= Solution
For $\epsilon>0$, the quartic interaction has a nonzero limiting value $\lambda_*=O(\epsilon)$ instead of disappearing under coarse-graining. Long-wavelength <thermal fluctuations> are therefore interacting, so the fluctuation-free <Landau approximation> cannot give the generic critical powers. For example, its ordinary <correlation-length critical exponent> is $1/2$, whereas the relevant thermal <eigenvalue> just derived gives
$$
\boxed{\nu=\frac1{y_t}=\frac12+\frac\epsilon{12}+O(\epsilon^2).}
$$
The tadpole is momentum independent, so the <anomalous dimension> begins at $O(\epsilon^2)$. Combining this with the scaling relations yields $\gamma=1+\epsilon/6+O(\epsilon^2)$, $\beta=1/2-\epsilon/6+O(\epsilon^2)$, and $\alpha=\epsilon/6+O(\epsilon^2)$, exhibiting the deviation from ordinary mean-field indices. These are controlled statements for small positive $\epsilon$, not exact numerical exponents at arbitrary dimension. The <Landau-Ginzburg theory> itself is not invalid: integrating its fluctuations is precisely how the <Wilson-Fisher fixed point> was obtained. The invalid step is replacing that interacting long-distance theory by a uniform saddle and ignoring its fluctuation corrections.
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