Coboundaries and eigenvalues of morphic subshifts

The paper’s Theorem 5.1 gives an exact characterization of additive eigenvalues for morphic subshifts as (W ∩ Δ_{Mσ}) v under the explicit hypotheses that σ is primitive and aperiodic and τ is recognizable in Xσ, with W and Δ_{Mσ} defined via the coboundary space of Xσ and the non‑contracting part of Mσ; the proof is routed through Proposition 5.2 and the S‑adic coboundary machinery of Section 4, and is internally consistent . The candidate solution states the same conclusion but (i) builds C using coboundaries of X_{τσ^ω} instead of Xσ, contrary to the paper’s setup, (ii) omits the aperiodicity hypothesis, and (iii) makes an incorrect step in the ⇐ direction by claiming that contraction implies δM_σ^N ≡ 0 mod 1 for some N, which does not follow; the paper instead obtains the requisite integrality/summability via Proposition 5.2 and Theorems 4.1–4.2 . Hence the paper is correct; the model’s proof is flawed/incomplete in key steps.