Convex and metric geometry·OpenAI Math·Result 089·Lean formalisation listed
Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs
One of the 372 results in OpenAI's public mathematics catalogue, filed under convex and metric geometry: 2 preprints, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the papers and abstracts, and embeds the animation.
Also known as Gupta–Newman–Rabinovich–Sinclair conjecture; GNRS conjecture; planar embedding conjecture; planar graphs embed into L1 with constant distortion
In one line: OpenAI reports a proof of Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs (convex and metric geometry) in 2 preprints, the latest dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Its own Short goes public 16 October 2026, 00:00 UTC; it will appear here then.
Chapter 57 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 19:27 · watch from there on YouTube →
What OpenAI claims
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 L1 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. (Lean)
Papers (2)
Planar Graph Metrics Embed into L1 with Constant Distortion
OpenAI's abstract 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.
L1 Embeddings of Graphs of Bounded Treewidth
OpenAI's abstract For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L1 with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.
Status
Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted 2 preprints, the latest dated 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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 L1 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. (Lean)
- Where is the proof?
- In the paper, Planar Graph Metrics Embed into L1 with Constant Distortion (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 57).
Also in
- Chapter 2 of 10 in Petty's Projection Conjecture, Log Brunn–Minkowski Conjecture and 8 More · starts at 0:22 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 57 from 19:27 · description, sources and chapters →
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