= Solution
The <energy> favoring <spin polarization> must compete with the increase in Fermi <kinetic energy>. Assume the usual three-dimensional zero-temperature parabolic <conduction band> at fixed $N$ and $V$. Write the <spin polarization> as $\zeta=(N_\uparrow-N_\downarrow)/N$, so $N_{\uparrow,\downarrow}=N(1\pm\zeta)/2$. The unpolarized <Fermi energy> is
$$
E_F=\frac{\hbar^2}{2m}\left(\frac{3\pi^2N}{V}\right)^{2/3}.
$$
The two <spin> populations have <Fermi energies> $E_F(1\pm\zeta)^{2/3}$. Summing their individual $3E_{F,\sigma}/5$ mean kinetic energies gives
$$
\frac{E_{\rm kin}}N=\frac{3E_F}{10}\left[(1+\zeta)^{5/3}+(1-\zeta)^{5/3}\right].
$$
The specified interaction is an <energy> \b[per <Electron>], $U(1-\zeta^2)/4$. Thus the <energy> change per <Electron> near the unpolarized state is
$$
\Delta f=\left(\frac{E_F}{3}-\frac U4\right)\zeta^2+\frac{E_F}{81}\zeta^4+O(\zeta^6).
$$
The quadratic coefficient changes sign at the <Stoner threshold for an interaction normalized per electron>,
$$
\boxed{U_c=\frac43E_F=\frac{2\hbar^2}{3m}(3\pi^2)^{2/3}\frac{N^{2/3}}{V^{2/3}},\qquad U_c\propto N^{2/3}V^{-2/3}.}
$$
The positive quartic term gives a continuous onset in this model. More globally, $f''(\zeta)=(E_F/3)[(1+\zeta)^{-1/3}+(1-\zeta)^{-1/3}]-U/2$. <Convexity> of $x^{-1/3}$ makes this nonnegative for $U\leq4E_F/3$, so no lower polarized minimum appears before that threshold.
At fixed <number density> $N/V$, the critical $U$ is independent of the total sample size. A comparison restricted to a fully polarized state gives a different numerical coefficient, $U>\frac{12}{5}E_F(2^{2/3}-1)$, but the same requested density scaling; it misses the partially polarized onset. One must also distinguish this $U$, with units of <energy>, from a contact coupling multiplying an interaction-energy density $g n_\uparrow n_\downarrow$: here $g=U/(N/V)$. Importing a differently normalized <Stoner criterion> would introduce an erroneous extra density factor.
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