OpenAI · 212-01 · 2026-09-24 · Probability and statistical mechanics
No bigeodesics in planar first-passage percolation
preprints/No-bigeodesics-in-planar-first-passage-percolation-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 planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
Preprint source and Lean scope
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A Lean scope note is stored for family 212, but it does not name this preprint. It names 212-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.