Endpoint and continuous polarization thresholds of a contact electron gas ($r_{\mathrm{end}}=2^{5/3}-2,r_{\mathrm{onset}}=10/9$)

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.