{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-falconer-distance-conjecture-lean/","md":"https://postcutoff.com/e/2026-10-06-openai-falconer-distance-conjecture-lean/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":"2026-10-06-openai-falconer-distance-conjecture-lean","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the Falconer distance conjecture (Lean)","deck":null,"takeaway":"Family 073 of OpenAI's 6 Oct 2026 release proves that every compact set in ℝᵈ (d ≥ 2) of Hausdorff dimension greater than d/2 has a distance set of positive length. This is Falconer's 1985 conjecture, the continuous analogue of Erdős's distinct-distances problem; the best planar threshold before was 5/4. The 64-page proof comes with a Lean formalization whose statement uses Mathlib's own Hausdorff dimension and Lebesgue measure, and the community comparator run accepted it. A first follow-up already builds on its method.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: The Falconer distance conjecture in all dimensions (23 Sep)","url":"https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"Lean scope note for family 073","url":"https://github.com/openai/math/blob/main/lean/docs/073.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Lean comparator challenge: FalconerAllDimensions.lean","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/FalconerAllDimensions.lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2610.22222: Cheng, Fang, Li, Qu, Xiao, A quantitative form of the planar Falconer distance theorem","url":"https://arxiv.org/abs/2610.22222","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"oai-math-revised: comparator run on all challenges (LEAN_CHECKS.md)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"oai-math-revised: community notes on the Falconer manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release proves Falconer's distance conjecture (1985) in every dimension, with a Lean proof of the exact statement","summary":"\"The Falconer distance conjecture in all dimensions\" (OpenAI, 23 Sep 2026, family 073 of the 6 Oct release) proves: \"For every integer d ≥ 2, a compact subset of ℝᵈ with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure.\" Falconer had shown (d+1)/2 suffices; the best planar bound was 5/4 (Guth–Iosevich–Ou–Wang, 2020), and higher-dimensional bounds stayed above d/2 + 1/4 − 1/(8d+4). The manuscript uses a multiscale analysis of measures with radial projection and incidence inputs. The all-dimensional statement and the planar case are formalized in Lean and accepted by the comparator. No expert assessment has been published; a quantitative planar version by Cheng, Fang, Li, Qu and Xiao develops the method.","key_facts":["Abstract (verbatim): 'We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝᵈ with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure.' 64 pages, dated 23 Sep 2026","Prior state (paper, §1.1): Falconer's threshold (d+1)/2; Wolff's planar 4/3; Erdoğan's d/2 + 1/3 for d ≥ 3; Du, Guth, Ou, Wang, Wilson and Zhang 9/5 in dimension three; Du–Zhang d/2 + 1/4 + 1/(8d − 4); Guth, Iosevich, Ou and Wang dimH E > 5/4 in the plane; Du, Ou, Ren and Zhang d/2 + 1/4 − 1/(8d + 4) for d ≥ 3. Shmerkin and Wang handled sets whose Hausdorff and packing dimensions both equal d/2","Method (paper, §1.2): a multiscale decomposition of measures into regular pieces with scale-dependent angular tests, building on Keleti–Shmerkin, Shmerkin–Wang and Orponen–Shmerkin; the retained incidence input is the Orponen–Shmerkin theorem; a 'two-depth potential' pays the costs of the recursion","Lean: comparator challenge FalconerAllDimensions.lean, theorem `falconer_distance_conjecture`: for every d ≥ 2 and compact E ⊆ EuclideanSpace ℝ (Fin d) with d/2 < dimH E, `0 < volume {r : ℝ | ∃ x ∈ E, ∃ y ∈ E, dist x y = r}`; plus PlanarFalconer.lean for the earlier planar case. Only Mathlib definitions are used, so the statement reads as the standard conjecture (our reading). Both were accepted in the oai-math-revised comparator run (25–26 Oct)","Follow-up: Guozheng Cheng, Xiang Fang, Menghan Li, Hongdou Qu and Chengbo Xiao, 'A quantitative form of the planar Falconer distance theorem' (arXiv 2610.22222, 15 Oct): 'The proof develops the multiscale method of the September 2026 manuscript of OpenAI'; their comment line promises higher dimensions 'building on the October 6 release of the OpenAI preprint'. Their AI note: an AI tool was used for language editing, LaTeX and 'local mathematical reasoning'","Our 19 Oct entry on the harmonic-analysis claims noted that 073 is among the few families in that area with a Lean link (see 2026-10-06-openai-release-3d-restriction-bochner-riesz-claims); no press article or expert blog post on this family was found by 29 Oct"],"key_numbers":[],"tags":["math","harmonic-analysis","geometric-measure-theory","falconer-conjecture","distance-sets","lean","openai","openai-math-release"],"science":{"field":"mathematics","subfield":"harmonic analysis / geometric measure theory","problem":"Falconer's distance conjecture (1985): compact sets in ℝᵈ of Hausdorff dimension > d/2 have distance sets of positive Lebesgue measure","result":"Claimed in every dimension d ≥ 2, with a Lean proof of the exact statement (comparator accepted)","open_since":"1985","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","verification":{"key":"lean","label":"Lean-verified","text":"Lean formalization of the main theorem; no published human verification yet"},"status":"pending","shock":"The planar threshold had moved from 4/3 to 5/4 over two decades; the release goes straight to the conjectured d/2 in every dimension."},"body_md":"## What happened\n\nHow big must a set be before the distances between its points fill out an interval of positive length? Kenneth Falconer showed in\n1985 that Hausdorff dimension above (d+1)/2 is enough in ℝᵈ, and conjectured that d/2 suffices, which would be sharp. The problem\nis a continuous cousin of Erdős's distinct-distances problem and became a test case for Fourier restriction and decoupling. Progress\nwas slow: the planar threshold went from 4/3 (Wolff, 1999) to 5/4 (Guth, Iosevich, Ou and Wang, 2020).\n\nOpenAI's 64-page manuscript claims the full conjecture in every dimension. It works directly with the distance measure through a\nmultiscale decomposition of the set's measure, angular tests at each scale and a recursive potential argument, using known\nincidence and radial projection theorems as inputs. The main theorem is formalized in Lean against a statement written with\nMathlib's Hausdorff dimension and Lebesgue measure, and the comparator check passed.\n\n## Why it matters\n\nFalconer's conjecture is one of the central problems linking geometric measure theory and harmonic analysis. With a Lean proof of a\nfaithful statement it is among the better-checked claims in the release, but human understanding of the argument is only beginning.\n\n## Reactions\n\nThe first outside paper, by Guozheng Cheng, Xiang Fang, Menghan Li, Hongdou Qu and Chengbo Xiao, gives a quantitative planar version\nby developing the manuscript's multiscale method and promises the higher-dimensional case. We found no blog post, talk or press piece assessing the proof.","disputed":[{"label":"Verification","text":"Lean formalization of the main theorem; no published human verification yet","sources":[]}],"related":[{"id":"2026-10-16-ai-assisted-results-october-15-listing","url":"https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/","date":"2026-10-16","date_precision":"day","short_title":"arXiv results with disclosed AI help, 15 Oct 2026 listing","deck":"Monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more","takeaway":"One day's maths listing brought five results big enough for their own entries (Smale's mean value constant, Hopf with T³, class numbers ≤ 2000, the Delsarte rate, LLM-found quantum circuits) and about two dozen more with explicit AI statements.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":12,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","url":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 3D restriction and Bochner–Riesz conjectures","deck":null,"takeaway":"Beside quasi-RH and Unique Games, OpenAI's 6 Oct catalogue claims a whole cluster of central harmonic-analysis conjectures in three dimensions: Stein's restriction conjecture for the sphere, the diagonal extension conjecture, Bochner–Riesz, Sogge's local smoothing and the planar endpoint of Carleson's Schrödinger problem. None has a Lean formalization and no expert review has been published; the first arXiv paper to cite one appeared on 15 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":7,"filed":"2026-10-19","updated":"2026-10-26","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 001–125; manuscript introduction, Lean statement, comparator results and the follow-up arXiv paper read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 001–125; manuscript introduction, Lean statement, comparator results and the follow-up arXiv paper read","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-proves-falconer-distance"}