$A'$ be unobstructed solutions on $(X,\Phi)$ and $(X',\Phi')$. Let $U_{A}$ and $U_{A'}$ be neighborhoods of $[A]$ and $[A']$ in their respective moduli spaces. The limiting holonomy of the connections at the tetrahedral point is the Klein $4$-group $V$, whose commutant in $\SO(3)$ is also $V$. So we have moduli spaces of solutions with framing at $t$ and $t'$ in which $[A]$ and $[A']$ have neighborhoods $\tilde U_{A}$, $\tilde U_{A'}$, such that $U_{A} = \tilde U_{A}/ V$ and $U_{A'} = \tilde
    U_{A'}/V$. Gluing theory provides a model for the moduli space on the connected sum with a long neck, of the form $$\tilde U_{A} \times_{V} \tilde U_{A'}.$$ If the action of $V$ on $\tilde U_{A}$ is free and $U_{A'}$ consists of the single point $[A']$, then this local model is a finite-sheeted covering of $U_{A}$ with fiber $V/\Gamma_{A'}$, where $\Gamma_{A'}\subset V$ is the automorphism group of the solution $A'$.

In particular if $A'$ is the smallest energy solution on $(S^{4},\Phi_{2})$, then the fiber is a single point, while for $(S^{4}, \Phi_{3})$ the fiber is $2$ points. For the case of compact, zero-dimensional moduli spaces on the connected sum, these local models become global descriptions when the neck is long, and we 