Result 089, Convex and metric geometry

Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs

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 formalization Proof

The bigger picture

Why it matters

A network's distances can be complicated even when its edges never cross. These manuscripts claim that such distances, and those of networks with bounded tree-like complexity, can be represented in a standard geometric space without uncontrolled distortion.

What changes?

The manuscripts report that shortest-path distances in finite connected graphs with arbitrary positive real edge lengths embed into real L1, where distance measures total absolute difference. An embedding represents vertices with distances preserved up to a multiplicative factor after rescaling. For planar graphs, drawable without edge crossings, that factor is universal. For graphs of bounded treewidth, meaning a tree-like decomposition into small overlapping vertex groups, it depends only on the treewidth bound, not graph size or edge lengths.

What does that help mathematicians do?

The reported consequence is a uniformly bounded multicommodity flow-cut gap: bottleneck estimates obtained by cutting a network cannot overestimate how much demand can be routed simultaneously by an arbitrarily large factor. The bound is universal for planar graphs and depends only on the treewidth bound in the second class. Researchers can therefore relate cut-based obstructions to actual routing limits, rather than treating cuts as potentially very loose estimates.

Are there practical applications?

The immediate value is foundational: these claims identify graph families whose distances admit controlled L1 representations and whose routing limits are captured by cuts within bounded factors. That connection is relevant to the mathematical analysis of network optimization. It is not, by itself, a demonstrated faster routing algorithm or evidence of practical performance.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Planar Graph Metrics Embed into L1 with Constant Distortion

September 23, 2026 45 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026,
  author = {{OpenAI}},
  title = {{Planar Graph Metrics Embed into $L_1$ with Constant Distortion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf}{OAI:Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026}},
  year = {2026}
}

L1 Embeddings of Graphs of Bounded Treewidth

September 23, 2026 30 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026,
  author = {{OpenAI}},
  title = {{$L_1$ Embeddings of Graphs of Bounded Treewidth}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf}{OAI:L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/089.md.

Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The planar case of the Gupta–Newman–Rabinovich–Sinclair embedding problem asks for a universal distortion bound in L1L_1. The formalized result gives one constant CC for every finite connected planar graph with arbitrary positive real edge lengths: its weighted shortest-path metric embeds into real L1L_1 with distortion at most CC. The constant is independent of the number of vertices and the ratios between edge lengths; singleton graphs are included.

The formalized result gives a uniform L1L_1 embedding bound for each bounded-treewidth class. For every bag-size bound k≥2k\ge2, there is a constant C(k)C(k) such that the weighted shortest-path metric of every nonempty finite connected graph with a tree decomposition of bag size at most kk embeds into finite-dimensional real L1L_1 with distortion at most C(k)C(k). Edge lengths may be arbitrary positive real numbers, and the constant is independent of the graph and decomposition size.

Comparator links

Result Comparator statement
Planar graph metrics in L1L_1 PlanarL1.lean
Bounded-treewidth graph metrics in L1L_1 BoundedTreewidthL1.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.