= Solution
Sum the diagonal mixed <anticommutators> in the <Super-Poincaré algebra>. Their spatial terms have zero Pauli trace, giving
$$
\boxed{H=P_0=\frac14\sum_{\alpha=1}^2\{Q_\alpha,Q_\alpha^\dagger\}.}
$$
For every normalized state $|u\rangle$ in the common operator domain,
$$
\langle u|H|u\rangle=\frac14\sum_{\alpha=1}^2\left(\|Q_\alpha|u\rangle\|^2+\|Q_\alpha^\dagger|u\rangle\|^2\right)\geq0.
$$
This proves <energy positivity in global supersymmetry>. The zero of energy is fixed by this algebra, rather than by an arbitrary independent subtraction.
A <supersymmetric vacuum> is invariant under every <supercharge> and its adjoint. Its energy therefore vanishes. Conversely, a normalized vacuum of zero energy makes every nonnegative norm on the right zero, so all supercharges annihilate it. Consequently
$$
\boxed{E_{\rm vac}>0\quad\Longrightarrow\quad\text{spontaneously broken global supersymmetry}.}
$$
For an infinite spatial volume the corresponding statement is about vacuum energy density, obtainable by first regulating the volume. It concerns rigid <supersymmetry>; the gravitational scalar potential in <supergravity> has additional terms and is not generally nonnegative.
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