Uniform power bound for matrix Fourier symbols (${\sup }_{\theta ,n\Delta t\le T}\Vert G(\theta {)}^n\Vert \le C_T$)

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 $n\Delta t\le T$. Uniform eigenvector conditioning or a suitable discrete energy estimate can establish the bound. Mesh-dependent diagonalizations cannot be treated as automatically harmless.