An exact three-point amplification theorem

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 is one three-dimensional reflection tensored with an identity, up to basis changes within spectral eigenspaces. The proof-critical computations use only integer and rational arithmetic; numerical optimization serves only to find the certificate. Amplification invariance is also proved for spectra on parallel lines, on perpendicular lines or on concentric circles, and whenever the matching distance is below half the least eigenvalue gap of one matrix. An amplification-independent lower bound and a noncommutativity condition on a smallest counterexample are also established.