= Solution
The <partition function> sums the <Boltzmann weights> of all statistical configurations. For discrete states $s$ of energy $E_s$, $Z=\sum_se^{-\beta_{\rm th}E_s}$; for a regulated scalar field it is a <functional integral>, $Z=\int\mathcal D\phi\,e^{-\beta_{\rm th}H[\phi]}$. It normalizes probabilities and generates thermodynamic averages. Its <Helmholtz free energy> is
$$
\boxed{F_{\rm tot}=-k_BT\log Z.}
$$
A quantum version uses $Z=\operatorname{Tr}e^{-\beta_{\rm th}\widehat H}$. In the following identification, $F(M)$ is a constrained free-energy density, so its units match those of a Hamiltonian density rather than a total extensive energy.
Fix a microscopic cutoff $\Lambda_0$. Split the field into retained modes $\phi_<$ with momenta below $\Lambda$ and eliminated modes $\phi_>$ between $\Lambda$ and $\Lambda_0$. Define the <Wilsonian coarse-grained statistical Hamiltonian> exactly by
$$
e^{-\beta_{\rm th}H_\Lambda[\phi_<]}=\int\mathcal D\phi_>\,e^{-\beta_{\rm th}H_{\Lambda_0}[\phi_<+\phi_>]}.
$$
Its coefficients and additive constant depend on $\Lambda$ so that integrating the retained modes gives precisely the original $Z$. With sources and observables carried along, the same construction preserves long-wavelength responses. Thus changing this intermediate <ultraviolet cutoff> is a change of description, not a change of the physical system. It is advantageous because one can integrate out fluctuations irrelevant to the observable length scale, organize a local <gradient expansion>, and use <renormalization-group flow> toward simpler universal fixed-point descriptions. Exact invariance retains all generated operators; truncating them is an approximation.
The quadratic gradient coefficient generally becomes $K_\Lambda$, not a constant. <Field renormalization> sets a chosen kinetic normalization, for example $\phi_R=\sqrt{K_\Lambda}\phi$ when this makes the gradient coefficient one in the energy convention. Sources and observables must be transformed inversely so that their physical meaning is preserved. Under rescaled coordinates $x'=x/b$, the critical field scaling is
$$
\phi'(x')=b^{x_\phi}\phi_<(bx'),\qquad x_\phi=\frac{D-2+\eta}{2}.
$$
Its anomalous part is the <anomalous dimension>; it gives $G(r)\sim r^{-2x_\phi}$ and the conjugate-field exponent $y_h=D-x_\phi$. Different conventions for the wave-function factor require corresponding changes in its logarithmic derivative. A fixed physical magnetization must not be confused with a cutoff-dependent dimensionless renormalized field.
To justify the <zero-cutoff identification of constrained free energy>, work first in a finite periodic volume $V$. Define the physical spatial mean $M=V^{-1}\int\phi(x)d^Dx$ and constrain that mean in the microscopic integral:
$$
Z_M=\int\mathcal D\phi\,\delta\left(V^{-1}\int\phi-M\right)e^{-\beta_{\rm th}H_{\Lambda_0}[\phi]},\qquad
F_c(M)=-\frac{k_BT}{V}\log Z_M.
$$
Retain the constant mode $M$ while eliminating every nonzero mode. When $\Lambda$ is below the smallest nonzero box momentum, the remaining energy is $H_\Lambda[M]=VU_\Lambda(M)$. Consequently $e^{-\beta_{\rm th}VU_\Lambda(M)}=Z_M$ with a consistently fixed, possibly field-independent measure normalization. This proves
$$
\boxed{F_c(M)=\lim_{\Lambda\to0}U_\Lambda(M).}
$$
For a uniform field, $U_\Lambda(M)$ is the Hamiltonian density $\mathcal H(\Lambda,M)$ in the question. The statement assumes that the physical zero mode is retained, all other modes are integrated exactly, field-rescaling factors are undone, and density and additive-normalization conventions agree. Taking the cutoff limit in the finite box before its <thermodynamic limit> makes this derivation unambiguous. With an applied uniform source, $Z(h)=\int dM\,e^{-\beta_{\rm th}V[F_c(M)-hM]}$ up to the same measure convention; its large-volume saddle minimizes $F_c(M)-hM$.
The <Landau approximation> further assumes that this potential can be approximated by a regular local polynomial on the homogeneous branches, and that the remaining fluctuations are negligible for the observables sought. It is not an assertion that an exact zero-cutoff free energy is the bare polynomial. In a coexistence region the exact constrained density can be convexified by domains, while a nonconvex Landau branch potential describes local phases and their metastability.
For the breakdown criterion use a short-range scalar theory with positive quadratic gradient stiffness. At the <Gaussian fixed point>, invariance of $\int(\nabla\phi)^2d^Dx$ gives field dimension $(D-2)/2$. A coupling $g_{2n}$ multiplying $\phi^{2n}$ therefore has engineering scaling exponent
$$
y_{2n}=D-2n\frac{D-2}{2}=2n-(n-1)D.
$$
It is irrelevant only above its <upper critical dimension>. For the ordinary critical point the leading stabilizing interaction is quartic, while at a tricritical point the renormalized quartic term is tuned away and the leading interaction is sextic. Therefore
$$
\boxed{D_c^{\rm ordinary}=4,\qquad D_c^{\rm tricritical}=3.}
$$
One can also see the failure through the <Ginzburg criterion>. Long-wavelength fluctuations averaged over a <correlation volume> obey $\langle(\delta M)^2\rangle_\xi\sim\xi^{2-D}$, up to temperature and stiffness factors. The quartic saddle has $M^2\sim\xi^{-2}$, giving a fluctuation/mean-square ratio proportional to $\xi^{4-D}$. The tricritical sextic saddle has $M^2\sim\xi^{-1}$, giving a ratio proportional to $\xi^{3-D}$. Both ratios grow without bound below their respective critical dimension, invalidating the fluctuation-free Landau exponents sufficiently close to a transition.
At equality the ratios are marginal, not power-law divergent. The loop expansion resolves this boundary: the quartic vertex correction contains $\int d^Dq/(q^2+\xi^{-2})^2$, logarithmic at $D=4$; the sextic vertex correction contains a two-loop three-propagator integral whose overall degree is $2D-6$, logarithmic at $D=3$. Marginal running therefore generally adds logarithmic corrections at these dimensions. Below them, where the corresponding transition exists, interactions control the long-distance critical behavior. Above them, leading thermodynamic powers are mean-field, although stabilizing <dangerously irrelevant couplings> explain why naive hyperscaling need not hold.
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