OpenAI · 211-03 · 2026-10-05 · Probability and statistical mechanics
A Linear Clock for Random Walk on Tree-Weighted Planar Maps
preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026
Family 211. Geometry, diffusion, and spectra of random planar maps.
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
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for and to the Brownian continuum random tree for , establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.
We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area -Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this same ensemble.
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.
A Lean scope note is stored for family 211, but it does not name this preprint. It names 211-07. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.
Emergence
A continuum random tree is not spacetime
Family 211 records finite-volume metric limits of critical Fortuin–Kasteleyn planar maps. For q>4 the stated limit is the Brownian continuum random tree. That metric-measure statement is not a quantum surface, not a Page curve, and not spacetime emergence. Oai211-A marks a conditioned block tree and does not verify the preprint. Hausdorff dimension two is not manifold dimension. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
EmergenceOai211-A path-sum certificateOai290-A typed recoveryEn07-C Hawking comparison