Polynomial Shape Adic Systems Are Inherently Expansive

The paper proves inherent expansiveness for polynomial-shape Bratteli–Vershik systems by contradiction via: (i) finding, for large i, an initial uncovered link between two level-(i+1) vertices visited by a hypothetical depth-i pair (Lemma 6.1), (ii) extending the link to a long straight distinguished chain of uncovered splitting vertices (Link Lemma 5.4), and (iii) using a simple coordinate/combinatorial observation at level i+2 (Lemma 6.2) plus the Vershik rule to force a contradiction; see Theorem 6.3 for the main statement and proof outline. These ingredients and their roles are visible in the paper’s statements and proof scaffolding: the main theorem and its construction of a distinguished chain (Theorem 6.3), the Link Lemma (Lemma 5.4), the chain-starting lemma (Lemma 6.1), uncovered-region facts (Lemma 4.9 and Corollary 4.10), and the final Vershik-based contradiction at level i+2. The candidate solution mirrors this structure closely: it defines polynomial-shape diagrams, codings, depth pairs, covered/uncovered vertices, constructs links and distinguished chains within uncovered regions, invokes a “lattice-arithmetic lemma” (the paper’s Lemma 6.2) to ensure an extra source in a shared vertex, and concludes expansiveness via contradiction. The only substantive gap in the candidate is the lack of an explicit bound N (the paper gives N = 4dq + 6q), but otherwise the logic and steps align. Citations: main theorem and chain construction ; Link Lemma 5.4 (statement and use) ; chain-starting Lemma 6.1 and its hypotheses ; uncovered-source-set results Lemma 4.9 and Corollary 4.10 ; definition of polynomial shape (Defs. 3.1–3.3) ; the concluding contradiction at level i+2 via Vershik dynamics and chain alternation .