What was already known
Benjamini and Schramm conjectured a nonempty nonuniqueness phase for nonamenable graphs with finitely many vertex orbits; Hutchcroft proposed the stronger ℓ² operator-threshold inequality.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Resolves both the nonuniqueness conjecture and the stronger operator-threshold question in this graph class.
Limitations
The statement is for Bernoulli bond percolation on locally finite quasi-transitive graphs; it is not a result for arbitrary dependent percolation.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision