= Solution
Use the <Fourier transform> for the spatial <Cauchy problem>, or its periodic analogue, and regard the two starting levels as independently perturbed data. A <Fourier mode> with spatial factor $e^{im\theta}$ has amplification roots $G$ satisfying
$$
G^2-(2\mu-1)(e^{i\theta}-1)G-e^{i\theta}=0.
$$
Put $c=2\mu-1$ and $G=e^{i\theta/2}q$. Then
$$
q^2-2ic\sin(\theta/2)q-1=0,\qquad
q_\pm=ic\sin(\theta/2)\pm
\sqrt{1-c^2\sin^2(\theta/2)}.
$$
If $|c|<1$, both roots have modulus one and their separation is bounded below uniformly in frequency:
$$
|G_+-G_-|\geq2\sqrt{1-c^2}>0.
$$
The <uniform power bound from separated amplification roots> now controls the two-level companion <matrix> for every time step. Its entries are uniformly bounded, and its eigenvector conditioning is bounded by the reciprocal root gap. The <Parseval identity> transfers this frequency-uniform bound to the spatial $\ell^2$ norm. This proves stability, rather than merely checking each root's modulus.
If $|c|>1$, the frequency $\theta=\pi$ has a root outside the unit disk, so there is exponential instability. If $|c|=1$, the two roots at $\theta=\pi$ coincide on the unit circle. The companion <matrix> is not a scalar <matrix> and has a nontrivial <Jordan block>; its powers grow linearly in the number of steps. Frequencies arbitrarily near that value produce the same lack of a uniform bound for localized Fourier packets, so this also invalidates Cauchy $\ell^2$ stability, not only periodic plane-wave stability. At $\mu=1$ the double amplification root is $-1$, and at $\mu=0$ it is $1$.
Therefore the full two-level stability range for a fixed positive Courant ratio is
$$
\boxed{0<\mu<1.}
$$
The endpoint $\mu=0$ is moreover not a positive time step. Bounds deteriorate as $\mu$ approaches either endpoint; the displayed range is not a uniform claim over ratios arbitrarily close to one. A prescribed starter that removes one special parasitic component can change behavior for selected initial data, but does not establish the requested unrestricted two-level stability.
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