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Research / Oct 6, 2026

The Factor-Two Hardness Threshold for Vertex Cover

Can the mathematical statement in The Factor-Two Hardness Threshold for Vertex Cover be proved and verified machine-checked in Lean?

Model / AI: OpenAI (unreleased)

machine-checked (formal proof)self-reported (formal proof)
Oct 6, 2026

Lean formalization in OpenAI math repository (Result 102)

Model / AI: OpenAI (unreleased)

machine-checked (formal proof)self-reported (formal proof)

OpenAI's internal frontier model proved and formalized in Lean the main result of 'The Factor-Two Hardness Threshold for Vertex Cover'. We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.

Source read 2026-10-08. Cataloged from OpenAI's formalization.yaml (Family 102). The Lean development was not rebuilt locally.