Exact orbit distance when one spectrum has two points

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 $t$ of one of the two eigenvalues of $A$ and the closed $t$-disks about them contain at least as many eigenvalues of $B$ as their multiplicities. Necessity follows from the two-eigenspace decomposition of $A$ and a dimension count; sufficiency from an explicit matching realized by aligning eigenbases. Block repetition multiplies counts and multiplicities by $k$ and leaves coverage unchanged, so $\delta_{nk}(I_k\otimes A,I_k\otimes B)=\delta_n(A,B)$ for every $k$ on this class, in which the other spectrum may be non-real and non-collinear.