AIProofWatch
← Explore interestsIndividual problem

Critical percolation on quasi-transitive graphs

At the critical threshold, can independently opened edges form an infinite cluster on an infinite graph with only finitely many vertex orbits?

Progress records 2

Understand the problem ↓

2 records · How we rate progress

Importance: major
VerificationAuthor claimAccessible artifact / proofFormal checking

Claude reports: no infinite critical cluster on integer lattices

The pinned Anthropic development reports θ(p_c)=0 for nearest-neighbour Bernoulli bond percolation on Zᵈ in every integer dimension d≥2. It proves Kozma–Nitzan’s Conjecture 3 through an additive gluing inequality and formalizes the reduction to the lattice statement.

Read original source Pinned Anthropic development: exact lattice statementgithub.com

Original report: 2026-08-28

Understanding the problem

Read the background

At the critical threshold, can independently opened edges form an infinite cluster on an infinite graph with only finitely many vertex orbits?

Why it matters

A precise target for comparing mathematical claims, their baselines, and the documented contribution of AI.

What counts as progress

A sourced theorem, construction, algorithm, bound, or substantive formalization with its assumptions and AI role stated.

What to watch for

  • Check the exact hypotheses and compare with the stated prior baseline.
  • Distinguish a source claim, an available proof, and a reproduced verification.
  • Retain human contributions, competing work, corrections, and withdrawals.

Background review: 2026-10-11. Selected sources:

Website timeline

Publication dates and editorial changes.