Let $A\in\C^{d\times d}$ and let denote its numerical range. For a bounded convex domain $\Omega\subset\C$ with boundary containing $\spec(A)$, consider the operator-valued boundary kernel \[ P_{\Omega}(\sigma,A)\;:=\;\Real\!\Bigl(n_{\Omega}(\sigma)\,(\sigma\Id-A)^{-1}\Bigr), \qquad \sigma\in\partial\Omega, \] where is the outward unit normal at . For convex with this kernel is strictly positive definite on and underlies boundary-integral functional calculi on convex domains. We analyze the opposite limiting regime . Along any convex exhaustion , if approaches with convergent outward normals and $\sigma_0\notin\spec(A)$, then and the corresponding min-eigenvectors converge (up to subsequences and phases) to the canonical subspace $(\sigma_0\Id-A)\mathcal M(n)$ determined by the maximal eigenspace of $H(n)=\Real(\overline{n}A)$. Quantitatively, we obtain two-sided bounds in terms of an explicit support-gap scalar, yielding a linear degeneracy rate under bounded-resolvent hypotheses and an explicit rate for outer offsets . For normal matrices we compute the eigenvalues of explicitly, showing that degeneracy may fail at spectral support points unless the supporting face contains multiple eigenvalues.