This proposal presents our plan to make a precision measurement of the
cross section for $\gamma \gamma \rightarrow \pi^0 \pi^0$ via the
Primakoff effect using the GlueX detector in Hall D. The aim is to
significantly improve the data in the low $\pi^0\pi^0$ invariant mass
domain, which is essential for understanding the low-energy regime of
Compton scattering on the $\pi^0$. In particular, the aim is to obtain
a first-ever experimental determination of the neutral pion
polarizability $\alpha_\pi - \beta_\pi$, which is one of the important
predictions of chiral perturbation theory and a key test of chiral
dynamics. Our goal is to measure
$\sigma(\gamma\gamma\rightarrow\pi^0\pi^0)$ to a precision of about
5.1\%, which will determine the combination of
$\alpha_{\pi^0}-\beta_{\pi^0}$ to a precision of 39\%. We expect this
experiment to run concurrently with the previously approved experiment
to measure the charged pion polarizability (CPP)~\cite{CPPexp} in Hall~D.