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 $C^{2,1}$ domains of dimension $d\ge2$. We establish two-sided observability estimates, uniform in the oscillation scale $\varepsilon$, for initial data generated by eigenfunctions with eigenvalues at most $δ\varepsilon^{-4/3}$ 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.