OPEN, in the site's label (page last edited 16 October 2025; proof-claims thread with no claim as of 6 October 2026); the site's remarks record neither result of September 2026, though Pálvölgyi announced the heptagon in the problem's discussion on 23 September 2026. The derived standing departs from the label because the classification of September 2026 postdates the site's last edit and is accepted here on its Lean proof. The standing in the frontmatter derives from the claim pages. OpenAI's classification of 23 September 2026, recorded on its claim page (OpenAI, 2026), answers the question: a finite set is Ramsey exactly when an explicit system of linear equations over the tensor square of the field generated by its coordinates has a solution; the proof is kernel-checked in Lean, built and axiom-audited here, which is the acceptance evidence its claim page lists, and no referee or outside reviewer has examined it. No decision procedure for that criterion is proved. Pálvölgyi's seven-point circle configuration of 20 September 2026, recorded on its claim page (Pálvölgyi, 2026), is a claimed, unreviewed partial result against Graham's conjecture that every spherical set is Ramsey; the accepted classification refutes that conjecture independently through its twelve-point set, built in Lean here. The classification paper also claims to refute the Leader–Russell–Walters conjecture, through its five-point Corollary 7.4 together with Corollary 2 of Leader, Russell and Walters's 2011 paper; the corpus has not built or reviewed that step, so that refutation stands as the release's claim. Four refereed results that decide the classification for classes of sets are recorded as accepted partial claims, each on the refereed publication alone: sphericity as a necessary condition and boxes as Ramsey, on EGMRSS 1973; nondegenerate simplices, on Frankl and Rödl 1990; subsoluble sets, regular polygons and the regular polyhedra, on Kříž 1991; and cyclic trapezoids, on Kříž 1992. The earlier results below give necessary conditions, positive classes and closure operations, and the August 2026 pyramid papers describe the classification as unresolved at their date.