TOEPLITZ SUBSHIFTS OF FINITE RANK

The paper proves all four claims (A)–(D): (A) a rank-2 Toeplitz subshift that is not strong rank-2 is constructed; the obstruction is the uniqueness of the centered left-asymptotic pair in any strong rank-2 constant-length, primitive, proper, recognizable 2-letter S-adic system (Lemma 3.7), contradicted by an explicit left-asymptotic pair with points that are aperiodic at −1 (Claim 3.11), yielding Theorem 3.9 (see the statement of Lemma 3.7 and its use in the proof around Theorem 3.9 , , and the wrap-up ). (B) Strong rank-2 Toeplitz subshifts are characterized via “strong rank-2 cuts” (Theorem 4.6) and shown to be generic (Theorem 4.9); the argument uses language-theoretic unique-building criteria and the universal-scale comeagerness of rank-2 Toeplitz systems (e.g., Theorem 4.6 and its proof; the genericity theorem and setup via Pavlov–Schmieding) , , . (C) On the space T2 of (aperiodic) rank-2 Toeplitz subshifts, conjugacy is hyperfinite (Theorem 5.8), and flip conjugacy and bi-factor are hyperfinite via finite-index arguments (Corollaries 5.9–5.10); non-smoothness is proved by E0 ≤_B reductions using single-hole systems (Lemma 5.11; Proposition 5.12) , , . (D) The automorphism group of any finite-rank Toeplitz subshift is Z ⊕ C with C finite cyclic (Proposition 7.1), and every finite cyclic C occurs for a strong finite-rank Toeplitz subshift (Theorem 7.3; scale analysis via Lemma 7.4) , , . The candidate solution’s outline mirrors the paper’s proofs and references these same intermediate lemmata and theorems. One minor nit: the candidate briefly states the scale is the “product of directive lengths,” whereas the paper’s precise statement is that the scale is lcm(d_n), where d_n = gcd(|τ[0,n+1)(a)|) (Lemma 3.8); for constant-length sequences this specializes to the supernatural product of the constant lengths along the directive, so the candidate’s phrasing is harmless in the strong (constant-length) setting but not the general form (cf. Lemma 3.8) .