Unitary-orbit distance under finite block repetition

For the classical three-point Krause pair $A=\operatorname{diag}(1,\tfrac{4+5i\sqrt3}{13},\tfrac{-1+2i\sqrt3}{13})$ and $B=-A$, an exact trace sum-of-squares certificate proves that the unitary-orbit distance of $A\otimes I_k$ and $B\otimes I_k$ is $\sqrt{27/13}$ for every $k\ge1$. The bottleneck eigenvalue-matching distance is $\sqrt{28/13}$, so the result lies outside the regime in which orbit distance equals spectral matching. The certificate also yields equality rigidity: every optimal amplified unitary …

The three-point pair $A_0=\operatorname{diag}(1,\tfrac{4+5i\sqrt3}{13},\tfrac{-1+2i\sqrt3}{13})$, $B_0=-A_0$, whose unitary-orbit distance is unchanged by every finite amplification, is extended to an open neighborhood in the space of six independently variable complex eigenvalues. Throughout this neighborhood, unitary-orbit distance is unchanged by every finite amplification, and every amplified minimizer is a repeated scalar minimizer up to spectral-block basis changes. The neighborhood may be chosen so …

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 …

For normal $A,B\in\mathbb C^{n\times n}$ with $A$ having at most two distinct eigenvalues (either matrix may play this role), the spectral-norm distance $\delta_n(A,B)=\min_U\|A-U^*BU\|_2$ between their unitary orbits equals the bottleneck distance $d_\infty(A,B)$, the least over bijections of the eigenvalue multisets of the largest matched distance. The common value is the least $t$ at which every eigenvalue of $B$ lies within …