# CAT(0) action counterexample library

Family 257: A hyperbolic group without a geometric CAT(0) action. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Record a hyperbolic group with a finite classifying space that lacks a proper cocompact action on any proper complete CAT(0) space.

## Applicability and commercial boundary

Hyperbolicity does not automatically imply this action property; the example does not invalidate restricted CAT(0) algorithms.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Extract the presentation and exact obstruction.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[A hyperbolic group with no geometric CAT(0) action](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf).