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Math-2632-A1 · revision r1 · on Math-2632

OpenAI claim summary for Math-2632

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: 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
  1. Uniform L1 distortion for bounded treewidth
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Uniform L1 distortion for bounded treewidth

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 each fixed treewidth class has a uniform $L_1$ distortion bound for arbitrary positive weighted shortest-path metrics.

Mathematical statement of the claimed result

Manuscript 1.
For every integer $k\ge2$, the claim gives a constant $C_k<\infty$ such that any finite connected graph whose tree decomposition has every bag cardinality bounded by $k$, with arbitrary strictly positive edge lengths, has a real $L^1$ embedding $F$ satisfying \[ d_{G,\ell}(u,v)\le\|F(u)-F(v)\|_1\le C_kd_{G,\ell}(u,v) \qquad(u,v\in V(G)). \] The constant depends only on bag size, not on vertex count, decomposition depth or edge-length ratios. Bag size $k$ corresponds to treewidth at most $k-1$.

Outline of the argument

For manuscript 1 (family 089; 2026-09-23), consistent cut laws are built directly on decomposition bags and glued by a Markov construction. A common-submeasure transport estimate preserves separation along long bag paths; threshold sweeps and protected anchors keep a useful distance scale alive.

Formal verification

For manuscript 1 (family 089; 2026-09-23), OpenAI lists Lean statements for this assertion: each fixed treewidth class has a uniform $L_1$ distortion bound for arbitrary positive weighted shortest-path metrics.

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