OpenAI · 223-06 · 2026-10-05 · Probability and statistical mechanics

Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces

preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026

Family 223. Conformal limits of square-lattice random-cluster interfaces.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves chordal $SLE_{κ}$ limits for critical square-lattice random-cluster interfaces for $0 < q \leq 4$ , with $κ = 4 π / arccos (- q /2)$ : bounded Jordan domains are allowed for $q \geq 1$ , and smooth Jordan domains for $q < 1$ , under the stated marked-boundary approximations. For $1 \leq q \leq 4$ , complete nested plane loops converge to $CLE_{κ}$ .

For every fixed $1 \leq q \leq 4$ , critical square-lattice random-cluster Dobrushin interfaces converge to chordal $SLE_{κ (q)}$ , where $κ (q) = 4 π / arccos (- q /2)$ , and the complete nested plane loop collections converge to whole-plane $CLE_{κ (q)}$ . The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

Preprint source and Lean scope

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