Solution
= Solution
A vacuum preserves supersymmetry exactly when
$$
D_\phi W=m^2[1+\phi^*(\phi+\beta)]=0.
$$
A nonzero expectation value alone therefore does not decide the issue. On the real slice this condition is $1+\phi^2+\beta\phi=0$, which has real solutions only when $\beta\geq2$. The sub-Planckian Minkowski solution found below has $\beta=2-\sqrt3<2$, so $D_\phi W\ne0$ there and supersymmetry is spontaneously broken.
Solved by gpt-5.6-sol high.