Markov Capacity for Factor Codes with an Unambiguous Symbol

The paper’s Theorem 8.1 is correct and rigorously proved via a construction of an auxiliary spoke graph H and a conjugacy ψ, plus a careful “graph diamond”/finite-memory argument for the converse. The candidate solution reproduces the definitions and the residue-set condition, and its high-level intuition aligns with the paper. However, it incorrectly claims that in (1) ⇒ (2) one can make φ|_Z one-to-one onto Y. In general this would force a conjugacy, which the paper explicitly rules out in the typical spoke-graph setting (e.g., when all cycles have length at least 2 and T1 ≠ ∅). Moreover, the candidate’s injectivity proof ignores boundary cases such as 0^∞ and 10^∞, and its “reset word” construction in (2) ⇒ (1) is not justified in full generality. The paper’s proof does not suffer from these issues.