# Diameter partition counterexample library

Family 156: Borsuk's conjecture fails in dimension nine. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Clustering researchers can test universal smaller-diameter partition claims using the rank-one-projector example in nine dimensions.

## Applicability and commercial boundary

The counterexample does not show ordinary finite clustering heuristics fail or provide an improved clusterer; covering constraints are specific.

## 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 a finite approximating configuration with the exact Frobenius metric.

## 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).

[A nine-dimensional counterexample to Borsuk's covering assertion](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf).