= Solution
Besides the <Gaussian fixed point>, solve the nonzero-coupling fixed-point equations. The quartic equation gives $\lambda_*=2\epsilon(1+\rho_*)^2/(3K_D)$. Substitute into the mass equation to get $2\rho_*+\epsilon(1+\rho_*)/3=0$, hence $\rho_*=-\epsilon/(6+\epsilon)$ within the displayed one-loop truncation. Since $K_4=2\pi^2/(2\pi)^4=1/(8\pi^2)$,
$$
\boxed{\rho_*=-\epsilon/6+O(\epsilon^2),\qquad
\lambda_*=16\pi^2\epsilon/3+O(\epsilon^2).}
$$
These are the <Wilson-Fisher fixed point> coordinates in this cutoff scheme. The higher terms obtained by solving the truncated equations exactly should not be regarded as a full higher-order epsilon calculation.
The attraction claim in the PDF needs a thermal qualification. The <stability matrix of a renormalization-group fixed point> at this solution has entries
$$
J=\begin{pmatrix}
2-\epsilon/3&K_D/[2(1+\rho_*)]\\
3K_D\lambda_*^2/(1+\rho_*)^3&-\epsilon
\end{pmatrix}.
$$
The lower-left entry is $O(\epsilon^2)$, so its <eigenvalues> are
$$
\boxed{y_t=2-\epsilon/3+O(\epsilon^2)>0,\qquad
y_g=-\epsilon+O(\epsilon^2)<0.}
$$
Increasing $\ell$ means approaching the infrared. The quartic scaling direction is attractive, but a perturbation in the thermal scaling direction grows as $e^{y_t\ell}$. Therefore the <Wilson-Fisher fixed point> is an infrared attractor \b[on its tuned <critical surface>], not an attractive fixed point of the entire two-dimensional coupling plane. In the plane it is a saddle; an arbitrary small thermal perturbation is a concrete counterexample to unconditional attraction. This is the <thermal relevant direction at the Wilson-Fisher fixed point>.
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