OpenAI · 191-01 · 2026-09-25 · Combinatorics
A power improvement in the Heilbronn triangle lower bound
preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
Family 191. A power improvement in the Heilbronn triangle problem.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
For every sufficiently large , constructs points in the unit square such that every triangle has area at least for one absolute . This disproves the conjectured almost- upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.
There are absolute constants such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least . Thus the almost n−2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 191: A power improvement in the Heilbronn triangle lower bound. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.