Horizontally Stationary Generalized Bratteli Diagrams

The paper proves that for class C (tridiagonal Toeplitz incidence with diagonal a_n≥1, off-diagonals 1, and ∑ 1/a_n < ∞) the ergodic probability tail-invariant measures are exactly the tail-saturation extensions of the odometer measures {μ̂_i : i∈Z}, via an inverse-limit analysis of stochastic matrices G(n,m) (Theorem 4.5) and the finiteness criterion of Theorem 4.1 (with σ(n)=2, so finiteness follows from ∑ 1/a_n < ∞) . The candidate’s solution gives a different, elementary proof using equal weights of cylinders with the same endpoint (tail invariance, cf. Definition 2.6(ii)) to compute the cotransitions P(X_n≠X_{n+1})=2/(a_n+2), then applies Borel–Cantelli to show eventual stabilization and identifies the product structure on the odometer tail, concluding ergodicity via Kolmogorov 0–1. This aligns with the paper’s classification, and its combinatorial steps (tower counts H^{(n)}=∏(a_k+2)) match the ERS row-sum structure used in the paper’s derivation . Hence, both are correct but follow different proof strategies.