# Nondegenerate planar point-set research design

Family 191: A power improvement in the Heilbronn triangle lower bound. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometric design researchers can explore point sets maximizing the smallest triangle area.

## Applicability and commercial boundary

The saving exponent is extremely small; the formal scope covers an unbounded sequence of sizes, while the paper claims all sufficiently large n. This is not a practical mesh-quality guarantee.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract actual constants and coordinates before proposing a finite sensor or mesh layout.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scopes were additionally inspected; explicit effectiveness passages were inspected for 076, 143 and 178. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[A power improvement in the Heilbronn triangle lower bound](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf).