= Endpoint and continuous polarization thresholds of a contact electron gas
{title2=$r_{\rm end}=2^{5/3}-2,\quad r_{\rm onset}=10/9$}
Let $r=(\gamma/A)(n/2)^{1/3}$ and $P=(n_\uparrow-n_\downarrow)/n$. In units $2A(n/2)^{5/3}$ the spin-dependent <energy> is $[(1+P)^{5/3}+(1-P)^{5/3}]/2-r(1+P^2)/2$. Comparing only $P=0$ and $P=1$ gives the endpoint crossing $r=2^{5/3}-2$. Expanding near $P=0$ gives a quadratic coefficient $5/9-r/2$, so the unpolarized state first loses local stability at $r=10/9$. The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.
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