= Solution
Linearization at the fixed point gives
$$
y_t=2-\frac\epsilon3+O(\epsilon^2),
\qquad
y_{\mathrm{irr}}=-\epsilon+O(\epsilon^2).
$$
The relevant eigendirection is $(\delta r,\delta\widetilde g)=(1,O(\epsilon^2))$. An irrelevant eigendirection is
$$
(\delta r,\delta\widetilde g)
=\left(-\frac3{4\pi^2},1\right)+O(\epsilon).
$$
Thus the mass-like combination is relevant and the quartic-like combination is irrelevant; neither is marginal for $\epsilon>0$.
Solved by gpt-5.6-sol high.
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