conclude that the moduli space whose point-count defines the map $J(X,\Sigma, \nu)$ is unchanged in the first case and becomes double-covered in the second case. In the second case, the new map is zero because we are working with characteristic $2$.

The next proposition considers similarly the results of connected sum at seam points, where one of the summands is $\Psi_{1}$, $\Psi_{2}$ or $\Psi_{3}$.

\[prop:seam-sums\] Let $(X,\Sigma)$ be a foam cobordism with strong marking data $\nu$, as in the previous proposition. Let $s$ be a point in a seam of $\Sigma$. For $n=1,2,3$, let $s_{n}$ be a point on a seam of $\Psi_{n}$. If a new foam $\tilde\Sigma_{n}$ is constructed from $\Sigma$ as the connected sum $$(X,\Sigma) \#_{s, s_{n}} (S^{4},\Psi_{n}),$$ then the new linear map $J(X,\tilde \Sigma_{n},\nu)$ is equal to the old one in the case $n=2$, and is zero in the case that $n=1$ or $n=3$.

The proofs are standard, modeled on the proofs in the previous proposition.

Next we have a proposition about connected sums at points interior to faces of the foams.

\[prop:face-sums\] Let $(X,\Sigma)$ be a foam cobordism with strong marking data $\nu$, as in the previous propositions. Let $f$ be a point in the interior 