First-order invariance under independent spectral splitting

Consider an explicit pair of $3\times3$ normal matrices with three distinct eigenvalues each. Its unitary-orbit distance is strictly below the eigenvalue-matching distance, and earlier rounds of this work show it is unchanged by every finite amplification. In an arbitrary finite repetition of this pair, all eigenvalues may be split independently. The first-order change of the squared unitary-orbit distance is exactly the minimum largest eigenvalue of a sum of four Hermitian unitary orbits, with explicit coefficients. A classical Horn block-averaging theorem makes that first-order quantity invariant under every finite amplification. For each fixed base repetition and amplification, the resulting squared-distance gain is $O(t^{3/2})$ as the splitting parameter $t$ tends to zero. The bound is not uniform in the amplification and does not assert equality at any nonzero splitting. The proof combines the earlier quantitative rigidity theorem, an exact positive curvature $1/52$, a local unitary slice and a Schur-complement argument.