= Solution
Let $A(M)=V(M)+hM$ be the zero-field <constrained order-parameter free energy>. At coexistence in a <conjugate field> $h_c$, two global minima $M_1<M_2$ obey
$$
A'(M_1)=A'(M_2)=h_c,\qquad
A(M_2)-A(M_1)=h_c(M_2-M_1).
$$
Thus the same line of slope $h_c$ is tangent to $A$ at both endpoints: the <common-tangent construction for phase coexistence>. Equivalently,
$$
\boxed{\int_{M_1}^{M_2}[A'(M)-h_c]\,dM=0.}
$$
This is the <Maxwell equal-area construction> on the equation-of-state curve $h=A'(M)$: the signed areas on either side of the horizontal coexistence line cancel. Equal slope alone finds stationary phases; the integral additionally enforces equal <free energy>.
If the mean <order parameter> is constrained between the endpoints, the system can separate into two macroscopic phases. With $\lambda=(M-M_1)/(M_2-M_1)$, their bulk <free-energy density> is $(1-\lambda)A(M_1)+\lambda A(M_2)$. The interface contributes subextensively in the <thermodynamic limit>. The resulting <constrained order-parameter free energy> is the convex envelope of $A$, with a straight coexistence segment. This explains why a nonconvex uniform <Landau free energy> must be convexified when describing stable constrained equilibrium, and why a <spinodal point> is not the <Maxwell construction> boundary.
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