{"id":"b3c1cfa1-9a21-49e8-a1f5-2f300445cfdf","arxiv_id":"2608.05114","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinitely many non-ball planar domains carry Neumann eigenfunctions that are constant on the boundary, so Schiffer's and Pompeiu's conjectures are false in R^2.","lead":"Authors construct infinitely many smooth, star-shaped, non-circular planar domains that admit a Neumann eigenfunction constant on the boundary, disproving Schiffer's conjecture in the plane. The construction also gives a negative answer to Pompeiu's problem through a classical reduction.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.2's Debye prefactor is inverted, making the stated asymptotic false; Proposition 4.5 relies on it, so Section 7 needs correction.","rationale":"The reader's weakest_assumption identified the uniform Debye-type Bessel asymptotics in Section 7 as the load-bearing layer, and my concern is a concrete prefactor error within exactly that layer, namely Lemma 5.2 and Eq. (7.15). I therefore partially agree with the reader's identification. However, my concern is more specific than the reader's: it is not merely that the estimates are imported without full proof, but that one of them is algebraically wrong as stated. The error is in the normalization of the asymptotic expansion for non-integer-order Bessel functions. The qualitative behavior needed by Proposition 4.5 -- that zeros of J_{lambda_0^{-1/2} rho + t}(rho) stay within O(rho^{-1}) of zeros of a cosine -- is preserved under the natural correction of the prefactor, so the overall architecture of the proof remains plausible and the correct response is to condition acceptance on fixing this lemma and rechecking the downstream density argument. I do not find an internal contradiction that would force rejection, and the rest of the construction, including the uniform bifurcation theorem and the R''(0) computation, appears coherent. Thus the reader's CONDITIONAL verdict stands unchanged, with the condition sharpened to include correction of Lemma 5.2.","tokens_in":42929,"tokens_out":18770,"duration_ms":205150,"concrete_test":"Use mpmath or scipy to evaluate L(rho) = (pi/(2 rho sqrt(1 - lambda_0^{-1})))^{1/2} J_{lambda_0^{-1/2} rho}(rho) for lambda_0 = 2 and rho = 10^3, 10^4, 10^5. If L(rho) decays like 1/rho rather than approaching 1, Lemma 5.2 is false as stated. Then verify the corrected prefactor (pi rho sqrt(1 - lambda_0^{-1})/2)^{1/2} gives a bounded oscillatory function, confirming that the fix rescues the density argument in Proposition 4.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing flaw I find is an internal prefactor inconsistency in Lemma 5.2 (Section 5.4) and its proof's Eq. (7.15). Debye asymptotic (7.3) gives J_nu(nu sec beta) = (2/(pi nu tan beta))^{1/2}(cos(nu(tan beta - beta) - pi/4) + O(nu^{-1})). Here nu = lambda_0^{-1/2} rho + t, and nu tan beta ~ rho sqrt(1 - lambda_0^{-1}), so the correct normalization making the cosine term O(1) is (pi rho sqrt(1 - lambda_0^{-1})/2)^{1/2}. Lemma 5.2 instead multiplies by (pi/(2 rho sqrt(1 - lambda_0^{-1})))^{1/2}, which is the reciprocal up to a factor of pi/2. Consistently, Eq. (7.15) claims the product of the two square-root prefactors is 1 + O(rho^{-1}), but the actual product is rho^{-1} (1 - lambda_0^{-1})^{-1/2} (1 + o(1)). Thus Lemma 5.2 is false as stated: its left-hand side decays like 1/rho rather than tending to cos(...). Since Proposition 4.5 uses Lemma 5.2 to place bifurcation crossings R* within epsilon of an integer, this is load-bearing. The likely fix is to replace the prefactor by (pi rho sqrt(1 - lambda_0^{-1})/2)^{1/2}; with that correction the density argument still works because zeros of J_nu(rho) remain within O(rho^{-1}) of zeros of the cosine, which is enough for the epsilon-dense argument. Nevertheless, the manuscript as written contains a false asymptotic lemma, so the Section 7 estimates must be corrected before the proof is sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a counterexample to Schiffer's conjecture in R^2: for infinitely many integers N >= 2 there exist non-ball domains Omega, with N-fold symmetry, simply connected, bounded, star-shaped, with C^infty boundary, admitting a nontrivial solution u of (1.1) with u=1 and normal derivative 0 on the boundary. The proof reformulates the Schiffer condition on a collar through a functional G(R,v), extends the symmetry order from an integer N to a real parameter R, proves a quantitative Crandall-Rabinowitz theorem whose branch size is uniform over a family of 'crossings' (R*,rho*,lambda*), shows that crossings can be found arbitrarily close to integers via Debye asymptotics of Bessel functions, and proves that the bifurcation branch bends downward by a uniform amount so that it reaches an integer radius. Corollary 1.2 converts this into a counterexample to Pompeiu's problem.","tokens_in":43295,"tokens_out":12573,"duration_ms":137433,"significance":"If correct, this settles two long-standing problems in the plane and introduces a genuinely new technique: relaxing the integer symmetry order to a real parameter and proving a uniform quantitative bifurcation theorem for the resulting functional. The paper also states that the main theorem and Corollary 1.2 have been formalized in Lean, which is a strong reproducibility feature if the formalization is audited. The main unresolved technical risk is the uniformity of the Bessel-asymptotic estimates, and in particular one displayed Debye prefactor is currently wrong; this must be corrected before the proof is sound.","major_comments":[{"comment":"Lemma 5.2 is false as stated. The Debye expansion (7.3) gives J_nu(nu sec beta) = (2/(pi nu tan beta))^{1/2}(cos(xi)+O(nu^{-1})), and with nu = lambda_0^{-1/2} rho + t, nu tan beta = sqrt(rho^2 - nu^2) = rho sqrt(1 - lambda_0^{-1}) + O(1). Hence the prefactor that makes the leading cosine term O(1) is (2/(pi rho sqrt(1-lambda_0^{-1})))^{1/2}, not (pi/(2 rho sqrt(1-lambda_0^{-1})))^{1/2}. The prefactor in Lemma 5.2 is the reciprocal of the correct one up to a factor of pi/2, so the left-hand side of (5.22) decays like rho^{-1/2} instead of converging to cos(omega_0 rho - theta_0 t - pi/4). Correspondingly, Eq. (7.15) claims that the product of the two square-root prefactors is 1+O(rho^{-1}), but the actual product is of order rho^{-1}. Since Proposition 4.5 invokes Lemma 5.2 to place crossings R* arbitrarily close to integers, this is a load-bearing error. With the corrected prefactor the zero-location argument still works, because zeros of J_nu(rho) remain within O(rho^{-1}) of zeros of the cosine, which suffices for the epsilon-density argument; nevertheless Section 7 must be revised before the proof is sound.","section":"§5.4, Eq. (5.22) and Eq. (7.15)"},{"comment":"The uniform Bessel-asymptotic layer is the most delicate part of the argument and is currently imported from DLMF and from Dunster's paper rather than proved in full. In particular, Proposition 4.5 requires uniformity of the Debye estimates over the crossing family C(R0), and Proposition 4.6 requires the uniform version of the order-derivative of zeros in Lemma 7.1. The prefactor error in Lemma 5.2 shows that these imported estimates need careful checking. The authors should either prove the needed uniform error bounds or quote them with precise statements, explicit non-asymptotic ranges, and verified constants, and should check that the corrected prefactors do not alter any constants used in Section 4.","section":"§7, Lemmas 5.2 and 7.1, Propositions 6.1 and 6.2"}],"minor_comments":[{"comment":"There is a typo: 'different from a a ball' should read 'different from a ball.'","section":"Theorem 1.1"},{"comment":"There are minor typographical errors: 'Driichlet' should be 'Dirichlet' and 'deffect' should be 'defect.'","section":"§5.5 and §6"},{"comment":"The square-root notation is easy to misread: the prefactor in (7.18) is sqrt(2 pi R sqrt(4-lambda)), not sqrt(2 pi R) times sqrt(4-lambda). The text should be typeset unambiguously, since the dimensionally different reading changes the subsequent ratio computation.","section":"Proposition 7.2, Eqs. (7.18)-(7.19)"},{"comment":"The sentence 'so that rho = nu sec beta' after defining beta = arccos(nu/rho) is correct but can be misread as an additional assumption; it would be clearer to write cos beta = nu/rho and hence rho = nu sec beta.","section":"Proof of Lemma 5.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick summary: if the proof holds, this resolves Schiffer and Pompeiu in the plane with infinitely many smooth simply connected counterexamples, and the real-N bifurcation framework is a genuinely new idea. I think the core strategy is right, but the stress-test criticism is correct: Lemma 5.2 has the Debye prefactor inverted, so the manuscript currently contains a false asymptotic statement in a load-bearing spot.\n\nWhat is new and good. The relaxed problem with real symmetry order R, the quantitative Crandall-Rabinowitz theorem with branch length uniform over the crossing family, and the equidistribution argument for crossings near integers are all substantial. The functional setup — anisotropic spaces, Dirichlet-to-Neumann collar, Fredholm index zero, explicit kernel and range, pitchfork behavior and the formula for R''(0) — is carefully assembled and internally consistent. I did not find a circular step: crossings are characterized by the kernel, not assumed. This is a different construction from Colbrook-Stepaniants' computer-assisted 10-fold example and is analytic and infinite.\n\nSoft spots, in proportion. First, Lemma 5.2 as stated is false: with their prefactor (π/(2ρ√(1-λ0^{-1})))^{1/2} the left side tends to 0 like 1/ρ, not to cos(...), and Eq. (7.15) miscomputes the product of prefactors. This matters because Proposition 4.5 uses it to get crossings arbitrarily close to integers. The fix is clear — use (πρ√(...)/2)^{1/2} — and the zero-location conclusion still follows from the correct Debye form, so I view this as a serious but repairable error rather than a fatal one. Second, the uniform Bessel estimates in Section 7 are mostly imported from DLMF/Olver/Dunster; the uniformity over the crossing family is the real content and needs referee checking. The Riccati/Volterra part for k≥2 is sketched but plausible. Third, the claimed Lean4 certification is not reproducible from the text: no commit or build details are given, so I treat it as a claim, not evidence. If the artifact checks out, that would substantially raise confidence.\n\nWho it is for. Spectral geometers, people working on overdetermined boundary problems, Pompeiu-type questions. It deserves serious peer review. My recommendation: engage with it, send to a careful referee, and return it to the authors with a request to correct the prefactor and make the Bessel asymptotics and Lean artifact verifiable.","headline":"A likely major result with a repairable but real asymptotic error in Lemma 5.2; worth sending to a serious referee.","tokens_in":43835,"tokens_out":4841,"would_cite":true,"duration_ms":54387,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35N25","35B32","35J05","35R35","33C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs infinitely many smooth planar domains, none of them balls, that admit a Neumann eigenfunction constant on the boundary, disproving the Schiffer conjecture in the plane.","keywords":["Schiffer conjecture","Pompeiu problem","overdetermined boundary value problem","bifurcation theory","Bessel functions","Debye asymptotics","Dirichlet-to-Neumann map","anisotropic spaces"],"falsifier":"Compute, for a sequence of crossings with $R_0\\to\\infty$, the quantities in Proposition 6.2 and the difference in (4.4): if the claimed uniform error rates fail for any crossing family with $R_*$ within $10^{-6}$ of an integer, the branch may not reach the integer and the construction collapses. Alternatively, numerically integrate the branch equations for the $N=28$ example and check whether $R(s)$ actually passes through the integer value 28 before the uniform $s$-bound is exceeded.","tokens_in":42703,"feed_emoji":"📐","tokens_out":7696,"duration_ms":89541,"temperature":0.7,"pith_summary":"This paper claims that the Schiffer conjecture is false in the plane: there exist infinitely many smooth, bounded, simply connected, star-shaped planar domains, none of them balls, on which the Laplacian has a Neumann eigenfunction that is constant on the boundary. Because of the classical Williams equivalence, the same construction gives a negative answer to Pompeiu's problem in the plane, with the explicit test function $f(x)=e^{i\\sqrt{\\lambda}x_1}$. The strategy is to replace the discrete symmetry order $N$ by a real parameter $R$ and study a relaxed problem in which Bessel functions of non-integer order are allowed; bifurcation branches from such crossings are shown to have size independent of $R$, and a density result puts crossings arbitrarily close to integers. A uniform downward bend of the branch then forces it to hit an integer $R=N$, yielding genuine $N$-fold symmetric counterexamples. The authors note that their infinite family needs no computer assistance, in contrast to a concurrent computer-assisted construction with 10-fold symmetry.","feed_headline":"Infinite family of non-circular domains disproves Schiffer conjecture","feed_subtitle":"A bifurcation branch on a real-symmetry relaxation reaches integer symmetries, refuting Schiffer and Pompeiu together.","key_machinery":"The load-bearing object is the relaxed functional $G(R,v):X\\to Y$ defined on anisotropic Hölder spaces on a collar, whose zero set contains the trivial ball solutions $G(R,0)=0$ and whose integer-$R$ zeros unfold into $N$-fold Schiffer domains. Its linearization at a crossing $R_*$ is the Bessel-mode operator $L_R$, with kernel spanned by $\\Phi_*(y)\\cos\\psi$, where $\\Phi_*$ is built from the noninteger-order Bessel function $J_{R_*}(\\sqrt{\\lambda_*}(R_*-y))$; crossings are triples $(\\rho,R,\\lambda)$ with $J_1(\\rho)=J_R(\\rho)=0$ and $\\lambda=\\rho^2/R^2\\in[2,3]$. The proof then runs on three uniform ingredients: a quantitative Crandall–Rabinowitz theorem (Proposition 3.5) that gives a branch of size independent of the crossing; Debye-type asymptotics (Lemma 5.2 and Propositions 6.1–6.2) showing that crossings accumulate near integers and that the non-resonance and symbol estimates hold uniformly; and the second-order computation $R''(0)=-\\frac{\\sqrt{\\lambda}}{4R\\,\\partial_\\nu j_{\\nu,m}}\\left(2+\\rho\\,J'_{2R}(\\rho)/J_{2R}(\\rho)\\right)$, which is asymptotically a negative universal constant depending only on $\\lambda$, forcing the branch to bend downward by a uniform amount.","core_discovery":"The central claim is Theorem 1.1: for infinitely many integers $N\\ge 2$ there are $\\lambda>0$ and a domain $\\Omega\\subset\\mathbb{R}^2$, different from a ball, with smooth boundary, simply connected, bounded and star-shaped, such that $\\Delta u+\\lambda u=0$ in $\\Omega$, $u=1$ and $\\partial_\\nu u=0$ on $\\partial\\Omega$, and $\\Omega$ is $N$-fold symmetric, with $u$ smooth up to the boundary. The construction works by considering a collar reformulation of the problem for real symmetry $R$, with a Dirichlet-to-Neumann map encoding the interior, and proves that the functional $G(R,v)$ has nontrivial zeros when $R$ is an integer. The authors show that the linearized operator at a crossing, where $J_1(\\rho)=0$, $J_R(\\rho)=0$ and $\\lambda=\\rho^2/R^2\\in[2,3]$, has a one-dimensional kernel, obtain a quantitative bifurcation branch whose length is uniform over all sufficiently large crossings, prove that crossings exist arbitrarily close to integers, and compute that the branch starts flat, $R'(0)=0$, and bends downward with $R''(0)$ uniformly bounded away from zero. Taylor's theorem then places the branch below the integer floor, so by continuity the branch value $R(s)$ takes an integer value, and Theorem 2.7 converts that zero into an actual Schiffer domain.","pith_inferences":["A natural testable extension is to check whether every sufficiently large integer $N$ admits such a counterexample, not merely infinitely many; the density of crossings alone does not guarantee that every large $N$ is reached, and the uniform bend may fail for some residue classes.","The same collar-plus-Dirichlet-to-Neumann strategy could be adapted to other symmetry groups, such as dihedral or higher-dimensional rotational symmetries, where real-order Bessel functions would again remove the integer-order obstruction.","The uniform bifurcation theorem may be of independent use in free-boundary problems: it quantifies how large a branch must be to cross a discrete parameter value, which is exactly the information needed to turn non-integer formal problems into genuine solutions.","Because the domains have high symmetry and explicit Bessel data, one could compute the first few examples, such as the $N=28$ case shown in the paper, to high precision as a numerical check of the asymptotic constants in (4.4)."],"forward_implications":["If Theorem 1.1 is correct, the Schiffer conjecture is false in the plane, settling a problem from Yau's list in the negative.","Pompeiu's problem is also answered negatively in the plane: with $f(x)=e^{i\\sqrt{\\lambda}x_1}$, every rigid motion of the constructed domain integrates $f$ to zero.","The construction provides an infinite sequence of symmetry orders $N$, all giving smooth, bounded, star-shaped, simply connected domains, so the counterexamples are not isolated.","The relaxed-real-$R$ bifurcation framework transfers to other overdetermined elliptic problems in which classical perturbations are blocked by the absence of common zeros of integer-order Bessel functions.","The paper's main theorem and the Pompeiu corollary are accompanied by a Lean certification, so the logical steps are machine-checked."],"supporting_citations":[{"why":"Supplies the classical bifurcation-from-simple-eigenvalue theorem that the paper quantifies into a uniform version.","marker":"[11]"},{"why":"Introduces the anisotropic spaces and bifurcation framework for Schiffer-type problems that the paper adapts to the collar and Dirichlet-to-Neumann setting.","marker":"[20]"},{"why":"Provides the Debye-type asymptotics for Bessel functions of noninteger order that underlie the crossing and symbol estimates.","marker":"[30]"},{"why":"Gives the uniform large-order asymptotics for derivatives of Bessel functions with respect to order, used for the zero-derivative estimate.","marker":"[16]"},{"why":"Records the classical fact that distinct integer-order Bessel functions have no common nontrivial zeros, motivating the relaxation to real order.","marker":"[33]"},{"why":"Provides the uniform oscillatory Debye expansions for Bessel functions of large order used in the symbol and density estimates.","marker":"[31]"},{"why":"Supplies the Schauder estimates used to show the harmonic collar operator is a uniform isomorphism.","marker":"[22]"},{"why":"Gives the equivalence between the Schiffer and Pompeiu properties in simply connected planar domains, used to derive the Pompeiu counterexample.","marker":"[38]"}],"fun_headline_variants":["Non-circular domains break Schiffer conjecture","Counterexamples found to Schiffer's conjecture","Planar domains disprove Schiffer and Pompeiu","Bifurcation yields non-ball Schiffer domains","Infinite counterexamples to Schiffer conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof rests on the assumption that certain asymptotic formulas for Bessel functions of large real order hold uniformly over every allowed crossing family, with errors that vanish as the radii grow; if the uniformity fails, a branch might start near an integer but curve back before hitting it.","fun_headline_variants_meta":{"raw":{"variants":["Non-circular domains break Schiffer conjecture","Counterexamples found to Schiffer's conjecture","Planar domains disprove Schiffer and Pompeiu","Bifurcation yields non-ball Schiffer domains","Infinite counterexamples to Schiffer conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1340,"prompt_tokens":1048,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":222}},"tokens_in":664,"tokens_out":292,"duration_ms":3616,"temperature":1.0,"reasoning_tokens":222,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:59:45.015429+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a sequence of crossings with $R_0\\to\\infty$, the quantities in Proposition 6.2 and the difference in (4.4): if the claimed uniform error rates fail for any crossing family with $R_*$ within $10^{-6}$ of an integer, the branch may not reach the integer and the construction collapses. Alternatively, numerically integrate the branch equations for the $N=28$ example and check whether $R(s)$ actually passes through the integer value 28 before the uniform $s$-bound is exceeded.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical bifurcation-from-simple-eigenvalue theorem that the paper quantifies into a uniform version."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the anisotropic spaces and bifurcation framework for Schiffer-type problems that the paper adapts to the collar and Dirichlet-to-Neumann setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Debye-type asymptotics for Bessel functions of noninteger order that underlie the crossing and symbol estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the uniform large-order asymptotics for derivatives of Bessel functions with respect to order, used for the zero-derivative estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Records the classical fact that distinct integer-order Bessel functions have no common nontrivial zeros, motivating the relaxation to real order."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the uniform oscillatory Debye expansions for Bessel functions of large order used in the symbol and density estimates."},{"cited_title":"Gilbarg and N","cited_arxiv_id":null,"evidence_quote":"Supplies the Schauder estimates used to show the harmonic collar operator is a uniform isomorphism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the equivalence between the Schiffer and Pompeiu properties in simply connected planar domains, used to derive the Pompeiu counterexample."}],"review_version":1}