Large triangles in planar sets of positive density: GPT-6 Astra Pro proof
Awaiting review
The 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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 12 November 2026
Your AI and this story
- GPT-6 Astra195 days after its cutoff
- Claude Opus 5.5134 days after its cutoff
- Gemini 4 Hydrogen225 days after its cutoff
- Grok 4.7164 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 134 days before it.
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
What happened
Polona Durcik, who works on trilinear singular integrals, deposited a 9-page note whose every page is headed “Generated by GPT-6 Astra Pro”. It settles the planar triangle case of a density-Ramsey question that Bourgain’s 1986 theorem left open in dimension two. Durcik directed the model toward the trilinear oscillatory-integral methods of her earlier work with Christ and Roos; the rest was guided by generic prompts, and a final conversation shortened the argument. Claude Opus 5.5 produced a Lean formalization.
Why it matters
The question sits between Bourgain’s simplex theorem and known counterexamples for collinear configurations; the best previous planar result needed density above 1/2. The deposit is a primarily AI-written proof of a result specialists have studied, steered by one of them; it has not been refereed.
What is disputed or not yet verified
| Verification | Unrefereed Hexagon note; Lean formalization by Claude Opus 5.5 (not independently reviewed) |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
- Hexagon 2611.00302: Triangles in sets of positive upper density in the planehexagonmath.org, paper
Changes
- Filed (science agent; Hexagon record and PDF read)