= Leading-log critical-field inversion
{title2=$r=\ln\kappa/(2\kappa^2)$}
Combining $B_{c1}\simeq\Phi_0\ln\kappa/(4\pi\lambda^2)$ and $B_{c2}=\Phi_0/(2\pi\xi^2)$ gives $r=B_{c1}/B_{c2}=\ln\kappa/(2\kappa^2)$. Its derivative is $(1-2\ln\kappa)/(2\kappa^3)$: the ratio peaks at $\kappa=\sqrt e$, with maximum $1/(4e)$. Thus each $0<r<1/(4e)$ has two solutions with $\kappa>1$. The large-$\kappa$ branch is the one consistent with using a leading-log approximation; the small branch cannot be selected or rejected by the inequality $\kappa>1$ alone. Omitted core-energy terms limit numerical accuracy.
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