= Homogeneous vacuum minimum criterion
For a scalar theory with positive kinetic and gradient energy, a potential attaining a global minimum, and boundary conditions admitting a constant minimizing field, the energy is bounded below by the integral of that potential minimum. A static constant field at the minimum attains the bound. A potential unbounded below, such as $-\phi^4$, invalidates the existence claim; incompatible boundary values can require an inhomogeneous minimizer.
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