Estimate the spectrum of affine dynamical systems from partial observations of a single trajectory data

The paper proves, for discrete-time affine systems, that q_{ASΩ,w} is uniquely determined from {SΩ x_t} and gives the per-eigenvalue equivalence (1)⇔(2)⇔(3); it reduces affine to homogeneous via first differences and argues through Jordan/Krylov decompositions (Theorem 3) . The candidate solution follows the same reduction, but develops uniqueness via the recurrence-operator ideal on Δy_t and proves the equivalences using a shift-operator elimination argument; this is consistent with the paper’s claims (Prop. 1, Lemma 4, Theorem 1/3) though presented through different mechanics . I find both correct; the model’s proof is a valid alternative emphasizing sequence recurrences and difference operators.