OpenAI · 232-03 · 2026-10-06 · Probability and statistical mechanics

Uniform real Lipschitz surfaces on the triangular lattice

preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026

Family 232. Gaussian fields and SLE interfaces for Lipschitz heights.

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 Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments $0, \pm 2$ and two-arc boundary values $\pm 1$ , and zero-boundary integer Lipschitz heights weighted by fixed $x \in [1/ 2, 1]$ . Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal $SLE_{4}$ , establishing Schramm’s real-field/interface predictions.

We prove the Gaussian free field and SLE $_{4}$ scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE $_{4}$ . We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.

Preprint source and Lean scope

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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.