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Math-413-A2 · revision r1 · on Math-413

OpenAI claim summary for Math-413

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 at unsolvedmath.com: 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.

Planar graphs and each fixed treewidth class satisfy the claimed embedding bound. The problem includes every proper minor-closed class, so a bound only for those two settings does not establish its universal quantifier.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. GNRS Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

GNRS Conjecture

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; In manuscript 2 (family 089; 2026-09-23), the asserted result is that arbitrary positive edge lengths on finite planar graphs admit a common constant-distortion embedding into real $L_1$.

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 $k-1$.
Manuscript 2.
Assign arbitrary positive real lengths to the edges of any finite planar graph that is connected. The claim supplies a real $L^1$ embedding and one constant $C<\infty$, common to all these graphs, satisfying \[ d_G(x,y)\le\|f(x)-f(y)\|_1\le C d_G(x,y)\quad(x,y\in V(G)). \] No dependence on vertex count, particular planar embedding or edge-length ratios occurs in $C$.

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.

For manuscript 2 (family 089; 2026-09-23), measurable labels define cut distances whose upper comparison holds deterministically. Random chart changes, portals, and component ancestry supply a lower separation probability, and packing control permits the cut families to be assembled at all scales.

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.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 2 (family 089; 2026-09-23), OpenAI lists Lean statements for this assertion: arbitrary positive edge lengths on finite planar graphs admit a common constant-distortion embedding into real $L_1$.

This coverage does not establish a formal proof of the complete problem as stated here.

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