  The proton elastic form factor ratio can be measured either via Rosenbluth
separation in an experiment with unpolarized beam and target, or via the use of
polarization degrees of freedom. However, data produced by these two approaches
show a discrepancy, increasing with $Q^2$. The proposed explanation of this
discrepancy - two-photon exchange - has been tested recently by three
experiments. The results support the existence of a small two-photon exchange
effect but cannot establish that theoretical treatments at the measured
momentum transfers are valid. At larger momentum transfers, theory remains
untested, and without further data, it is impossible to resolve the
discrepancy. A positron beam at Jefferson Lab allows us to directly measure
two-photon exchange over an extended $Q^2$ and $\epsilon$ range with high
precision. With this, we can validate whether the effect reconciles the form
factor ratio measurements, and test several theoretical approaches, valid in
different parts of the tested $Q^2$ range. In this proposal, we describe a
measurement program in Hall A that combines the Super BigBite, BigBite, and
High Resolution Spectrometers to directly measure the two-photon effect. Though
the limited beam current of the positron beam will restrict the kinematic
reach, this measurement will have very small systematic uncertainties, making
it a clean probe of two photon exchange.
