BRATTELLI–VERSHIK MODEL FROM BASIC SET

The paper proves exactly that every basic set B in an invertible zero‑dimensional system arises as the set of minimal paths in a Bratteli–Vershik model, by (i) building a Kakutani–Rokhlin (K–R) refinement from any quasi‑section (Theorem 3.3) and (ii) passing from the K–R refinement to an ordered Bratteli diagram with a Vershik homeomorphism and ϕ(B)=E_{0,∞,min} (Proposition 4.3), yielding Theorem 4.4 . The candidate solution mirrors this construction (nested clopen sections → K–R towers → ordered Bratteli diagram → Vershik map → identification of B with minimal paths). The only fix needed is a minor technicality ensuring the chosen clopen neighborhoods are nested so that U_{n+1}⊂U_n; with this adjustment, the model’s proof is sound. Coverage of X by towers from a complete section is standard (cf. Lemma 2.1) .