{ "abstract": "A Lean 4 development around the tensor criterion of the classification of finite Euclidean Ramsey configurations (openai/math, family 172). With the criterion's definitions restated character for character from that project's challenge file: every set of six distinct points on a circle satisfies the criterion, by an explicit certificate (the product of the two bilinear forms through the first three and the last three points of a homogeneous parametrisation of the circle, scaled and conjugated by the parametrisation); the sharp dimension theorem: every spherical set of at most m + 4 distinct points, m the dimension of its affine span, satisfies the criterion in its own space and coordinate field (either the augmented vectors admit real isolating quadrics and the sphere lifts by the adjugate, or a dependence among them is supported on six concyclic points while every other point adds a dimension, and the six-point certificate is carried into the ambient space, symmetrised, adjusted to the common sphere, extended point by point and descended from R tensor R to F tensor F), so every spherical set of at most six points satisfies it in any dimension with any affine span; every spherical set with algebraic coordinates satisfies it, of any size, by the separability idempotent of F tensor F; the criterion always implies sphericity, so for algebraic coordinates it is exactly sphericity. Given the sufficiency half of the classification as an explicit hypothesis (quantified as the source states its Theorem 1.1), every spherical set of at most six points, every spherical set of at most m + 4 points and every spanning spherical set with algebraic coordinates is Ramsey; given both halves, a spanning set with algebraic coordinates is Ramsey if and only if it is spherical. The source proves the Ramsey property for at most five circle points; six is the first case its method of independent quadratic rows cannot reach. With the seven-point non-Ramsey circle set of A cyclic non-Ramsey heptagon (arXiv:2609.23327), the least size of a non-Ramsey spherical set is seven, given the classification, and, for m at least 2, m + 5 the least size of one with affine span of dimension m, since the heptagon with m - 2 further sphere points fails the criterion and is not Ramsey; the research note in note/ states this, verifies by an exact derivation identity that the heptagon fails the criterion, and proves the sharpness on paper (nothing in Lean says that a set fails the criterion). The classification is not proved here. The restated definitions and a few adapted lemmas derive from Apache-2.0 work of OpenAI (NOTICE, LICENSES/Apache-2.0.txt); the rest is MIT.", "authors": [ { "name": "JD Jones" } ], "challenge_render": { "artifact_path": "renders/PALOMAR-2026-10-07-000011-v1/1349f50b76e2368e53b336a7dbcbd1aa29a13835f3b9d0d058e9b6bc85e3a935/", "artifact_tree_sha256": "1349f50b76e2368e53b336a7dbcbd1aa29a13835f3b9d0d058e9b6bc85e3a935", "bwrap_source_tag": "v0.12.0", "entrypoint": "Challenge/index.html", "format": "verso-html", "rendered_at": "2026-10-07T15:55:51Z", "renderer_commit": "d4e41c1d5b0d114c4859e6e5831dc6d3ad1d0d44", "verso_commit": "9f8096e40b31715b1d8d5997f15a0bd832f7e37d" }, "classification": { "arxiv": [ "math.CO", "math.MG" ], "msc2020": [ "05D10", "52C10" ] }, "first_registered_on": "2026-10-07", "formalization": { "challenge_path": "Challenge.lean", "comparator_config_path": "comparator.json", "definition_names": [ "SixConcyclic.Space", "SixConcyclic.Congruent", "SixConcyclic.Ramsey", 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