ON THE STABILITY AND SHADOWING OF TREE-SHIFTS OF FINITE TYPE

The paper proves exactly the equivalence claimed—A tree-shift X is of finite type if and only if it has the pseudo-orbit-tracing property (Theorem 3.6)—and does so by the same two-direction argument used by the candidate. The model’s “coherence along words” Fact 2 is the paper’s Lemma 3.2, and the construction s(w)=t(w)(ε) for the finite-type⇒POTP direction is identical to the paper’s proof. For the converse, the model glues local (n+1)-blocks from X along a pseudo-orbit and uses POTP with m=1 to obtain a global s∈X, which matches the paper’s proof up to a harmless off-by-one in the chosen block height (the paper uses n while the model uses n+1, both yielding a finite forbidden list). See Theorem 3.6 and its proof, including the coherence lemma and the finite-type characterization via bounded-height blocks, in the paper’s Section 3 and Lemma 2.4 .