Non-Realizability of the Poisson Boundary

The candidate solution tracks the paper’s architecture almost step-for-step: reduction to an ICC quotient via the hyper-FC centre; abundance of switching/superswitching elements along the walk; a non-randomized stopping-time that produces F-switching supports; assembling an Erschler–Kaimanovich ladder forest and showing convergence to its boundary; then constructing a bounded µ_τ-harmonic function h that fails µ-harmonicity via an optional-stopping/Cauchy–Schwarz argument. These are precisely the ingredients of Theorem 1.4 and its proof in Sections 3–5 (including Lemma 3.1–3.3, Lemma 4.6, Prop. 4.7, Cor. 4.8, and Prop. 5.1) and the ICC reduction at the start of §5. Minor imprecision in the model’s description of the final inequality (it’s an average under µ_τ^n, not µ_τ) and omission of the “records eventually simple” condition on p do not affect the core correctness or alignment with the paper’s proof. See the paper’s statements and proofs of Def. 1.2, Thm. 1.4, the ICC reduction, the ladder construction, and the h_n/Cauchy–Schwarz estimate for details .