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Math-2597-A1 · revision r1 · on Math-2597

OpenAI claim summary for Math-2597

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.

Algorithmic strong thin trees: The manuscript’s result handles finite multigraphs on at least one vertex, k>=1, all cuts and one absolute C. It returns a tree in deterministic time polynomial in total binary encoding, explicitly allowing binary parallel multiplicities and the empty one-vertex tree. Loops are excluded by the paper's multigraph convention.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

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

Manuscript

Contents
  1. Deterministic polynomial construction of strong thin spanning trees
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Deterministic polynomial construction of strong thin spanning trees

OpenAI's claim

Algorithmic strong thin trees: The algorithmic manuscript claims a deterministic procedure polynomial in the binary input length that returns a $C/k$-thin spanning tree in a finite $k$-edge-connected loopless multigraph. Binary-encoded parallel multiplicities and the one-vertex case are included.

Mathematical statement of the claimed result

Mathematical claim 1.
Given a finite $k$-edge-connected undirected multigraph $G$ and integer $k\ge1$, a deterministic algorithm is claimed to return a spanning tree $T$ using a number of bit operations polynomially bounded by the full input encoding length such that \[|\delta_T(S)|\le(C/k)|\delta_G(S)|\quad(\varnothing\ne S\subsetneq V(G)),\] where $C$ is absolute and $\delta_G(S)$ is the edge cut. The bound applies to explicitly listed edges and binary-encoded multiplicities; for one vertex the output is the empty tree.

Outline of the argument

Algorithmic strong thin trees: An effective metric-cone hierarchy constructs a rational shortcut form and many low-resistance candidate edges. A second frame preserves the ranks needed for spanning-tree packing, and rational padding supplies signing certificates without irrational whitening. Iterated simultaneous sparsification controls every cut and packing inequality, with a separate reduction handling binary multiplicities.

Formal verification

Family 174 describes Lean coverage for thin-tree existence and the companion deterministic construction with binary multiplicities.

Algorithmic strong thin trees: The manuscript links AlgorithmicThinTrees.lean, StrongThinTree.lean (declaration OAI.StrongThinTree.strongThinTree).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources