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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

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2026-08-05 00:43 UTC pith:UR6PXP65

load-bearing objection 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. the 1 major comments →

arxiv 2608.01579 v1 pith:UR6PXP65 submitted 2026-08-03 math.AP math-phmath.MPmath.SP

A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

Matthew J. Colbrook , George Stepaniants This is my paper
classification math.AP math-phmath.MPmath.SP MSC 35J0535N2535P0542B1047J0565G20
keywords Pompeiu problemSchiffer conjectureoverdetermined Neumann eigenvalue problemconformal mappingdisk polynomialscomputer-assisted proofinterval arithmeticFourier zero set
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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 whose value and normal derivative both vanish 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.

Core claim

The central claim is that for a specific ten-fold symmetric domain Omega = phi(D), where phi 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 the normal derivative of u equal to zero 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 phi', 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 mono

What carries the argument

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 l1 sums; the same positivity makes the infinite tails monotone, permitting finite enumeration plus rigorous 'everything beyond is smaller

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

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.

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 (1)
  1. [Appendix A.1/A.5] 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.
minor comments (4)
  1. [§2.2, after Eq. (11)] 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.
  2. [§3.5.3, Eqs. (55)–(58)] 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).
  3. [Table 1] 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.
  4. [Figure 1 caption] 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.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained and the a posteriori contraction proof does not reduce to its inputs.

full rationale

The paper's central claim is an existence theorem proved by an a posteriori Newton–Kantorovich contraction argument. The numerical centre x∘ and frozen approximate inverse R are computational inputs; they are not fitted from the target solution, and the proof verifies that an exact zero of F(g,p)=g+|p|^2(1+Kg) lies within a small ball around that centre. The reduction from the PDE to the cubic coefficient equation is a genuine equivalence proved in Lemmas 2.4 and 2.5 via an explicit inverse K, whose formula is taken from Arioli–Koch [3] and independently verified in the paper; this is not a self-citation and does not assume the conclusion. The positive linearisation lemma is cited from Koornwinder [25,26] and used as an external algebraic fact. The radii-polynomial constants are computed from interval arithmetic and explicit tail bounds, not from the existence of the desired solution. The Pompeiu and Schiffer consequences follow from the constructed eigenfunction via Lemmas 1.2 and 1.3 and standard trace/analyticity arguments, none of which use the conjectures. The only genuinely load-bearing assumption is the correctness and reproducibility of the shipped computational certificate (2471×2471 inverse, MPFR enclosures, exact-rational checker). That is an implementation-verification risk, not a circularity risk, and the paper mitigates it with byte-identical audits and a standalone verifier. No derivation step is equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The construction uses no new physical entities and no fitted physical constants. The ledger entries are computational proof parameters and standard external mathematical facts. The center and inverse are inputs to the contraction argument, not outputs of the theorem, so they carry no circularity burden.

free parameters (5)
  • Weight exponent rho in coefficient norm = 1.05
    Hand-selected to make the weighted coefficient algebra and tail estimates work; the proof constants depend on it, but it is not fitted to any observed data.
  • Truncation parameters (L, S, R_p, J) = (60, 40, 30, 30)
    Finite block cutoffs chosen so that the radii polynomial is negative at the candidate radius. They are computational choices, not conclusions.
  • Candidate radius r = 10^-6
    Radius at which the radii inequalities are evaluated; chosen to clear the residual and defect margins.
  • Numerical centre coefficients (g^o, p^o) = 2471 dyadic coefficients in data/center_L60_S40_R30.hex
    Approximate zero from exploratory continuation; used as the contraction centre. It is an input to the proof, not an output of the theorem.
  • Frozen approximate inverse R = 2471 x 2471 binary64 dyadic entries
    Computed numerically and then frozen; treated as exact dyadic rationals. The proof needs the norm bound (45), not any particular form of R.
axioms (6)
  • standard math Koornwinder positive linearisation and the Arioli-Koch Zernike inverse recurrence (Lemma 2.2, Lemma 2.4).
    These external results give the coefficient algebra norm constant one and the inverse bound ||K|| = 1/8, which are used throughout the validation.
  • standard math Sobolev trace theory and Green's formula characterize the compatible Laplacian range (Lemma 2.5).
    The equivalence between zero Dirichlet and Neumann traces and the coefficient range G_rho is essential for representing the unknown as v = Kg.
  • domain assumption Conformal covariance transfers the free boundary problem to the fixed unit disc (Proposition 2.1).
    The Helmholtz equation on the unknown domain is pulled back to an equation with coefficient |p|^2 on the disc; this change of variables is assumed to preserve the Sobolev and trace structure needed.
  • standard math Cauchy-Kowalevski and Morrey-Nirenberg analytic elliptic regularity extend U across the boundary (Section 4.3).
    Used to promote the Sobolev solution to a real-analytic function on a neighbourhood of the closed disc, giving the real-analytic boundary conclusion.
  • domain assumption Directed MPFR interval arithmetic and IEEE-754 FMA error bounds are correctly implemented (Appendix A.1-A.2).
    The numerical enclosures Y, Z, and the matrix norm bound rely on correct directed rounding and on the stated floating-point error analysis.
  • domain assumption The supplied verifier and reproduction script correctly reconstruct the theorem bounds from the interval enclosures (Appendix A.5).
    The leaf-to-theorem Python checker performs exact rational arithmetic; the central claim depends on this code being free of implementation errors.

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Cite this review

Pith. "Pith review of A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures." pith.science (2026). https://pith.science/paper/UR6PXP65

@misc{pith2026260801579,
  author       = {Pith},
  title        = {Pith review of: A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UR6PXP65}},
  note         = {Machine review of arXiv:2608.01579}
}
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read the original abstract

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.

Figures

Figures reproduced from arXiv: 2608.01579 by George Stepaniants, Matthew J. Colbrook.

Figure 1
Figure 1. Figure 1: Boundary of the finite conformal approximation used as the numerical centre. The [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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