Constrained polynomial roots and a modulated approach to Schur stability

The paper’s Theorem 5.2 states exactly the criterion the model proves: after substituting the original recurrence into a single coordinate (specifically the first nonzero coefficient index j, though the paper notes any coordinate also works), if the ℓ1-sum of the iterated coefficients ∑|b_i| is < 1, then the original monic polynomial p is Schur-stable. The paper’s proof uses a dynamical embedding plus its ℓ1-condition (Proposition 5.1) to conclude global stability of the iterate implies stability of the original system, and it explicitly remarks the same result holds for substitution in any variable . The model gives two routes: (A) an operator/characteristic-polynomial factorization q_r(z) = (z^r − a_{n−r})p(z), which the paper does not spell out, and (B) the same dynamical tail-embedding argument used in the paper. Aside from a minor inequality slip in the model’s elementary ℓ1 proof, both arguments are sound and reach the same conclusion. The paper’s exposition is complete for its claims and matches the model’s result and its “any coordinate” variant .