  An operatorial polynomial polyhedron is a set of the form
$$B_{\delta}(\mathcal{B}(\mathcal{H}))=\{X\in \mathcal{B}(\mathcal{H})^d :
\Vert\delta(X)\Vert<1\}$$ where $\mathcal{B}(\mathcal{H})$ denotes the space of
bounded operators on a separable Hilbert space $\mathcal{H}$, and $\delta$ is a
matrix of polynomials in $d$ noncommuting variables. These sets appear
throughout the literature on noncommutative function theory. While much of what
has been written involves matricial polynomial polyhedra, there do exist
$\delta$ such that the associated $B_{\delta}(\mathcal{B}(\mathcal{H}))$ is
non-empty but contains no matrix points. Algebraicity of operatorial
noncommutative functions has been established in the case that the domain
$B_{\delta}(\mathcal{B}(\mathcal{H}))$ is a balanced set (hence contains the
matrix point 0). In this paper, we dispense of such assumptions on the domain
and prove that an operatorial noncommutative function on any
$B_{\delta}(\mathcal{B}(\mathcal{H}))$ is weakly algebraic in the sense that
its value at each operator tuple $Z$ lies in the weak operator topology closure
of the unital algebra generated by the coordinates of $Z$.
