{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/","md":"https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-28-schiffer-pompeiu-convex-higher-dimensions","date":"2026-09-28","date_precision":"day","short_title":"Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol","deck":null,"takeaway":"On 28 Sep 2026 Jizhou Guo posted what he says are the first counterexamples to the Schiffer and Pompeiu conjectures in dimension three and higher.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2609.35419: Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen","url":"https://arxiv.org/abs/2609.35419","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: aster2024/schiffer-pompeiu-counterexamples (paper and verification)","url":"https://github.com/aster2024/schiffer-pompeiu-counterexamples","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"GitHub: aster2024/schiffer-pompeiu-r4 (earlier dimension-four version)","url":"https://github.com/aster2024/schiffer-pompeiu-r4","type":"code","group":"primary","domain":"github.com"}],"official":3,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Jizhou Guo"],"title":"Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol","summary":"On 28 Sep 2026 Jizhou Guo posted what he says are the first counterexamples to the Schiffer and Pompeiu conjectures in dimension three and higher. They are also the first convex counterexamples in any dimension: bounded convex non-ball domains in R^3, R^4, R^6, R^8, R^10 and R^14, with computer-assisted existence proofs. The paper says the work 'was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol)'.","key_facts":["arXiv 2609.35419 (87 pages, v1 28 Sep 2026); supersedes the author's Zenodo preprint on dimensions four and six (doi 10.5281/zenodo.22999505)","Dimensions 4, 6, 8, 10 and 14: adjoint-invariant domains in the rank-two compact Lie algebras u(2), so(4), su(3), so(5) and g2. Harish-Chandra's radial-part formula reduces each case to a planar Helmholtz problem, and Kostant's theorem reduces convexity to the Cartan section","Dimension 3: an axisymmetric convex domain, plus a separate strictly star-shaped non-convex example","Existence by Newton–Kantorovich contractions in weighted coefficient spaces: finite blocks checked in Arb ball arithmetic, infinite tails bounded analytically; verification code and certificates are in the arXiv ancillary files and on GitHub","Acknowledgement, verbatim: 'This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol). The author takes full responsibility for the content of this paper.' The paper does not break down which parts each model did","Builds on the August 2026 planar counterexamples of Colbrook–Stepaniants and Cao-Labora–de Dios Pont","Status: single-author preprint, not peer-reviewed"],"key_numbers":[],"tags":["math","pde","spectral-theory","counterexample","computer-assisted-proof","claude-opus-5-5","gpt-6-astra"],"science":{"field":"mathematics","subfield":"spectral theory / overdetermined PDE","problem":"Schiffer and Pompeiu conjectures in dimensions n ≥ 3 and for convex domains","result":"Bounded convex non-ball domains with real-analytic spherical boundary in dimensions 3, 4, 6, 8, 10 and 14 that admit a nonconstant solution of Δu + u = 0 with u = 1 and ∇u = 0 on the boundary, so both conjectures fail there, even for convex domains.","open_since":"1929","ai_system":["Claude Opus 5.5","GPT-6 Astra","GPT-6 Sol"],"human_role":"Single author; AI assistance disclosed without detail on its extent","verification":{"key":"preprint","label":"Preprint only","text":"Computer-assisted proof with public certificates; preprint, not peer-reviewed"},"status":"pending","shock":""},"body_md":"## What happened\n\nEight weeks after the planar disproofs, a single author extended the counterexamples to higher dimensions and to convex domains. He used Lie-theoretic symmetry to reduce each case to a planar problem and certified the solutions with interval arithmetic.\n\n## Why it matters\n\nIf it holds up, the Schiffer and Pompeiu conjectures fail even for convex domains, which was one of the natural fallback forms of the conjectures. It is also an example of a September 2026 paper that names the newest frontier models (Claude Opus 5.5 was released on 22 Sep) without saying what each one did.","disputed":[{"label":"Verification","text":"Computer-assisted proof with public certificates; preprint, not peer-reviewed","sources":[]}],"related":[{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-22-claude-opus-5-5","url":"https://postcutoff.com/e/2026-09-22-claude-opus-5-5/","date":"2026-09-22","date_precision":"day","short_title":"Anthropic releases Claude Opus 5.5","deck":"Fable-5.1-level performance at $4/$20, first model of the Claude 5.5 family","takeaway":"Opus 5.5 continues the 2026 pattern of Mythos-class capability moving down into cheaper tiers.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":36,"official":16,"filed":"2026-09-29","updated":"2026-10-18","orgs":["Anthropic"]},{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-08-03-schiffer-pompeiu-conjectures-disproved","url":"https://postcutoff.com/e/2026-08-03-schiffer-pompeiu-conjectures-disproved/","date":"2026-08-03","date_precision":"day","short_title":"Planar Schiffer and Pompeiu conjectures disproved by two independent groups","deck":"One proof has a Lean certificate written by GPT-5.6","takeaway":"Schiffer's conjecture is a classic rigidity question and appears on Yau's list of open problems.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":7,"official":5,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Matthew Colbrook","George Stepaniants","Gonzalo Cao-Labora","Jaume de Dios Pont"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-09-30","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":151,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":90,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":181,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":120,"in_training_data":false}],"short_url":null}