What was already known
Earlier finite-gap and slow-decay criteria did not cover geometric progressions. A separate OpenAI manuscript established the special ratio q=1/2.
AI involvement
OpenAI authors the geometric-ratio manuscript and credits its internal model with the release. Kan Jiang and Lifeng Xi later explain and extend the finite-tree construction in their own work.
OpenAI institutional authorship and the release's internal-model production account cover the geometric manuscript, without a session-level basis for AI-autonomous.
Importance
This extends the dyadic case to every fixed geometric ratio, while leaving the general Erdős similarity conjecture open.
Limitations
Patrick White reports computational spot checks of the general-ratio argument, not a complete proof check. The Lean formalization covers only the separate dyadic theorem. Jiang and Xi later proved a stronger variable-ratio result by a separately credited argument.
Technical verification
Patrick White reports computational spot checks of the general-ratio mechanism, while expressly treating the all-q theorem as paper-only. The source-reported Lean formalization and independent replay concern a distinct dyadic manuscript, so they do not count as a machine check of this assessed argument. Jiang and Xi give a stronger new human proof but do not certify the original argument. No complete rigorous check of the October theorem is documented.
Human understanding
Jiang and Xi's October 8 human-authored extension explicitly explains the original finite-tree routing, local windows, grid stability, and scale-count mechanism and adapts it to variable ratios. This supports understanding of the central construction, independently of whether they checked every line of OpenAI's proof.
Correction and publication history
This extends the dyadic case to every fixed geometric ratio, while leaving the general Erdős similarity conjecture open.
Read this published revision