What does this equation mean?

This is an exact analytic evaluation of the formula derived above for stated, idealized parameters — not a simulation, not a measurement, and not the output of any code run for this article. It is offered as illustrative, and it carries a built-in consistency check: because $v_r$=$\hbar$ $k_{\rm eff}$/m by definition of a recoil kick, the same number equals $\hbar$ $k_{\rm eff}^2$T/m , the ordinary recoil phase used to extract $\hbar$/m in precision atom-interferometry measurements through the general path-integral formalism for atomic interferometry [ 14 ] . The two expressions are algebraically identical, so this is a check that the loop construction reproduces a quantity the field already measures by an independent route, not new physics. Inverting the formula returns $m_{\rm op}$[$\mathcal L_{\rm G}$]=$\hbar$$\Phi$/($v_r$b)=m exactly, because the phase was defined from m in the first place. That circularity is the point of this section: it isolates what an idealized, confound-free evaluation looks like, so the next two sections can show exactly what breaks the idealization.