MULTIBAND LINEAR CELLULAR AUTOMATA AND ENDOMORPHISMS OF ALGEBRAIC VECTOR GROUPS

The paper’s Theorem F states exactly the four items (i)–(iv) the model set out to prove, and the paper establishes them via a trace-based embedding ι built from a compatible family of normal basis generators and relative traces; see the statement of Theorem F and its proof, including Proposition 1.2.2 for (i)–(ii) and the argument for (iii)–(iv) using the skew-polynomial classification and a subshift-of-finite-type lemma . The model’s construction of ι via a global Fp-linear functional L agrees in spirit with the paper’s approach (L is, in effect, tr(α·) for a suitable α coming from an inverse-limit normal-basis choice, cf. Proposition 1.1.2 ), and its steps (i)–(iii) are essentially correct. However, the model’s Step 5 (σ confined ⇒ gσ confined) is incomplete: it assumes away the possibility of non–shift-periodic fixed points by a shifting argument, but that argument requires a structural input the model omits. The paper supplies precisely this via Lemma 2.1.1 (a subshift-of-finite-type lemma) to conclude finiteness and hence confinedness for gσ from confinedness of σ . Consequently, the paper’s proof is correct, while the model’s proof has a substantive gap in Step 5.