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 positive definite on and underlies boundary-integral functional calculi and spectral-set bounds in the sense of Delyon--Delyon and Crouzeix.We analyze the opposite limiting regime . Along any convex exhaustion , if approaches a non-spectral boundary point $\sigma_0\in\partial W(A)\setminus\spec(A)$ with convergent outward normals , then and the associated min-eigenvector directions 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 the support-gap scalar $\delta(\sigma,n)=\Real(\overline{n}\,\sigma)-\lambda_{\max}(H(n))$, yielding a linear degeneracy rate under bounded-resolvent hypotheses and an explicit rate for outer offsets . Under a spectral-isolation hypothesis for , we characterize the entire collapsing eigenvalue cluster under non-tangential offsets: exactly eigenvalues decay as with a computable slope spectrum given by the eigenvalues of an explicit Gram matrix , while the remaining eigenvalues stay uniformly bounded away from $0C^1\delta\sigma_0\in\spec(A)\cap\partial W(A)1/\varepsilon\Ker(\sigma_0\Id-A)O(\varepsilon)\mathcal M(n)\ominus\Ker(\sigma_0\Id-A)O(1)0P_\Omega(\sigma,A)$ explicitly, recovering a simple dichotomy at spectral support points in terms of whether the supporting face contains multiple eigenvalues. Finally, we include reproducible numerical experiments (Python) validating the predicted slopes and splittings.