Unbounded polynomial norm ratios for super-identical pseudospectra

We give explicit nonnegative nilpotent weighted shifts of order $(m+1)^2$
whose singular values agree after every complex scalar shift, but whose norms
under a common polynomial have ratio at least $(4/5)\sqrt m$. Thus no
dimension-independent comparison constant exists. A two-by-two transfer
identity proves equality of the shifted singular values; a residue-class
decomposition gives the norm bounds. The proof uses only finite matrices and
elementary linear algebra. It does not determine the optimal
dimension-dependent bound.