Signed circulants at the Ramanujan bound

For the circulant graph $C_n(1,2)$ with $n\ge10$ even, the $\F_2$ 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 $\{\pm2\sqrt{\cos^2\theta_k+\cos^2 2\theta_k}\}$ and spectral radius exactly $2\sqrt2$, well below the Kesten bound $2\sqrt3$; 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 $\rho_-(n)=2\sqrt{\cos^2(\pi/n)+\cos^2(2\pi/n)}<2\sqrt2$. 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.