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Independent preprints give planar counterexamples to the Schiffer and Pompeiu conjectures

Two independent preprints construct noncircular bounded simply connected planar domains with a Neumann eigenfunction constant on the boundary, contradicting Schiffer’s conjecture and yielding counterexamples to the associated planar Pompeiu conjecture.

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Stable ID
math-schiffer-pompeiu-counterexamples-2026
Revision
math-schiffer-pompeiu-counterexamples-2026.v1
Field
Mathematics · Spectral geometry, partial differential equations and harmonic analysis
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional counterexamples
Last checked

AI role

Colbrook and Stepaniants used ChatGPT 5.5 Codex to debug early certificate code. Cao-Labora and de Dios Pont disclose GPT 5.5/5.6 and Claude models for numerical checks, proof drafts and exposition; GPT 5.6 wrote their Lean 4 verification. The authors state that outputs were checked and accept responsibility.

Record details

Problem or result
The planar Schiffer conjecture and the equivalent rigidity direction of the Pompeiu problem
Authors
Matthew J. Colbrook and George Stepaniants; independently, Gonzalo Cao-Labora and Jaume de Dios Pont
Institutions
University of Cambridge; California Institute of Technology; École Polytechnique Fédérale de Lausanne; New York University
Result date
Preprints submitted August 3 and August 5, 2026

Why it matters

The Pompeiu problem dates to 1929, and Schiffer’s spectral formulation became a prominent rigidity conjecture. The two groups reach the same negative conclusion using different constructions: an exact computer-assisted certificate and an analytic bifurcation family with Lean verification.

Limits

Neither manuscript has been peer reviewed. One argument depends on a rigorous computational certificate and the other on a new bifurcation analysis plus a claimed Lean check; both require specialist scrutiny. The stated counterexamples concern planar simply connected domains.

Sources

  1. Primary: Colbrook and Stepaniants, A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures
  2. Independent preprint: Cao-Labora and de Dios Pont, Counterexamples to Schiffer’s Conjecture
  3. Open data: Validated computational certificate for the Colbrook–Stepaniants construction

Correction and revision history

  1. 2026-08-08 — Added after primary-source review as a provisional claim; the scope, evidence level and material limitations are stated explicitly.

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