{"schema":"postcutoff/event@1","as_of":"2026-10-21T14:07:00+02:00","url":"https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/","md":"https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/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":"2026-10-20-double-cap-conjecture-witsenhausen-lean-ai","date":"2026-10-20","date_precision":"day","short_title":"Double-cap conjecture claimed in all dimensions, with a separate AI-built Lean proof","deck":null,"takeaway":null,"category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.27523: An orthogonality inequality in RP^(n−1) and the double-cap conjecture","url":"https://arxiv.org/abs/2610.27523","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: allenibiapina/LeanShadow (Lean 4 proof)","url":"https://github.com/allenibiapina/LeanShadow","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-21","updated":"2026-10-21","orgs":["Universidade Federal do Ceará","Universidade de São Paulo","OpenAI"],"title":"Double-cap conjecture (Witsenhausen 1974, Kalai) claimed in all dimensions, with a separate AI-built Lean proof","summary":"Ibiapina, Gomes, Pio, Belo and Ferreira (Brazil) claim that caps minimize the orthogonal neighbourhood in RP^(n−1), which yields the double-cap conjecture: for n ≥ 3, a measurable subset of the sphere S^(n−1) with no orthogonal pair has measure at most that of two opposite caps of radius π/4. This answers Witsenhausen's 1974 problem. AI \"played an assistive role\" in the paper, and a separate Lean 4 proof by a different argument was built \"almost entirely using AI-assisted tools\" (Codex agent per its metadata).","key_facts":["Claim (arXiv 2610.27523): 'a measurable subset of S^(n−1) containing no orthogonal pair has measure at most that of two opposite open caps of radius π/4', for n ≥ 3, with rigidity up to null sets; also the measurable fractional chromatic number of the spherical orthogonality graph","History: Witsenhausen (Amer. Math. Monthly 1974) proved α_n ≤ 1/n; the double-cap conjecture α_n = δ_n is attributed to Kalai; the best earlier bound on S² was α_3 < 0.313","AI statement (abstract): 'AI played an assistive role in preparing this paper. The formalization and verification workflow for a separate Lean 4 proof of the main inequality and double-cap result, based on a different argument, were carried out almost entirely using AI-assisted tools.'","Paper's caveat: the Lean development 'does not constitute a verification of the proof or the additional applications presented here'","LeanShadow repo: 138 modules, sorry-free, standard axioms; Comparator matched 12 main results (incl. double_cap_conjecture, s2_double_cap_bound 1 − √2/2) against a reference challenge; Lean and nanoda kernels; formalization.yaml lists automation 'agent, framework: Codex' for packaging, adapters and tooling; review status 'self-assessed'"],"key_numbers":[],"tags":["math","geometry","isoperimetric","double-cap","witsenhausen","kalai","lean","codex"],"science":{"field":"mathematics","subfield":"spherical geometry / geometric measure theory","problem":"Double-cap conjecture / Witsenhausen's problem on spherical sets without orthogonal pairs","result":"For n ≥ 3, sets in S^(n−1) with no orthogonal pair have measure at most that of two opposite caps of radius π/4, with equality only for double caps (claimed)","open_since":"1974","ai_system":["Codex","AI-assisted tools (unspecified)"],"human_role":"Human-led paper with an assistive AI role; the separate Lean proof was produced almost entirely with AI-assisted tools","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; separate Lean 4 proof (different argument) with Comparator and two kernels, statements written by the authors, self-assessed"},"status":"pending","shock":""},"body_md":"## What happened\n\nThe double-cap conjecture asks for the largest measurable set on a sphere that contains no two orthogonal points. Witsenhausen\nposed the problem in 1974, and Gil Kalai's survey on Borsuk's problem made it widely known. The conjectured answer is two\nopposite caps of angular radius π/4.\n\nOn 20 Oct 2026 five mathematicians from Fortaleza and São Paulo posted a proof for all n ≥ 3. It runs through a sharp\n\"orthogonal neighbourhood\" inequality in real projective space, proved with the Funk transform, rotation comparisons and\nsecond-variation arguments. A separate Lean 4 development (LeanShadow, public since 11 Sept 2026) proves the same main inequality\nand the double-cap result by a different argument. The authors say they built it \"almost entirely using AI-assisted tools\", and\nits metadata names a Codex agent.\n\nThe paper is unrefereed, and the Lean proof was self-assessed. No expert\nreaction was found on 21 Oct.\n\n## Why it matters\n\nIf confirmed, a 52-year-old geometry problem falls in every dimension, and a machine-checked proof of the statement already\nexists. Experts can check that Lean statement against the classical formulation faster than a long analytic paper. This is\nanother case of AI tools mainly turning a human result into a formally verified one.","disputed":[{"label":"Verification","text":"Unrefereed preprint; separate Lean 4 proof (different argument) with Comparator and two kernels, statements written by the authors, self-assessed","sources":[]}],"related":[{"id":"2026-10-21-ai-assisted-results-october-21-listing","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"A ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-21","type":"filed","text":"Created (PDF, LeanShadow README and formalization.yaml read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-21","run":null,"sources_read":"PDF, LeanShadow README and formalization.yaml read","updated":"2026-10-21","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":173,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":112,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":203,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":142,"in_training_data":false}],"short_url":"https://postcutoff.com/s/double-cap-conjecture-all"}