= Solution
Use a spatial <discrete Fourier mode> $u_{k,j}^n=a_ne^{i(k\xi+j\eta)}$. Its recurrence and <amplification polynomial of a multilevel finite difference scheme> are
$$
a_{n+1}=2is\,a_n+a_{n-1},\qquad G^2-2isG-1=0,\qquad
s=\mu(\sin\xi+\sin\eta).
$$
The <polynomial roots> are $G_\pm=is\pm\sqrt{1-s^2}$. For $|s|<1$ both have modulus one and are separated. Since $|\sin\xi+\sin\eta|\le2$, a fixed $0<\mu<1/2$ bounds their separation below by $2\sqrt{1-4\mu^2}$. The two-level <companion matrix> is therefore diagonalizable with uniformly bounded <eigenvector> <matrix> and inverse. To see the uniform bound explicitly, write $V=\begin{pmatrix}G_+&G_-\\1&1\end{pmatrix}$. Its determinant is $G_+-G_-$; all its entries have modulus one, and those of $V^{-1}$ are bounded by the reciprocal of the root separation. The companion matrix powers are $V\operatorname{diag}(G_+^n,G_-^n)V^{-1}$, bounded independently of frequency and step number. <Parseval's identity> converts this into a mesh-independent discrete $L^2$ bound for arbitrary perturbations at the two initial levels.
If $\mu>1/2$, choose $\xi=\eta=\pi/2$. Then $s=2\mu>1$, and one <polynomial root> has modulus $s+\sqrt{s^2-1}>1$. On periodic meshes with the number of points divisible by four, this is an actual grid mode, producing exponential <linear instability>.
At $\mu=1/2$ the same phases give $(G-i)^2$. The <companion matrix> is not a <scalar matrix>, so the double <polynomial root> has a nontrivial <Jordan block>. The solution $a_n=ni^n$ has bounded starting amplitudes but grows like $n$. On a fixed physical time interval $n$ is of order $1/k$, so no mesh-independent <stability> bound exists. Consequently
$$
\boxed{0<\mu<\tfrac12}
$$
is the <stability> interval under the standard two-level definition. Testing only <polynomial root> moduli would misleadingly include the endpoint. A special startup selecting the non-growing branch at that endpoint restricts the perturbations and does not establish <stability> of the full recurrence. This is the <two-dimensional leapfrog stability threshold>.
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