 Solution

Sum the diagonal spinor entries of the $\mathcal{N}=1$ Super-Poincaré algebra, obtainingThe expectation in any state is thereforeThis 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 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.