Two-level stability at the leapfrog Courant boundary ($\mu =1:a_n=(A+Bn)i^n$)

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.