The Benjamini-Schramm nonuniqueness conjecture: p_c < p_u for bond percolation on nonamenable quasi-transitive graphs

Let $G$ be an infinite connected locally finite quasi-transitive graph and consider Bernoulli bond percolation with critical parameter $p_c(G)$ (infinite clusters appear) and uniqueness threshold $p_u(G)$ (the infinite cluster becomes unique almost surely). On a regular tree of degree at least three, $p_c<p_u=1$ . Benjamini and Schramm (1996, Conjecture 6) conjectured that nonamenability alone always separates the two thresholds. It was known under extra hypotheses: planar one-ended transitive graphs, highly nonamenable or large-girth graphs, graphs with nonconstant harmonic Dirichlet functions (Gaboriau), Gromov hyperbolic and nonunimodular graphs (Hutchcroft), and Cayley graphs for some suitably chosen generating set. Hutchcroft further conjectured the stronger strict inequality $p_c<p_{2\to2}$ for the $\ell^2$ operator threshold. Is $p_c(G)<p_u(G)$ for every nonamenable quasi-transitive graph $G$ , so that some interval of $p$ has infinitely many infinite clusters?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Percolation on nonamenable graphs
Posed by: Itai Benjamini and Oded Schramm
Year posed: 1996
Years open: 30y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 66 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every infinite connected locally finite quasi-transitive graph with positive vertex isoperimetric constant, $\sup_{p<p_c}\|T_p\|_{2\to2}=\|T_{p_c}\|_{2\to2}<\infty$ and $p_c<p_{2\to2}\le p_u$ ; there are $p_c<p_1<p_2<1$ such that, in the uniform-label coupling, almost surely infinitely many infinite clusters exist simultaneously for all $p\in[p_1,p_2]$ . Corollary 1.2 gives the same for every finite symmetric generating set of every nonamenable finitely generated group. Further consequences: the triangle condition at $p_c$ , mean-field critical exponents, and exponential decay of connections below $p_{2\to2}$ . It concerns bond percolation only; the site-percolation form of the conjecture and the related $p_u<1$ question for one-ended graphs are not addressed. Constants and the interval depend on $G$ .

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 24, 2026) is the whole family.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Conjecture 6 of Benjamini-Schramm; they state the conjecture for bond percolation in full generality (quasi-transitive, nonamenable via positive vertex isoperimetric constant) and Hutchcroft's stronger $p_c<p_{2\to2}\le p_u$ . The proof was not refereed. The challenge lean/ComparatorChallenges/BenjaminiSchramm.json (solution module OAI.Probability.BenjaminiSchramm.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; its statement BenjaminiSchramm.lean was read here. full_main asserts, for every infinite connected locally finite quasi-transitive bond graph with $h_V>0$ , that the critical two-point operator is bounded, $p_c<p_{2\to2}\le p_u$ , and an interval $p_c<p_1<p_2<1$ on which, in the standard monotone coupling, there are almost surely infinitely many infinite clusters for all $p\in[p_1,p_2]$ ; cayley_main covers every finite symmetric generating set; critical_laws states the mean-field critical exponents. This states the headline claim. CayleyPercolation.json is a second challenge. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The proof uses as inputs Hutchcroft's theorem that critical percolation dies under exponential growth and the Haggstrom-Peres-Schonmann simultaneous-phase theory.

The Benjamini-Schramm nonuniqueness conjecture: p_c < p_u for bond percolation on nonamenable quasi-transitive graphs

What was actually shown

What the AI did

Verification

Sources

Discussion