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Math-2632 · mathematics · Geometry

Uniform L1 distortion for bounded treewidth

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each positive integer k, does a finite C(k) bound the $L^{1}$ embedding distortion of all finite connected graphs with positive real edge lengths and a tree decomposition having at most k vertices per bag? The constant is independent of graph size and lengths, and embeddings may use an arbitrary real $L^{1}$ target and positive scale.

Short English statement

For each positive integer k, does a finite C(k) bound the $L^{1}$ embedding distortion of all finite connected graphs with positive real edge lengths and a tree decomposition having at most k vertices per bag? The constant is independent of graph size and lengths, and embeddings may use an arbitrary real $L^{1}$ target and positive scale.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2632-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2632

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).

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

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