= Uniform power bound from separated amplification roots
{title2=$\inf_\theta|g_1(\theta)-g_2(\theta)|>0$}
For a uniformly bounded two-level Fourier companion <matrix> $C(\theta)$ with distinct roots $g_1,g_2$ of modulus at most one, its powers satisfy
$$
C^n=\frac{g_1^n(C-g_2I)-g_2^n(C-g_1I)}{g_1-g_2}.
$$
A positive frequency-uniform gap therefore gives a uniform bound on every power. The <Parseval identity> then supplies spatial <L2 norm> stability. When the gap closes at a repeated unit root, a <Jordan block> can produce growth proportional to the number of steps despite both roots having modulus one.
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