What was already known
The OpenAI classification paper established the criterion and handled at most five circle points; a seven-point circle configuration fails it. This project supplies the previously untreated six-point case and a dimension-dependent sufficient condition for the criterion.
AI involvement
JD Jones directed the work. The disclosure credits Claude with the theorem development and Lean proofs; a GPT model proposed the m+4 extension and proof outline, later formalized with Claude.
JD Jones directed the project; Claude generated the theorem development and Lean proof, while GPT proposed the m+4 extension and outline formalized with Claude.
Importance
Fills the six-point boundary of a new classification method and supplies a broader criterion theorem for small spherical configurations.
Limitations
The selected unconditional result is the tensor criterion, not an unconditional Ramsey theorem. The repository’s Ramsey corollaries take classification sufficiency as an explicit hypothesis and do not prove it. The seven-point sharpness argument is written/certificate-supported, outside Lean. No independent human review is reported.
Technical verification
The source reports fourteen completed proofs with exact definition and statement comparison, standard-axiom audit and local proof build; ramsey consequences retain explicit hypotheses. This applies to the selected claim and is not an independent reproduction by AIProofWatch.
Formal checking
The source reports completed Lean proofs for fourteen compared statements, exact definition matching to the OpenAI criterion, a standard-axiom audit and local build. Ramsey theorems retain the unproved hsuff or hclass argument.
- Checker
- Lean 4 kernel and documented local statement, axiom or certificate audits run by repository authors
- Scope
- Every set of six distinct points on a circle satisfies the FieldCriterion tensor condition of the OpenAI Euclidean Ramsey classification. More generally, every spherical finite set of at most m+4 points, where m is its affine-span dimension, satisfies that criterion in its own coordinate field; every spherical set with algebraic coordinates also satisfies it.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No independent human review or explanatory account by a human of the tensor-certificate mechanism is documented. Few is the evidence default.
Correction and publication history
Claim-level primary-source review of exact theorem, prior baseline, AI contribution and documented checking scope.
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