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.

Main source

Nonuniqueness of percolation on nonamenable quasi-transitive graphs.