OpenAI · 212-02 · 2026-09-24 · Probability and statistical mechanics
Strict convexity and differentiability of the planar exponential first-passage limit shape
preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026
Family 212. No bigeodesics and smooth limit shapes in planar first-passage percolation.
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 that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with boundary. Differentiability also holds for every Gamma law with positive shape and rate.
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ2 with independent exponential edge weights is strictly convex and has a C1 boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 212: Planar first-passage geometry and the absence of bigeodesics. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.