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

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

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. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments

What the papers state

Theorem 1.1

In Planar Graph Metrics Embed into L1 with Constant Distortion, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is a finite universal constant C\ge1 with the following property. For every finite connected planar graph G=(V,E) with strictly positive real edge lengths, there exist a measure space (\Omega,\mathcal F,\mu) and a map f:V\to L_1(\Omega,\mathcal F,\mu;\R) such that

d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y) \qquad(x,y\in V).

Theorem 1.1

In L1 Embeddings of Graphs of Bounded Treewidth, 23 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

For every integer k\ge2 there is a finite constant C_k\ge1 with the following property. If a finite connected graph G=(V,E) admits a tree decomposition with bags of size at most k, then, for every edge-length function \ell:E\to(0,\infty), there is a map F from V into a real L_1 space such that

d_{G,\ell}(u,v) \le \|F(u)-F(v)\|_1 \le C_k\,d_{G,\ell}(u,v) \qquad(u,v\in V).

The constant is independent of the number of vertices, the depth of the decomposition, and all ratios of edge lengths.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

L1 Embeddings of Graphs of Bounded Treewidth

23 September 2026, current edition.

Read it herePDF

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.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

BoundedTreewidthL1

The paper states

Theorem 1.1

In L1 Embeddings of Graphs of Bounded Treewidth, 23 September 2026

For every integer k\ge2 there is a finite constant C_k\ge1 with the following property. If a finite connected graph G=(V,E) admits a tree decomposition with bags of size at most k, then, for every edge-length function \ell:E\to(0,\infty), there is a map F from V into a real L_1 space such that

d_{G,\ell}(u,v) \le \|F(u)-F(v)\|_1 \le C_k\,d_{G,\ell}(u,v) \qquad(u,v\in V).

The constant is independent of the number of vertices, the depth of the decomposition, and all ratios of edge lengths.

The words they share: decomposition, real, map, tree, integer, connected, edge, length, bags, finite, graph, embeddings, number, size, independent, constant.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Bounded-treewidth graph metrics in $L_1$

BoundedTreewidthL1.lean, the release lists it as a main result.

def MainStatement : Prop :=  ∀ k : ℕ, 2 ≤ k → ∃ C : ℝ, 1 ≤ C ∧    ∀ (V : Type) [Fintype V] [DecidableEq V] [Nonempty V]      (G : SimpleGraph V),      G.Connected → HasTreeDecomposition G k →      ∀ ℓ : G.edgeSet → ℝ, (∀ e, 0 < ℓ e) → HasL1Embedding G ℓ Ctheorem main_theorem : MainStatement := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

PlanarL1

The paper states

Theorem 1.1

In Planar Graph Metrics Embed into L1 with Constant Distortion, 23 September 2026

There is a finite universal constant C\ge1 with the following property. For every finite connected planar graph G=(V,E) with strictly positive real edge lengths, there exist a measure space (\Omega,\mathcal F,\mu) and a map f:V\to L_1(\Omega,\mathcal F,\mu;\R) such that

d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y) \qquad(x,y\in V).

The words they share: universal, planar, real, positive, connected, measure, edge, space, finite, lengths, graph, embed, distortion, metrics, constant.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Planar graph metrics in $L_1$

PlanarL1.lean, the release lists it as a main result.

def MainStatement : Prop :=  ∃ C : ℝ, 1 ≤ C ∧    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (length : Fin n → Fin n → ℝ),      G.Connected → IsPlanar G →      (∀ u v, length u v = length v u) →      (∀ u v, G.Adj u v → 0 < length u v) →      ∃ (Ω : Type) (measurable : MeasurableSpace Ω)          (μ : @MeasureTheory.Measure Ω measurable)          (f : Fin n → @MeasureTheory.Lp Ω ℝ measurable _ 1 μ),        ∀ x y, graphDistance G length x y ≤ ‖f x - f y‖ ∧          ‖f x - f y‖ ≤ C * graphDistance G length x ytheorem planar_graph_metrics_embed_L1 : MainStatement := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: Both L1-embedding theorems are stated; the flow-cut-gap corollary is not. Planarity is a hand-rolled topological drawing predicate.

Definitions the group asks readers to compare with the paper: IsPlanar is a custom 'drawing by simple plane arcs’ predicate (injective points, arc u v h : C(unitInterval, ℝ × ℝ) per oriented edge, interiors of arcs of distinct unoriented edges disjoint); HasTreeDecomposition and graphDistance := sInf (range walkLength) are also hand-rolled; both read as standard

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

The gap they name: Only the flow–cut-gap sentence, a classical corollary of the stated embeddings (LLR 1995 / Aumann–Rabani 1998); no flow or cut object appears in either challenge.

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The planar case of the Gupta–Newman–Rabinovich–Sinclair embedding problem asks for a universal distortion bound in $L_1$. The formalized result gives one constant $C$ for every finite connected planar graph with arbitrary positive real edge lengths: its weighted shortest-path metric embeds into real $L_1$ with distortion at most $C$. 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 $L_1$ embedding bound for each bounded-treewidth class. For every bag-size bound $k\ge2$, there is a constant $C(k)$ such that the weighted shortest-path metric of every nonempty finite connected graph with a tree decomposition of bag size at most $k$ embeds into finite-dimensional real $L_1$ with distortion at most $C(k)$. Edge lengths may be arbitrary positive real numbers, and the constant is independent of the graph and decomposition size.

AI-drafted comparisons, with no recorded human review

  • Bounded-treewidth graph metrics in $L_1$ BoundedTreewidthL1

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the treewidth paper: for each bag-size bound k ≥ 2, a constant C(k) bounds the L1 distortion of every weighted graph with a tree decomposition of bag size at most k.

  • Planar graph metrics in $L_1$ PlanarL1

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one constant bounds the L1 distortion of the shortest-path metric of every finite connected planar graph with positive edge lengths.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Bounded-treewidth graph metrics in $L_1$ BoundedTreewidthL1

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 108 s; peak memory 6.3 GiB).11 October 2026Original report

Planar graph metrics in $L_1$ PlanarL1

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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Offer to write an explanationCompare one Lean statement with its paper: PlanarL1, BoundedTreewidthL1Suggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

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  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 089 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 089 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form