Leading-log critical-field inversion ($r=\ln \kappa /(2{\kappa }^2)$)

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.