Observability conditions for neural state-space models with eigenvalues and their roots of unity

The paper’s Theorem 4 asserts that if Ψj is diagonal with the stated entries and the pairwise kernel non-containment ker(CVΨj1V−1) ⊄ ker(CVΨj2V−1) holds for all j1 ≠ j2, then the system is observable “with high probability,” but the proof only sketches a Fourier-domain heuristic and invokes a pigeonhole-style intuition without the algebra needed to guarantee that the intersection over j of those kernels is trivial. In particular, pairwise non-containment alone does not imply a trivial intersection, and the proof never imposes a separating sample size condition (such as L ≥ n) or distinctness of the induced spectral parameters that would make the reduction rigorous (see the theorem statement and proof sketches around equations (45)–(47), and the ensuing discussion of “row rank diversity” and the Fourier link in the text and appendix) . By contrast, the model’s solution provides an explicit eigenbasis analysis tied to the PBH/Hautus criterion, proves a clean frequency-separation lemma that requires L ≥ n and pairwise distinct transformed poles, and derives the exact identity ⋂j ker(CVΨjV−1) = V·span{ei : Cvi = 0}. Under these standard, checkable hypotheses, this yields the precise equivalence: (C,A) observable ⇔ ⋂j ker(CVΨjV−1) = {0}, sharpening the paper’s high-probability claim. The paper itself notes only that Cv ≠ 0 is necessary (and not sufficient in the repeated-eigenvalue case), and motivates, rather than proves, the kernel-diversity route to observability . Overall, the paper’s argument is incomplete at key steps, while the model’s proof is correct under clearly stated additional assumptions.