Planar Schiffer and Pompeiu conjectures disproved by two independent groups; one proof has a Lean certificate written by GPT-5.6

2026-08-03★★★★after cutoffscienceMatthew ColbrookGeorge StepaniantsGonzalo Cao-LaboraJaume de Dios Pontconfidence: high

In August 2026 two groups independently disproved Schiffer's conjecture (Yau's Problem 80) and the planar Pompeiu problem (1929). Colbrook and Stepaniants (3 Aug) built one explicit ten-fold symmetric domain with a computer-assisted proof, using ChatGPT/Codex only to debug code. Cao-Labora and de Dios Pont (5 Aug) built infinitely many counterexamples by bifurcation theory. They used GPT-5.5/5.6, Claude Opus 4.8 and Claude Fable 5 for estimates and drafts, and GPT-5.6 wrote their Lean 4 certificate.

Key facts

Schiffer's conjecture: if a smooth domain has a Neumann eigenfunction that is constant on the boundary, the domain is a ball (Problem 80 in Yau's 1982 problem list). The planar Pompeiu problem dates to Pompeiu's papers of 1929
Colbrook & Stepaniants (arXiv 2608.01579, 3 Aug 2026): one bounded, simply connected, non-circular domain with real-analytic boundary, close to an explicit degree-301 polynomial conformal map with ten-fold symmetry; eigenvalue k in (31.967007261, 31.967007293); existence certified by an interval-arithmetic contraction argument
Colbrook & Stepaniants AI declaration: 'ChatGPT 5.5 in the form of codex was used to help debug an early version of some of the code. All of its outputs were checked line by line by the authors.'
Cao-Labora & de Dios Pont (arXiv 2608.05114, 5 Aug 2026): infinitely many N-fold symmetric counterexamples for large N. They relax N to a real parameter and use bifurcation theory with branch size uniform in N. No computer assistance is needed in the proof
Cao-Labora & de Dios Pont AI usage: 'LLMs, in the form of GPT 5.5 and 5.6, Claude Opus 4.8 and Claude Fable 5' were used to numerically verify asymptotic estimates, draft first versions of the Bessel-function estimate proofs and help with exposition; 'The Lean4 verification of the proof was written by GPT 5.6'
Lean repository jaumededios/Schiffer (created 4 Aug 2026) contains challenge files for Schiffer's conjecture and for the Pompeiu statement in Google DeepMind's Formal Conjectures repository. It follows a slightly different route from the paper because Mathlib lacks elliptic regularity
Follow-up (arXiv 2608.08953, 9 Aug): Colbrook, Sadeghi and Stepaniants disproved the unrestricted planar Berenstein conjecture (the Dirichlet counterpart). Their AI declaration says ChatGPT 5.6 'suggested ideas contributing to early versions of Lemma 3.2 and Proposition 3.5, as well as to a revised version of the code', starting from a warm start of the authors' own draft and code
Colbrook presented the result as 'A Shortcake Counterexample to the Planar Pompeiu and Schiffer Conjectures' at Princeton (PACM seminar, 16 Sep 2026)
Status: preprints, not yet peer-reviewed; the Cao-Labora–de Dios Pont proof is Lean-checked

Science result

Field: mathematics / spectral theory / overdetermined PDE / integral geometry
Problem: Schiffer's conjecture (Yau Problem 80) and the planar Pompeiu problem (open since 1929)
Result: Non-circular bounded planar domains carrying a nonconstant Neumann eigenfunction that is constant on the boundary, which disproves both Schiffer's conjecture and the Pompeiu conjecture in the plane (one explicit domain, and an infinite family).
AI system: GPT-5.6, GPT-5.5, Claude Opus 4.8, Claude Fable 5, ChatGPT/Codex
Human role: Human-led. Colbrook–Stepaniants: AI only helped debug code. Cao-Labora–de Dios Pont: humans devised the bifurcation strategy; LLMs checked estimates numerically and drafted some lemma proofs, and GPT-5.6 wrote the Lean formalisation
Verification: Computer-assisted interval certificate (Colbrook–Stepaniants); Lean 4 formalisation (Cao-Labora–de Dios Pont); not yet peer-reviewed
Status: pending
Why surprising: Two independent disproofs of a well-known rigidity conjecture appeared on arXiv two days apart.

What happened

On 3 August 2026 Matthew Colbrook and George Stepaniants posted a counterexample to the planar Pompeiu and Schiffer conjectures. They found a domain numerically, then proved that an exact counterexample exists near it with an interval-arithmetic contraction argument. Two days later Gonzalo Cao-Labora and Jaume de Dios Pont posted an independent construction of infinitely many counterexamples. Their method is bifurcation theory with a real-valued symmetry parameter. It is purely analytic, and their Lean certificate was written by GPT-5.6. Cao-Labora and de Dios Pont note in their paper that Colbrook and Stepaniants posted an independent construction two days earlier. On 9 August Colbrook's group used the same machinery to disprove the planar Berenstein conjecture, this time with ChatGPT 5.6 contributing ideas to some lemmas.

Why it matters

Schiffer's conjecture is a classic rigidity question and appears on Yau's list of open problems. The episode shows the range of AI roles in mid-2026 papers on the same problem. In one paper it only debugged code. In another it checked estimates, drafted proofs and wrote the full formal verification. In a third it proposed ideas that the authors then developed.

Changelog

2026-09-30: created

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Sources (7)

id: 2026-08-03-schiffer-pompeiu-conjectures-disproved · updated 2026-09-30 · open in the interactive timeline