= Solution
Sum the diagonal spinor entries of the $\mathcal N=1$ <Super-Poincaré algebra>, obtaining
$$
\boxed{H=\frac14\sum_{\alpha=1}^2\{Q_\alpha,Q_\alpha^\dagger\}.}
$$
The expectation in any state is therefore
$$
\langle H\rangle=\frac14\sum_\alpha\bigl(\|Q_\alpha|v\rangle\|^2+\|Q_\alpha^\dagger|v\rangle\|^2\bigr)\ge0.
$$
This proves <energy positivity in global supersymmetry>, with the energy origin fixed by the algebra.
If a <ground state> is invariant under <supersymmetry>, all its <supercharge> variations vanish and its energy is zero. Conversely, zero energy makes every nonnegative squared norm above zero, so every <supercharge> annihilates that vacuum. Thus \b[strictly positive vacuum energy implies supersymmetry breaking]: some <supercharge> fails to annihilate the vacuum. A finite-volume regulator can be used before passing to vacuum-energy density. This is a statement about the global <supersymmetry algebra>; it is not a positivity theorem for the general <supergravity F-term potential>.
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