What does this equation mean?

Magdalena Zych and Časlav Brukner’s 2018 paper supplies the anchor the new work builds directly on top of: a classical body’s equivalence principle, they argue, says nothing at all about whether the quantum version holds, because a quantum system’s mass is not one number but an operator, and equivalence has to be stated as equivalence between its rest, inertial, and gravitational internal-energy operators rather than between their average values [ 1 ] . Model that operator, call it $\hat M$ , as $\hat M$ = m(1 + $\hat H_{int}$/m$c^2$) , where $\hat H_{int}$ is the system’s internal Hamiltonian, and let the gravitational mass be $m_g$ = $m_i(1 + \hat\eta)$ for some violation operator $\hat\eta$ acting on the internal, and for two-body tests the joint, Hilbert space. Expand that operator and, the paper argues, exactly five structurally distinct pieces fall out: