# Finite-representation separation counterexample

Family 252: A torsion-free hyperbolic group that is neither residually finite nor linear over any field. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Test the assumption that finite quotients or finite-dimensional linear representations separate every element of a torsion-free hyperbolic group.

## Applicability and commercial boundary

The specified element and group do not rule out separation in every group class.

## 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

Read the finite presentation and exact non-separation witness.

## 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 torsion-free hyperbolic group that is not residually finite](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf).