# Planar geometric incidence benchmark

Family 167: Planar distinct distances and unit-distance bounds. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Computational-geometry researchers can test distinct pinned distances and unit-distance counts against new asymptotic limits.

## Applicability and commercial boundary

All-but-o(n) and unspecified positive power savings give no usable finite runtime or numeric threshold.

## 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

Inspect effective constants before claiming faster geometric computation.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The weak pinned planar distance theorem](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf).