= Solution
Use the <Witten-Reshetikhin-Turaev invariant> with the source's <normalized Kirby color> on all $2g+1$ surgery components. Since the <signature> of the <linking matrix> is zero, the framing-anomaly factor is one, and the <TQFT partition function> is
$$
Z(\Sigma_g\times S^1)=\frac1{\mathcal D}\,\langle L(\Omega,\ldots,\Omega)\rangle.
$$
All evaluations in this part use the <isotopy normalization of anyon diagrams> and preserve the indicated <blackboard framing>.
Expand the long component as $\mathcal D^{-1}\sum_a d_a a$ and fix its charge $a$. Each pair of auxiliary components is in the <Borromean rings> arrangement with this component. Eliminate such a pair using the <two-strand Kirby projector> from <2/c>: bend the long component to expose its two oppositely oriented portions, viewed as equally oriented strands labeled $a,\bar a$. The first encircling <normalized Kirby color> loop replaces these by a vacuum cup-cap with coefficient $\mathcal D/d_a$. This vacuum connection opens the second encircling loop so that the same <two-strand Kirby projector> applies once more. Removing the resulting vacuum edges and straightening the bends restores the original charge-$a$ strand. Thus the <Borromean handle-pair evaluation> multiplies the bare strand by
$$
H_a=\left(\frac{\mathcal D}{d_a}\right)^2.
$$
There is no residual twist. This is a relative strand evaluation: the final closure of the long component is counted separately. As a normalization check, when $a=1$ the two auxiliary components become two unlinked <normalized Kirby color> loops, each of value $\mathcal D$, giving $H_1=\mathcal D^2$.
The $g$ pairs occupy disjoint portions of the long component, so their factors multiply. Closing that component gives its <quantum dimension> $d_a$. Combining this with its <normalized Kirby color> coefficient $d_a/\mathcal D$ gives
$$
\begin{aligned}
\langle L(\Omega,\ldots,\Omega)\rangle
&=\sum_a\frac{d_a}{\mathcal D}\,d_a\left(\frac{\mathcal D^2}{d_a^2}\right)^g\\
&=\mathcal D^{2g-1}\sum_a d_a^{2-2g}.
\end{aligned}
$$
Applying the overall surgery prefactor yields
$$
\boxed{Z(\Sigma_g\times S^1)=\mathcal D^{2g-2}\sum_a d_a^{2-2g}.}
$$
Finally, the <topological quantum field theory> assigns the identity map to a cylinder over $\Sigma_g$. Closing its time direction takes the <trace> of that identity, so
$$
\boxed{\operatorname{GSD}(\Sigma_g)=\dim V(\Sigma_g)=Z(\Sigma_g\times S^1).}
$$
This independently reproduces <2/b>. For $g=0$ the surgery link is one zero-framed <normalized Kirby color> <unknot>: its value $\mathcal D$ cancels the surgery prefactor and gives $Z(S^2\times S^1)=1$. For $g=1$ the result is the number of simple <anyons>, as required for the <torus>.
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