= Three-body decay energy endpoints
{title2=$m_1\leq E_1\leq[M^2+m_1^2-(m_2+m_3)^2]/(2M)$}
Use <natural units> $c=1$ and assume open <phase space>, $M>m_1+m_2+m_3$. In the parent's <rest frame>, define $s_{23}=(p_2+p_3)^2$. <Four-momentum conservation> gives
$$
E_1=\frac{M^2+m_1^2-s_{23}}{2M},\qquad (m_2+m_3)^2\leq s_{23}\leq(M-m_1)^2.
$$
The upper recoil-mass limit follows from $E_1\geq m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair <rest frame> and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2E_3(1-\cos\theta_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.
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