{ "abstract": "Theorem D: for real theta, the nine points (cos k theta, sin k theta), k = 0, ..., 8, of the Euclidean plane are distinct, lie on the unit circle and are not Euclidean Ramsey when tan(theta/2) is transcendental; and whenever they are distinct, they are Ramsey if and only if cos theta and sin theta are algebraic (that is, iff e^(i theta) is algebraic), if and only if tan(theta/2) is algebraic. This is an iff for a whole one-parameter family, where earlier explicit examples with a transcendental parameter (OpenAI's Proposition 7.7 and Corollary 7.5) settle single sets: the same nine-point pattern is Ramsey or not exactly as e^(i theta) is algebraic or transcendental. Nine is the exact threshold (eight consecutive multiples are Ramsey for every theta; by hand, in spherical-ramsey), and at theta = e the question becomes whether e^(i e) is algebraic, which appears to be open. Ramsey is OpenAI's definition from result 172 (for every r >= 2 there is D such that every r-colouring of R^D contains a monochromatic congruent copy). The statements are given in OpenAI's vocabulary plus the two-line definition of the nine points, and again with no definitions beyond Mathlib. The non-Ramsey direction mirrors OpenAI's nine-point proof (their Proposition 7.6): the coordinate field is Q(tan(theta/2)), its tensor square embeds in Q(T, U), and the denominator-cleared 9x9 evaluation matrix is nonsingular, certified at (T, U) = (2, 3) by an explicit integer triangular factorization checked by decide +kernel; OpenAI's linear obstruction concludes. The converse uses the theorem that finite spherical sets with algebraic coordinates are Ramsey (prior work of JD-Jones-ASES, Rogerhu12 and atimics; proved again here in Lean on OpenAI's development). The proof builds on OpenAI's complete Lean proof of the classification (openai/math fd4aeeb2), whose 35 files used here are included with header-only changes for the Lean module system. This repository is a module-system port, with no statement or proof line changed apart from a namespace rename, of Lean already public as rope AV3 of spherical-ramsey v1.1.0 (doi:10.5281/zenodo.23396387), packaged so that Theorem D can be checked on its own. 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