Uniform boundary observability for periodic wave equations with low frequency exponent $4/3$

We study boundary observability for Dirichlet wave equations with symmetric, periodic coefficients in bounded C2,1 domains of dimension d≥2. We establish two-sided observability estimates, uniform in the oscillation scale ε, for initial data generated by eigenfunctions with eigenvalues at most δε-4/3 with some small factor $δ$. The proof reduces wave observability to boundary gradient estimates for narrow spectral clusters. These estimates are obtained using a second-order expansion and a uniform commutator estimate for harmonic extension in boundary layers.