Factorizable embeddings and the period of an irreducible sofic shift

The paper’s Theorem 3.9 proves exactly the criterion the candidate states: for a factor code π: X→Y from an irreducible SFT X of global period p and a subshift Z with h(Z) < h(Y), there exists an embedding Z↪Y factoring through π iff Z is p-periodic and q_{np}(Z) ≤ r_{np}(π) for all n. The necessity matches the candidate’s argument (injectivity on Q_{np}(Z) into Q_{np}(Y) ∩ π(Q_{np}(X))), and the sufficiency in the paper is obtained by constructing W ⊆ X with π|_W injective and periodic-point counts dominating those of Z (via Theorem 3.4), and then embedding Z→W using a generalization of Krieger’s theorem for irreducible SFT targets (Proposition 3.7), finally composing into Y (Theorem 3.9; see the proof excerpt) . The candidate’s proof uses a different but standard reduction: pass to σ^p on cyclic components (mixing SFTs), apply the p=1 case (MacDonald’s theorem) to obtain embeddings that factor through the restricted codes, and then assemble across the cyclic decomposition; this aligns with the structure of the paper (MacDonald’s theorem is recalled as Theorem 1.3) . Minor issues: in the assembly step, the candidate’s displayed formula uses ϕ_i on σ^{-i}z∈Z_0, which is a notational slip; replacing it by Φ(z)=σ^i(ϕ_0(σ^{-i}z)) fixes it. Aside from this small slip, the logic is sound and reproduces the same necessary-and-sufficient conditions, though the paper’s sufficiency proof follows a different construction via an intermediate SFT W and Proposition 3.7 .