\cup D_{n}$$ is a foam in $S^{4}$. This description does not specify the topology of $\Psi_{n}$ uniquely when $n$ is large, because there are different combinatorial configurations of real projective lines. The cases of most interest to us are $\Psi_{0}$ (which is just the real projective plane $R$ in $S^{4}$, with $R\cdot
R=2$), and the foams $\Psi_{1}$, $\Psi_{2}$ and $\Psi_{3}$. The foam $\Psi_{2}$ has a single tetrahedral point where $\partial D_{1}$ meets $\partial D_{2}$ in $R$, while $\Psi_{3}$ has three tetrahedral points.

The formal dimension of the moduli space of anti-self-dual bifold connections of action $\kappa$ on $(S^{4}, \Psi_{n})$ is given by $$8 \kappa - (1 - n/2)^{2}.$$ In particular, we have

1.  $8\kappa - 1$ for $\Psi_{0}$;

2.  $8\kappa - 1/4$ for $\Psi_{1}$;

3.  $8\kappa $ for $\Psi_{2}$;

4.  $8\kappa - 1/4$ for $\Psi_{3}$;

The dimension formula in general is given in [@KM-jsharp Proposition \[prop:dimension\]], and for foams in $S^{4}$ it reads $$8 \kappa + \chi(\Psi_{n}) + \frac{1}{2}( \Psi_{n} \cdot
            \Psi_{n} ) - \tau(\Psi_{n}) - 3,$$ where the self-intersection number $\Psi_{n} \cdot \Psi_{n}$ is computed face-by-face using the framing of the double-cover of the boundary obtained from the seam [@KM-jsharp]. For $\Psi_{n}$, there is a contribution of $+2$ from 