= Two-level stability at the leapfrog Courant boundary
{title2=$\mu=1:\ a_n=(A+Bn)i^n$}
For the centered two-level advection recurrence, the Fourier roots at Courant number one coalesce at $i$ for phase $\pi/2$. The resulting nontrivial Jordan block admits linearly growing solutions despite both eigenvalues having modulus one. Uniform two-level stability for arbitrary starting perturbations therefore requires the strict inequality $0<\mu<1$. A specially consistent startup can select the exact translating branch at the boundary, which is a restriction on the data rather than a uniform bound on the full update.
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