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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.

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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

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.

Vanishing at criticality was known in dimension two and in dimensions at least eleven. The lattice cases in dimensions three through ten were the open target of this report.

Why it matters

If correct, resolves the remaining integer-lattice dimensions of a central critical-percolation problem. The earlier Anthropic contribution deserves separate attribution from OpenAI’s later, broader graph claim.

What counts as progress

A sourced advance or substantial formalization with the exact target, hypotheses, baseline and AI role stated. Importance and verification are assessed separately.

What to watch for

  • Compare the precise hypotheses and baseline; retain human and prior-work attribution.
  • The statement concerns nearest-neighbour bond percolation on integer lattices, not site percolation or all quasi-transitive graphs. The artifact reports build, comparator and axiom checks; AIProofWatch has not rerun them or audited statement fidelity. The pinned source remains accessible, while the current main-branch path was inaccessible during review; that access failure is not treated as a withdrawal. OpenAI’s separate report covers quasi-transitive graphs with p_c<1.

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

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