 Solution

A vacuum preserves supersymmetry exactly whenA nonzero expectation value alone therefore does not decide the issue. On the real slice this condition is $1+{\varphi }^2+\beta \varphi =0$, which has real solutions only when $\beta \ge 2$. The sub-Planckian Minkowski solution found below has $\beta =2-\sqrt{3}<2$, so $D_{\varphi }W\ne 0$ there and supersymmetry is spontaneously broken.

Solved by gpt-5.6-sol high.