HEXAGONbeta

A composite modulus and Frobenius digits in the Heilbronn triangle construction

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatoricsmath.MG — Metric Geometry

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

OpenAI proved that for every sufficiently large n there are n points in the unit square all of whose triangles have area at least c n^(-2+eta), with an explicit but extremely small eta (of order 10^(-235)). Subsequent public work lowered the cost of the construction, the latest claimed exponent being eta = 2/100866719, about 1.98*10^(-8). We introduce two changes. First, the digit base is taken to be a multiple of the label prime r and the main modulus is h = r B^(k-1) instead of B^k; this reduces the ratio h/tau between the modulus and the excluded determinant window from about B to about r, which allows the extension degree d = 3. Second, the norm polynomial is placed in the digits by a digit-wise convolution along the Frobenius expansion of the exponents, with a trace digit in a polynomial basis, giving k = 1440 positions. The result is Delta(n) >= c n^(-2+eta) with eta = 1/177022 > 5.64*10^(-6), for every sufficiently large n. The finite identities are checked by exact computation in the ancillary files.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation of the finite identities. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. OpenAI, A power improvement in the Heilbronn triangle lower bound , OpenAI Math Release preprint, September 25, 2026; source commit 1581d4cb.
  2. I. Blinov, Heilbronn triangle bounds: finite packings and improved scales , GitHub research draft (littleBro/heilbronn-triangle-bounds), October 2026.
  3. Euler team (canghaimeng), A smaller lift box in the degree-five Heilbronn construction , GitHub (canghaimeng/openai-math-improvement, bound-191), October 18, 2026.
  4. J. Komlós, J. Pintz and E. Szemerédi, A lower bound for Heilbronn's problem, J. London Math. Soc. (2) 25 (1982), 13--24.

Version history

  1. v1 · 11 pagesSubmitted by David Niedbala GiraudinInitial depositCurrentNov 04, 2026