this moduli space. If we stretch the neck at the connected sum, and if we obtain in the limit a solution on $(X,\Sigma)$ and a solution on $(S^{4},
    \Psi_{2}^{-})$, then the action of the solutions on these two summands must be (respectively) $\kappa$ and $0$. The moduli space on $(X,\Sigma)$ with action $\kappa$ is zero-dimensional. The moduli space on $(S^{4}, \Psi_{2}^{-})$ with action $0$ consists of a unique $V$-connection $A_{V}$, but the formal dimension of the corresponding moduli space is $-1$, because there is a $1$-dimensional obstruction space $H^{2}(A_{V})$ in the deformation complex. The gluing parameter is $O(2)/V = S^{1}$, so the description of the moduli space on $(X,\Sigma)$ is as the zero-set of a real line bundle $h$ over an $S^{1}$ bundle over the moduli space associated to $(X,\Sigma)$. From this description, we see that $$J(X,\tilde\Sigma,\nu) = \epsilon J(X,\Sigma, \nu)$$ where $\epsilon\in \{0,1\}$ is the evaluation of $w_{1}(h)$ on the $S^{1}$ fibers.

To describe $w_{1}(h)$, let us write the $\SO(3)$ vector bundle $E$ for the connection $A_{V}$ as a sum of three line bundles, $$E = L_{0} \oplus L_{1} \oplus L_{2}.$$ In this decomposition, let $i$, $j$, $k$ be the non-identity elements of $V$, chosen to have the form 