OpenAI claim summary for Math-2643
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 089; 2026-09-23).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Uniform L1 distortion of planar graph metrics
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 089; 2026-09-23), the asserted result is that arbitrary positive edge lengths on finite planar graphs admit a common constant-distortion embedding into real $L_1$.Mathematical statement of the claimed result
Manuscript 1.
Assign arbitrary positive real lengths to the edges of any finite planar graph that is connected. The claim supplies a real $L^1$ embedding and one constant $C<\infty$, common to all these graphs, satisfying \[ d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y)\quad(x,y\in V(G)). \] No dependence on vertex count, particular planar embedding or edge-length ratios occurs in $C$.Outline of the argument
For manuscript 1 (family 089; 2026-09-23), measurable labels define cut distances whose upper comparison holds deterministically. Random chart changes, portals, and component ancestry supply a lower separation probability, and packing control permits the cut families to be assembled at all scales.Formal verification
For manuscript 1 (family 089; 2026-09-23), OpenAI lists Lean statements for this assertion: arbitrary positive edge lengths on finite planar graphs admit a common constant-distortion embedding into real $L_1$.The central assertion is within the documented formal scope; the additional applications retain their own scope limits.
The linked family Comparator checks are BoundedTreewidthL1.lean, PlanarL1.lean.
Their named declarations include OAI.BoundedTreewidthL1.main_theorem, OAI.PlanarL1.planar_graph_metrics_embed_L1.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Planar Graph Metrics Embed into L${}_{1}$ with Constant Distortion, 2026-09-23; citation key
OAI:Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 089: Lean scope.
- Comparator BoundedTreewidthL1.json.
- Comparator BoundedTreewidthL1.lean.
- Comparator PlanarL1.json.
- Comparator PlanarL1.lean.
- OpenAI catalogue entry.