Borromean handle-pair evaluation ($H_a={\mathcal{D}}^2/d_a^2$)

For a charge-$a$ component threading a pair of zero-framed auxiliary components in the standard Borromean rings surgery arrangement, color both auxiliary components by the normalized Kirby color. Relative to the bare $a$ strand, eliminating this pair contributes $H_a={\mathcal{D}}^2/d_a^2$. The two-strand Kirby projector applied to the two oppositely oriented portions of the $a$ component supplies a factor $\mathcal{D}/d_a$ at each of the two eliminations. This local evaluation underlies the surgery calculation of the handle-cut formula for anyon ground-state degeneracy.