GENERIC CONTROLLABILITY OF EQUIVARIANT SYSTEMS AND APPLICATIONS TO PARTICLE SYSTEMS AND NEURAL NETWORKS

The paper establishes Theorem 1.3 (generic controllability in strata of M/G) and Theorem 1.5 (simultaneous controllability in strata of dimension ≥ 2) via a constructive scheme on M: it proves a strong bracket-generation result for equivariant fields on M (Theorems 3.3–3.4), leverages the slice theorem and the structure of strata (Lemma 3.9; Corollary 3.10), and then deduces controllability on MG by Chow–Rashevskii, as stated in Theorem 1.3 and extended to simultaneous control in Theorem 1.5. These statements and their proofs are clearly present and coherent in the PDF . By contrast, the model’s solution sketches a jet-transversality argument on each stratum and on configuration spaces but relies on an unproven key step: that arbitrary finite jets compatible with the quotient structure can be realized by G-equivariant vector fields (and that the parametric jet map from (Vec_G(M/G))^k is sufficiently submersive to apply Thom–jet transversality). Without a precise and verified description of the admissible jet space in the equivariant class and a proof of surjectivity/submersion of the r-jet map, the transversality claim—and hence the generic bracket-generation conclusion—does not follow. Therefore, the paper’s argument is correct, while the model’s proof outline has critical gaps.