What was already known
Earlier work proved sphericity necessary and established positive cases such as simplices and selected symmetric configurations. A seven-point circular counterexample refuted the claim that sphericity alone suffices; no complete classification was available in the manuscript’s account.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Supplies a necessary-and-sufficient algebraic test for every finite Euclidean Ramsey configuration.
Limitations
The tensor criterion relies on exact coordinate-field relations, not numerical approximation to arbitrary real coordinates. It concerns congruent copies at the original scale under arbitrary finite colorings.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A finite Euclidean configuration is Ramsey for monochromatic congruent copies at its original scale exactly when its coordinate field admits the paper’s rational-tensor matrix satisfying the pointwise evaluation identities and identity-matrix product identities; the criterion covers every affine dimension.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision