|\\nabla u|=1$ holds on $\\partial\\Omega$ even though $u$ is not sign-definite inside.","The construction provides a $D_{13}$-symmetric, non-centrally-symmetric example, so rigidity theorems restricted to centrally symmetric convex domains do not apply."],"supporting_citations":[{"why":"supplies the conformal fixed-disc framework, disk-polynomial Banach algebra, inverse formula for modes s≥1, and the validated-tail computational architecture on which the present proof builds.","marker":"[7]"},{"why":"formulates the unrestricted Berenstein conjecture that Theorem 1.1 disproves.","marker":"[2]"},{"why":"provides the interval-arithmetic and verified floating-point framework used to certify the finite bounds.","marker":"[33]"},{"why":"gives the moving-plane symmetry theorem used to conclude that any counterexample must be sign-changing.","marker":"[36]"},{"why":"supplies the exterior-uniqueness facts behind the Fourier-circle equivalence in Proposition 2.1.","marker":"[8]"},{"why":"provides the single-layer potential jump relations used in the same equivalence.","marker":"[24]"},{"why":"introduces the radii-polynomial validated continuation criterion adapted in Theorem 4.1.","marker":"[13]"},{"why":"adapts the radii-polynomial method to weighted analytic sequence spaces in the contraction step.","marker":"[20]"},{"why":"defines the Jacobi polynomials and orthogonality conventions underlying the disk-polynomial basis.","marker":"[37]"}],"fun_headline_variants":["Computer-assisted counterexample to planar Berenstein conjecture","Verified non-disc domain refutes planar Berenstein conjecture","Planar Berenstein conjecture false: computer-verified construction","13-fold symmetric domain disproves Berenstein conjecture","Berenstein conjecture disproved by non-circular domain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the correctness of the authors' own computer verification: the bounds $Y\\le 3.6219\\times 10^{-9}$, $Z\\le 0.4732202748897342$, and the contraction checks were produced by their own interval-arithmetic code and aggregated by their own checker, with no independent formal verification reported.","fun_headline_variants_meta":{"raw":{"variants":["Computer-assisted counterexample to planar Berenstein conjecture","Verified non-disc domain refutes planar Berenstein conjecture","Planar Berenstein conjecture false: computer-verified construction","13-fold symmetric domain disproves Berenstein conjecture","Berenstein conjecture disproved by non-circular domain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000986,"raw_usage":{"total_tokens":4289,"prompt_tokens":1161,"completion_tokens":3128,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":777,"completion_tokens_details":{"reasoning_tokens":3052}},"tokens_in":777,"tokens_out":3128,"duration_ms":28296,"temperature":1.0,"reasoning_tokens":3052,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:20:45.000794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an independent implementation of the certificate checks: if it returns $Y>3.6219\\times10^{-9}$ or $Z>0.4732202748897342$ at the stated centre, or if two reproductions of the directed-rounding runs disagree, the existence theorem fails.","supporting_citations":[{"cited_title":"McLean.Strongly Elliptic Systems and Boundary Integral Equations","cited_arxiv_id":null,"evidence_quote":"provides the single-layer potential jump relations used in the same equivalence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"formulates the unrestricted Berenstein conjecture that Theorem 1.1 disproves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the moving-plane symmetry theorem used to conclude that any counterexample must be sign-changing."},{"cited_title":"Colton and R","cited_arxiv_id":null,"evidence_quote":"supplies the exterior-uniqueness facts behind the Fourier-circle equivalence in Proposition 2.1."},{"cited_title":"Day, J.-P","cited_arxiv_id":null,"evidence_quote":"introduces the radii-polynomial validated continuation criterion adapted in Theorem 4.1."},{"cited_title":"Szegő.Orthogonal Polynomials, volume 23 ofAmerican Mathematical Society Colloquium Publications","cited_arxiv_id":null,"evidence_quote":"defines the Jacobi polynomials and orthogonality conventions underlying the disk-polynomial basis."}],"review_version":1}