# Nonamenable network phase reference

Family 214: The Benjamini–Schramm nonuniqueness conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Graph researchers can explore the interval of multiple infinite clusters and the stated triangle-condition context.

## Applicability and commercial boundary

The result assumes infinite nonamenable quasi-transitive graphs. It is not a business network resilience guarantee or a finite reliability algorithm.

## Initial business decision

Defer direct product. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Extract an explicit graph example and explain which finite observables can honestly approximate the phases.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[Nonuniqueness of percolation on nonamenable quasi-transitive graphs](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf).