COBOUNDARIES AND EIGENVALUES OF FINITARY S-ADIC SYSTEMS

The paper’s Theorem 6.6 proves exactly the constant-length S-adic eigenvalue characterization the candidate solves: (i) necessity of h = lim_n λ^{Q_n} and a constant coboundary under a fully essential 2-letter word, (ii) sufficiency under finitary plus (one-sided prefix-straight & recognizable) or (two-sided straight & recognizable), with all continuous eigenvalues rational and MEF the odometer (Zh̃,(q_n), +1) where h̃ is coprime to every q_n and divides ∏(q−1). This appears verbatim in the paper (Theorem 6.6; and its proof sketch, including the MEF statement) . The model’s solution uses the same tower/odometer and coboundary machinery as the paper (Sections 2.4, 4.1–4.3, 6.1), and reconstructs a continuous eigenfunction via tower addresses/recognizability in the same spirit (compare Theorems 4.4/4.5/4.11/4.12 and the constant-length specialization) . Minor differences: the model’s sufficiency step glosses convergence details that the paper secures either via rationality in the constant-length case (Lemma 6.2 within Theorem 6.6) or via summability criteria in the general S‑adic case (Theorem 4.8) . Overall, arguments match in structure and conclusions.