First-forbidden unique beta-decay phase space ($C(E)\propto p_e^2+E_{\nu }^2$)

A first-forbidden unique beta decay has $|\Delta J|=2$ with a parity change. Its leading lepton shape factor contains two additional powers of momentum relative to an allowed beta decay. In the massless, no-Coulomb limit, ${\int }_0^Q{E}^2(Q-E{)}^2[E^2+(Q-E{)}^2]dE=2Q^7/105$. Thus its microscopic reduced nuclear matrix element multiplies a seventh-power phase-space factor, rather than the fifth power in the Sargent rule. A fifth-power comparison across different forbiddenness classes instead extracts convention-dependent effective strengths.