Almost automorphic subshifts with finiteness conditions for the boundary of the separating cover

The solver’s target is exactly Theorem 5.2 in the paper: under the isolation hypothesis for a value (ω,a), construct a sliding block factor Ψ of a non-periodic Toeplitz subshift X_x with the same odometer and with B_Ψ(X) = {ω}. The paper proves this by selecting long words U using Lemma 5.1 and a two-symbol code that marks precisely the occurrences tied to ω, then establishing Ω′ = Ω via divisibility in both directions and pinning down the boundary using equations (6) and (7) (Theorem 5.2 and its proof). The candidate solution achieves the same conclusion via a different local rule (“exception + default”) that reads a fixed p_{l2}-periodic coordinate j0 and overrides it only along the ω-cylinder when the central letter equals a. Minor gaps (address recognizability from a window and a brief justification that the period structure remains the same) can be filled using the paper’s Lemma 5.1 and the odometer-factor argument in the proof of Theorem 5.2, so the model’s method is sound and distinct in construction while yielding the same result. See Theorem 5.2 and the surrounding discussion of the odometer factor and equations (6)–(7) in the paper for the canonical proof and all needed lemmas, including the definition of π_x and address alignment (Proposition 2.3) and the factor-odometer relationship (Proposition 2.1) as summarized in Section 2.2.