of a face of $\Sigma$. Let $f_{n}$ be a point in a face of $\Psi_{n}$. If a new foam $\tilde\Sigma_{n}$ is constructed from $\Sigma$ as the connected sum $$(X,\Sigma) \#_{f, f_{n}} (S^{4},\Psi_{n}),$$ then the new linear map $J(X,\tilde \Sigma_{n},\nu)$ is equal to the old one in the case $n=0$, and is zero in all other cases.

Again, this is now straightforward.

We consider next a different type of connect sum. Let $\Psi_{2}^{-}$ be the mirror image of $\Psi_{2}$. It has self-intersection number $-1$, Euler number $2$, and one tetrahedral point. In the description of $\Psi_{2}$ in , the surface $R$ is divided into two components by the seams of the foam. Let $f_{2}\in R$ be a point in one of those two components.

\[prop:sum-with-Psi-2-minus\] Let $(X,\Sigma)$ be a foam cobordism with strong marking data $\nu$. Let $f$ be a point in the interior of a face of $\Sigma$. Let $f_{2} \in
    \Psi_{2}^{-}$ be as above. If a new foam $\tilde\Sigma$ is constructed from $\Sigma$ as the connected sum $$(X,\Sigma) \#_{f, f_{2}} (S^{4},\Psi^{-}_{2}),$$ then the new linear map $J(X,\tilde \Sigma,\nu)$ is equal to the old one.

Consider a $0$-dimensional moduli space on $(X,\tilde\Sigma)$. Let $\kappa$ be the action of solutions in 