{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro/","md":"https://postcutoff.com/e/2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro/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":"2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro","date":"2026-11-11","date_precision":"day","short_title":"Large triangles in planar sets of positive density: GPT-6 Astra Pro proof","deck":null,"takeaway":"The planar case of Bourgain's simplex theorem, previously known only for density above 1/2, is claimed in a 9-page AI-written note steered by a harmonic analyst who works on trilinear estimates; Claude Opus 5.5 wrote a Lean formalization.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2611.00302: Triangles in sets of positive upper density in the plane","url":"https://hexagonmath.org/abs/2611.00302","type":"paper","group":"primary","domain":"hexagonmath.org"}],"official":1,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","Anthropic","Chapman University"],"title":"Every planar set of positive upper Banach density contains all large dilates of any triangle (up to rotation): a note 'generated by GPT-6 Astra Pro', deposited by Polona Durcik, with a Lean formalization by Claude Opus 5.5","summary":"Hexagon 2611.00302 (11 Nov 2026), deposited by Polona Durcik (Chapman University), proves that every measurable set E ⊂ R² with positive upper Banach density contains, for all sufficiently large λ, a translate of a rotation of λΔ for any fixed noncollinear triangle Δ. Bourgain's 1986 theorem gives this for simplices only in higher ambient dimension, and Falconer, Kovač and Yavicoli had the planar case for density above 1/2. The text is headed \"Generated by GPT-6 Astra Pro\"; the Lean formalization was generated by Claude Opus 5.5. Unrefereed.","key_facts":["Theorem 1.1: Δ ⊂ R² noncollinear, E measurable with upper Banach density d*(E) > 0 ⇒ there is λ₀ with a translate of a rotation of λΔ inside E for every λ ≥ λ₀","Prior results cited in the note: Furstenberg–Katznelson–Weiss (every positive-radius neighbourhood of E contains such copies), Ziegler (all finite configurations, neighbourhood version), Bourgain 1986 (exact copies of large triangles in dimension ≥ 3), Falconer–Kovač–Yavicoli (planar case for d*(E) > 1/2); Castro-Silva conjectured the planar statement up to congruence","Method: density Hales–Jewett theorem plus an operator-norm continuity estimate for the triangle-counting form, from a trilinear smoothing theorem in the style of Christ and of Christ–Durcik–Roos","Provenance (Hexagon): 'The manuscript was generated over three conversations with GPT-6 Astra Pro via Codex on November 4–5, 2026. The model was directed by the submitter toward trilinear oscillatory integral methods … The Lean formalization was generated by Claude Opus 5.5 with medium reasoning effort.' Text 'Primarily AI-generated'; human understanding 'some'","Status: unrefereed 9-page note; Lean formalization stated; no outside reaction found by 12 Nov"],"key_numbers":[],"tags":["math","harmonic-analysis","geometric-ramsey","euclidean-ramsey","gpt-6-astra-pro","codex","claude-opus-5-5","lean","hexagon"],"science":{"field":"mathematics","subfield":"harmonic analysis / Euclidean Ramsey theory","problem":"Large similar (rotated) copies of a triangle in planar sets of positive upper density (planar case of Bourgain's simplex theorem)","result":"Claimed: all sufficiently large dilates of any noncollinear triangle occur, up to translation and rotation, in every planar set of positive upper Banach density","open_since":"","ai_system":["GPT-6 Astra Pro","Codex","Claude Opus 5.5"],"human_role":"Submitter chose the method (trilinear oscillatory integrals) and prompted; model wrote the proof; Claude wrote the Lean proof","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed Hexagon note; Lean formalization by Claude Opus 5.5 (not independently reviewed)"},"status":"pending","shock":""},"body_md":"## What happened\n\nPolona Durcik, who works on trilinear singular integrals, deposited a 9-page note whose every page is headed \"Generated by GPT-6\nAstra Pro\". It settles the planar triangle case of a density-Ramsey question that Bourgain's 1986 theorem left open in dimension two.\nDurcik directed the model toward the trilinear oscillatory-integral methods of her earlier work with Christ and Roos; the rest was\nguided by generic prompts, and a final conversation shortened the argument. Claude Opus 5.5 produced a Lean formalization.\n\n## Why it matters\n\nThe question sits between Bourgain's simplex theorem and known counterexamples for collinear configurations; the best previous\nplanar result needed density above 1/2. The deposit is a primarily AI-written proof of a result specialists have studied, steered by\none of them; it has not been refereed.","disputed":[{"label":"Verification","text":"Unrefereed Hexagon note; Lean formalization by Claude Opus 5.5 (not independently reviewed)","sources":[]}],"related":[{"id":"2026-11-11-ai-assisted-results-november-11-listing","url":"https://postcutoff.com/e/2026-11-11-ai-assisted-results-november-11-listing/","date":"2026-11-11","date_precision":"day","short_title":"AI-assisted results, 11 Nov: arXiv and Hexagon","deck":null,"takeaway":"Own entries today: Smale's constant = 4 (Goel, Lean), Talagrand's Ultimate Matching Conjecture (GPT-6), the Benson–Etingof–Ostrik conjecture (GPT-6 Astra), large triangles in planar dense sets (GPT-6 Astra Pro) and explicit RIP matrices (GPT-6 Astra). Below: the rest, quoted from Hexagon provenance and arXiv PDFs; nothing here is refereed.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":113,"official":113,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","Anthropic","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-12","type":"filed","text":"Created (science agent; Hexagon record and PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-12","run":null,"sources_read":"Science agent; Hexagon record and PDF read","updated":"2026-11-12","human_review":null,"version":{"date":"2026-11-12"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":195,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":134,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":225,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":164,"in_training_data":false}],"short_url":null}