OpenAI · 089-01 · 2026-09-23 · Convex and metric geometry
Planar Graph Metrics Embed into L1 with Constant Distortion
preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026
Family 089. Bounded-distortion embeddings of planar and bounded-treewidth graphs.
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
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.
We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L1 with a universal distortion bound. This resolves the planar embedding conjecture positively.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 089: Bounded-distortion <i>L</i><sub>1</sub> embeddings of planar and bounded-treewidth graphs. 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.