Subshifts of finite type and matching for intermediate β-transformations

The paper proves that SFT implies matching and that on each fixed-β fiber the matching parameters can be approximated by SFT parameters (yielding density equivalence), and it supplies an explicit example with matching but not SFT. The candidate solution states exactly these results and explains the lexicographic characterization; its proof sketch uses a finite-memory/lex-bound intuition, whereas the paper’s proof uses periodic kneading invariants and kneading determinants. Hence both are correct, with different proof styles.