  It is well-known that the traditional treatment of radiative corrections that
utilizes the "true" neutrino momentum $\vec{p}_\nu$ in the differential decay
rate formula could lead to a $\sim \alpha/\pi$ systematic error in certain
observables due to the mistreatment of 4-body kinematics. We investigate the
theory structure of one of the proposed solutions, the "$\nu'$-formalism", in
the non-recoil limit appropriate for neutron and nuclear beta decays. We derive
an elegant master formula for the 4-body phase space and use it to re-analyze
the spectrum-dependent "outer" radiative corrections to the beta decay of a
polarized spin-half nucleus; a complete set of analytic expressions is provided
for readers to straightforwardly obtain the final numerical results. We compare
it to the "recoil formalism" where the energy of the recoil nucleus is fixed.
