Uniform boundary observability for periodic waves, I: up to the $\varepsilon^{-1}$ eigenvalue scale

Let $\mathcal{L}_\varepsilon=-\operatorname{div}(A(x/\varepsilon)\nabla)$ be a scalar divergence-form operator on a bounded $C^3$ domain, where $A$ is symmetric, uniformly elliptic, Lipschitz continuous, and periodic. For initial data spectrally truncated at the eigenvalue level $N$, we establish whole-boundary observability and projected exact controllability, uniformly in the range $N\leqδ\varepsilon^{-1}$. This extends the cutoff $N\lesssim\varepsilon^{-1/2}$ obtained by Lin and Shen (J. Eur. Math. Soc. 2022). The key input is a two-sided boundary estimate for fixed-width spectral packets of $\sqrt{\mathcal{L}_\varepsilon}$ at frequencies that satisfy $\varepsilonμ^2\ll1$. Its lower bound combines quantitative Dirichlet corrector estimates with a forcing-stable microscopic boundary Cauchy estimate. Miller's spectral packet criterion and the Hilbert uniqueness method then convert this packet estimate into the corresponding wave results, with constants independent of $\varepsilon$ and $N$.