What’s Decidable about Discrete Linear Dynamical Systems?

The paper’s Theorem 4 states the undecidability of the problem “given k, a semialgebraic S, rational M, and a rational hyperplane H, decide whether ∃x∈S with |{n∈N : M^n x∈H}|≥k,” via a reduction from a variant of Hilbert’s tenth problem where the unknowns are required to be distinct natural numbers; it also sketches the key construction that realises (M^n x)_1 as a degree-(d−1) polynomial in n with prescribed roots, and explains how to obtain M from the linear-recurrence representation of u_n=∏(n−y_i) and encode initial conditions via S (see Theorem 4 and its proof sketch, plus the LRS-to-matrix discussion in the surrounding text ). By contrast, the model proposes a different reduction using a unipotent Jordan block and an L-residue trick to ensure distinct times, but its reverse implication incorrectly infers y∈Z from the integrality of r_j=L y_j+j; from r_j∈Z one only gets y_j∈(1/L)Z, not y_j∈Z. Without the paper’s “distinct natural number” variant (or an alternative sound integrality enforcement), the model’s reduction does not conclude H10 over Z. Hence the paper’s claim stands while the model’s solution has a fatal gap.