= Zero-cutoff identification of constrained free energy
{title2=$F_c(M)=\lim_{\Lambda\to0}U_\Lambda(M)$}
In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the <Wilsonian coarse-grained statistical Hamiltonian>. Below the smallest nonzero momentum, the remaining effective energy is $VU_\Lambda(M)$ and its <Boltzmann weight> is precisely the <constrained order-parameter free energy> weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare <statistical Hamiltonian> and an unconstrained equilibrium <free energy>. An analytic polynomial <Landau free energy> further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an <upper critical dimension> can invalidate that analytic approximation.
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