Remark. What the structure theorem leaves unresolved at rank deficiency [ftip-00BJ]
Remark. What the structure theorem leaves unresolved at rank deficiency [ftip-00BJ]
In the ambient matrix space, full-column-rank matrices form an open dense subset whose complement has Lebesgue measure zero. A probability-one claim requires the random gradient law to be absolutely continuous and not confined to a lower-rank set; random initialization alone does not supply that premise.
Theorem [ftip-00BH] therefore makes no classification claim on a general rank-deficient stratum. Equivariance still constrains values within each orthogonal orbit, and it forces \(\Phi (0)=0\), but it does not determine the rule there by continuity unless continuity is separately assumed. The source's generic-rank discussion therefore does not establish a classification at rank deficiency.