  We develop lower bounds for the energy of configurations in $\mathbb{R}^d$
periodic with respect to a lattice. In certain cases, the construction of sharp
bounds can be formulated as a finite dimensional, multivariate polynomial
interpolation problem. We use this framework to show a scaling of the
equitriangular lattice $A_2$ is universally optimal among all configurations of
the form $\omega_4+ A_2$ where $\omega_4$ is a 4-point configuration in
$\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is
universally optimal among all configurations of the form $\omega_6+L$ where
$\omega_6$ is a 6-point configuration in $\mathbb{R}^2$ and $L=\mathbb{Z}
\times \sqrt{3} \mathbb{Z}$.
