= Uniform power bound for matrix Fourier symbols
{title2=$\sup_{\theta,n\Delta t\leq T}\|G(\theta)^n\|\leq C_T$}
This uniform bound is the actual fixed-time stability requirement for a vector-valued periodic grid evolution. Eigenvalue moduli alone are insufficient for <nonnormal matrices>. A unit Jordan block with an order-one off-diagonal entry grows like $n$, while an entry proportional to $\Delta t$ can give bounded growth on $n\Delta t\leq T$. Uniform eigenvector conditioning or a suitable discrete energy estimate can establish the bound. Mesh-dependent diagonalizations cannot be treated as automatically harmless.
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