Probability and statistical mechanics·OpenAI Math·Result 214·Lean formalisation listed
The Benjamini–Schramm nonuniqueness conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under probability and statistical mechanics: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of the Benjamini–Schramm nonuniqueness conjecture (probability and statistical mechanics) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 47 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 15:56 · watch from there on YouTube →
What OpenAI claims
Proves pc < pu for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. (Lean)
Paper
Nonuniqueness of percolation on nonamenable quasi-transitive graphs
OpenAI's abstract We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing pc < p2 → 2 ≤ pu.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves pc < pu for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. (Lean)
- Where is the proof?
- In the paper, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 47).
Also in
- Chapter 2 of 10 in Benjamini–Schramm Conjecture, Mézard–Parisi Formula, Voronoi Percolation and 7 More · starts at 0:21 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 47 from 15:56 · description, sources and chapters →
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