---
id: "2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro"
url: "https://postcutoff.com/e/2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-11-11"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Lean-verified
sources: 1
editor: human editor
human_review: null
version: "2026-11-12"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro/

# Large triangles in planar sets of positive density: GPT-6 Astra Pro proof

Full 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

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

## 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)

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 195 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 134 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 225 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 164 days after its cutoff

## Sources

1. [Hexagon 2611.00302: Triangles in sets of positive upper density in the plane](https://hexagonmath.org/abs/2611.00302) (hexagonmath.org, paper)

## Changes

- 2026-11-12 (filed): Created (science agent; Hexagon record and PDF read)

## Related

- 2026-11-11: [AI-assisted results, 11 Nov: arXiv and Hexagon](https://postcutoff.com/e/2026-11-11-ai-assisted-results-november-11-listing/index.md)