Signed circulants at the Ramanujan bound

For the circulant graph $C_n(1,2)$ with $n\ge 10$ even, the ${\text{F}}_2$ system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is $+1$ on step-$1$ edges and $(-1{)}^i$ on step-$2$ edges has spectrum $\{\pm 2\sqrt{{\cos }^2{\theta }_k+{\cos }^{2}2{\theta }_k}\}$ and spectral radius exactly $2\sqrt{2}$, well below the Kesten bound $2\sqrt{3}$; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by $({\tau }_0,\alpha )$ and the spectral radius depends only on the Hamilton-cycle holonomy $\alpha$; and that the two twisted classes attain ${\rho }_{-}(n)=2\sqrt{{\cos }^2(\pi \mathrm{/}n)+{\cos }^2(2\pi \mathrm{/}n)}<2\sqrt{2}$. Exhaustive enumeration of all $2^{n+1}$ switching classes for $n\in \{8,10,12,14,16,18\}$ shows that ${\rho }_{-}(n)$ is the global minimum in every case, and we conjecture this for all even $n$; the lower bound is a flux-minimization statement in the sense of Lieb’s flux-phase theorem. For odd $n$ the quadrilateral system is inconsistent.