ALGEBRAIC CHARACTERIZATION OF DENDRICITY

The uploaded paper’s Theorem 1 states exactly the equivalence among (i) dendricity, (ii) all return sets being tame bases, and (iii) all return sets being free bases, and proves the nontrivial converse (iii)⇒(i) via Theorem 8 and a connectedness/complexity argument. It also sources (i)⇒(iii) (Return Theorem) and (i)⇒(ii) (tameness) from earlier literature, synthesizing them into the equivalence . The candidate solution asserts the same equivalence: (i)⇒(iii) by the Return Theorem, (i)⇒(ii) via known tame-basis results (framed through S-adic/tame automorphisms), (iii)⇒(i) by citing the 2024 result, and (ii)⇒(iii) trivially. The only minor nuance is that the model labels the tameness refinement as part of a “strong Return Theorem,” whereas the paper explicitly attributes tameness to [4] (Maximal bifix decoding) and not to the Return Theorem itself. Substantively, both are aligned, and the logical steps and hypotheses (minimality, uniform recurrence, left/first return words) match the paper’s framework and definitions .