Although our notation does not reflect this, the connected sum is *not unique* when we summing at a tetrahedral point or a seam. The cause of the non-uniqueness (in the case of the tetrahedral points, for example) is that we have to choose how to identify the $1$-skeleta of the two tetrahedra that arise as the links of $t$ and $t'$.

We consider a connected sum at a tetrahedral point in the case that $(X',\Phi')$ is either $(S^{4},
\Psi_{2})$ or $(S^{4}, \Psi_{3})$.

\[prop:sum-with-Psi-2-3\] Let $(X,\Sigma)$ be a foam cobordism with strong marking data $\nu$, defining a linear map $J(X,\Sigma,\nu)$. Let $t$ be a tetrahedral point in $\Sigma$.

1.  If a new foam $\tilde\Sigma$ is constructed from $\Sigma$ as a connected sum $$(X,\Sigma) \#_{t, t_{2}} (S^{4},\Psi_{2}),$$ where $t_{2}$ is the unique tetrahedral point in $\Psi_{2}$, then the new linear map $J(X,\tilde \Sigma,\nu)$ is equal to the old one.

2.  If a new foam $\tilde\Sigma$ is constructed from $\Sigma$ as a connected sum $$(X,\Sigma) \#_{t, t_{3}} (S^{4},\Psi_{3}),$$ where $t_{3}$ is any of the three tetrahedral points in $\Psi_{3}$, then the new linear map $J(X,\tilde \Sigma,\nu)$ is zero.

Consider a general connected sum at tetrahedral points, as in equation . Let $A$ and 