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The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture

Planar paper, Theorem 1.1: a universal constant $C\ge1$ such that every finite connected planar graph with positive edge lengths embeds into $L_1$ with distortion at most $C$ ; hence the undirected multicommodity flow-cut gap is uniformly bounded on planar graphs. Treewidth companion, Theorem 1.1: for each $k$ , a constant $C_k$ (no explicit estimate) for graphs with tree decompositions of bag size at most $k$ , hence also for graphs excluding a fixed planar minor. Together they give constant distortion for fixed-parameter almost-embeddable families. Not shown: the full GNRS conjecture for every minor-closed family (closure under clique sums is open), and no explicit value of $C$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture · GNRS conjecture: planar and bounded treewidth

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed
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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts (TeX source), read against the GNRS conjecture as cited. Lean-checked on the release's Comparator challenge PlanarL1 (declaration OAI.PlanarL1.planar_graph_metrics_embed_L1, listed in lean/formalization.yaml). Its statement was read here: it asserts one real $C\ge1$ such that every connected graph on Fin n with a planar drawing (injective vertex points, simple arcs with pairwise disjoint interiors) and symmetric positive edge lengths has a map into $L^1$ of some measure space with $d\le\|f(x)-f(y)\|\le C d$ for the walk-length distance $d$ . That is the planar headline. The companion challenge BoundedTreewidthL1 (OAI.BoundedTreewidthL1.main_theorem, also listed) states, for every bag-size bound $k\ge2$ , a constant $C(k)$ and an embedding into finite-dimensional $\ell_1$ for every finite connected graph with a tree decomposition of bag size at most $k$ ; that is the treewidth headline. Neither was rebuilt here. The papers state that the full GNRS conjecture (general clique-sum closure) is not established.

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The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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L1 Embeddings of Graphs of Bounded Treewidth (companion)

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paper link supplied by the VibeMathed registry.

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Lean Comparator challenge PlanarL1 (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean Comparator challenge BoundedTreewidthL1 (OpenAI math release)

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lean-proof link supplied by the VibeMathed registry.

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Gupta-Newman-Rabinovich-Sinclair 2004, Cuts, trees and l1-embeddings of graphs

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problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge PlanarL1 (OpenAI math release)

lean artifact · pending

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Lean Comparator challenge BoundedTreewidthL1 (OpenAI math release)

lean artifact · pending

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VibeMathed record: The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture evidence for this event

L1 Embeddings of Graphs of Bounded Treewidth (companion) evidence for this event

The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean Comparator challenge BoundedTreewidthL1 (OpenAI math release) evidence for this event

Lean Comparator challenge PlanarL1 (OpenAI math release) evidence for this event

Gupta-Newman-Rabinovich-Sinclair 2004, Cuts, trees and l1-embeddings of graphs evidence for this event

The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture parent of this event

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