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

Natural Occupation Measures for Critical Square-Lattice FK Interfaces

preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-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 $CLE_{κ}$ .

We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight $1 \leq q < 4$ . A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.