Local amplification invariance and quantitative rigidity

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 that orbit distance stays strictly below bottleneck eigenvalue matching. The proof transports a strictly positive trace sum-of-squares certificate. Exact rank calculations supply the compatibility this transport requires: the coefficient map has rank $107$ in a $109$-dimensional real trace space, and the unique reference optimality density has eigenvalues $5/28$ and $23/28$. Separately, a dimension-independent normalized Hilbert--Schmidt estimate with constant $2{,}705{,}974$ controls all near-minimizers of the original pair. The neighborhood is existential; no numerical radius is asserted.