{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude/","md":"https://postcutoff.com/e/2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude/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":"2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude","date":"2026-10-16","date_precision":"day","short_title":"Equiregular counterexample to the sub-Riemannian Sard conjecture","deck":"Claude Opus 5/5.5 wrote the construction and first proofs, Lean-formalized with Codex (Wang, Zelenko)","takeaway":"Lerario, Rizzi and Tiberio had disproved the sub-Riemannian Sard conjecture with a degenerate distribution, which left the natural equiregular case open. Wang and Zelenko close it with a rank-two distribution in dimension nine. Zelenko proposed the strategy; Claude generated the explicit structure and first proofs; the main theorems are fully formalized in Lean.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.24080: An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps","url":"https://arxiv.org/abs/2610.24080","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Zenodo: Lean 4 formalization (10.5281/zenodo.23310061)","url":"https://doi.org/10.5281/zenodo.23310061","type":"code","group":"primary","domain":"doi.org"}],"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Texas A&M University","Anthropic","OpenAI"],"title":"Equiregular counterexample to the sub-Riemannian Sard conjecture: Claude Opus 5/5.5 wrote the construction and first proofs, Lean-formalized with Codex (Wang, Zelenko)","summary":"arXiv 2610.24080 (16 Oct 2026) by Bangjie Wang and Igor Zelenko constructs an equiregular rank-two distribution on an open set diffeomorphic to R⁹, with growth vector (2, 3, 5, 8, 9), whose endpoint map from the origin has no regular values. This settles the equiregular case left open by Lerario, Rizzi and Tiberio's disproof of the Sard conjecture of Zhitomirskii and Montgomery. Zelenko proposed lifting their example; \"the mathematical implementation of this strategy was generated … in interactive sessions with Anthropic's Claude [Claude Opus 5 and Claude Opus 5.5]\". Theorems 1.1–1.3 are formalized in Lean 4 with Codex, with no unproved statements. Unrefereed.","key_facts":["Paper: 'An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps', Bangjie Wang and Igor Zelenko, arXiv 2610.24080 [math.DG], 16 Oct 2026","Result: equiregular rank-two distribution on an open set ≅ R⁹ with small growth vector (2, 3, 5, 8, 9), 'maximal and generic for rank-two distributions in dimension nine', whose endpoint map has no regular values. The earlier Lerario–Rizzi–Tiberio counterexample (rank two on R⁴) 'is neither equiregular nor real-analytic'","Announced, proofs deferred: an equiregular counterexample in dimension twelve with constant Tanaka symbol, to appear in a separate paper","AI disclosure (follows the Leiden Declaration), verbatim: 'The mathematical implementation of this strategy was generated in September and October 2026 in interactive sessions with Anthropic's Claude [Claude Opus 5 and Claude Opus 5.5], used through the Claude desktop application. This includes the explicit structure (1.2), the first versions of the proofs of Theorems 1.1–1.3, in particular the formulation and solution of the moment problem of Sections 4–6, the statements in Appendices A and B, the explicit dimension-twelve structure announced in Appendix A, and first drafts of most of the text'. ChatGPT (6 Astra) later reviewed drafts against the formalization reports","Human role: the lifting strategy and the Tanaka-symbol and abnormal-dynamics questions came from Zelenko; 'Both authors checked all arguments line by line'; Sections 4–6 replace a single section of the AI draft","Lean: Wang formalized Theorems 1.1–1.3 in Lean 4.30.0 / Mathlib with Codex; 'The development contains no unproved statements and uses only the standard axioms'; Appendices A and B are not formalized; the formalization 'detected two defects in an earlier draft'. Sources: Zenodo 10.5281/zenodo.23310061. The formal statements were written by the authors, who checked that they match the paper"],"key_numbers":[],"tags":["math","differential-geometry","sub-riemannian-geometry","control-theory","sard-conjecture","counterexample","claude-opus-5-5","codex","lean","leiden-declaration"],"science":{"field":"mathematics","subfield":"sub-Riemannian geometry / geometric control","problem":"Sard conjecture for sub-Riemannian endpoint maps (Zhitomirskii–Montgomery) in the equiregular case","result":"Claimed equiregular rank-two counterexample in dimension 9 whose endpoint map has no regular values.","open_since":"","ai_system":["Claude Opus 5","Claude Opus 5.5","Codex","ChatGPT 6 Astra"],"human_role":"AI-assisted: human strategy and line-by-line checking; Claude generated the explicit construction, the moment-problem solution and first proofs; Codex helped with the Lean formalization","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed preprint; complete Lean 4 formalization of Theorems 1.1–1.3 (author-written statements) on Zenodo"},"status":"pending","shock":"","short":"Sub-Riemannian Sard conjecture, equiregular case"},"body_md":"## What happened\n\nThe Sard conjecture in sub-Riemannian geometry asks whether the endpoints of abnormal (singular) horizontal curves form a set of\nmeasure zero. Lerario, Rizzi and Tiberio found a counterexample, but it relied on a degenerate distribution. Zelenko suggested\nlifting their example to higher dimension so that it becomes equiregular. Claude worked out the explicit structure and the hard\nanalytic moment problem. The authors rewrote the analytic core, and Wang formalized the main theorems in Lean with Codex.\n\n## Why it matters\n\nIt shows that the failure of the Sard property is not caused by degenerate growth. It is also one of the most detailed AI\ndisclosures so far, listing which parts came from Claude, which from the humans, and what the formalization caught.","disputed":[{"label":"Verification","text":"Unrefereed preprint; complete Lean 4 formalization of Theorems 1.1–1.3 (author-written statements) on Zenodo","sources":[]}],"related":[{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-06-02-leiden-declaration-ai-mathematics","url":"https://postcutoff.com/e/2026-06-02-leiden-declaration-ai-mathematics/","date":"2026-06-02","date_precision":"day","short_title":"Leiden Declaration on Artificial Intelligence and Mathematics sets community norms for AI in maths (4,000+ signatories)","deck":null,"takeaway":"It is the broadest grassroots statement of mathematicians' norms on AI: disclosure, attribution, integrity of proof, and independence from industry.","category":"policy-safety","category_label":"Policy & safety","importance":3,"confidence":"medium","status":{"key":"partly","labels":["Partly confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Lorentz Center","International Mathematical Union"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (PDF read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"PDF read in full text","updated":"2026-10-20","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":169,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":108,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":199,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":138,"in_training_data":false}],"short_url":null}