Threshold Hierarchy for Packet-Scale Boundary Cancellation of Dirichlet Eigenfunctions

We identify geometry–dependent minimal packet scales required for cancellation of boundary correlations of high–frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result is a threshold hierarchy: for zero–mean boundary weights, the energy–weighted packet average of boundary correlation coefficients vanishes once the packet length exceeds a scale determined by the vanishing order of curvature moments of the weight. In particular, the threshold $N_k\mathrm{/}{k}^{1-2\mathrm{/}d}\to \mathrm{\infty }$ suffices when ${\int }_{\mathrm{\partial }\mathrm{\Omega }}w,d\sigma =0$, while a strictly weaker threshold applies when additionally ${\int }_{\mathrm{\partial }\mathrm{\Omega }}H,w,d\sigma =0$, reducing in dimension $d=3$ to the minimal condition $N_k\to \mathrm{\infty }$. The thresholds follow from the boundary local Weyl law. As a structural consequence of the Rellich identity alone, the single–mode share of boundary energy within any sublinear spectral packet is of order $1\mathrm{/}{N}_k$. All estimates are independent of eigenvalue monotonicity and remain stable under eigenvalue crossings.