Periodic Signings of C_n(1,2): An Exact Band Edge and Short-Period Classification

Abstract

We study spectral-radius optimization for periodic real signings of the 4-regular circulants C_n(1,2). After gauging all step-1 edges positive, a period-8 triangle-flux pattern produces an explicit 8 × 8 Hermitian Bloch symbol. Its upper band edge occurs at phase z = 1, and the associated infinite operator and every finite quotient C_(8q)(1,2) have the same spectral radius, where t₀ is the largest root displayed below:

√t₀ = 2.7936044933…, t₀⁴ − 16t₀³ + 76t₀² − 96t₀ + 16 = 0.

The proof uses an exact Bloch determinant, rational Sturm certificates, and monotonicity on the full unit-circle phase interval. Independently, a symmetry-reduced certificate classifies all triangle-flux patterns of fundamental period at most 16: the period-8 pattern is the unique norm minimizer up to rotation, reflection, and global flux complement. As an application, its radius is strictly below Suvagiya's proposed twisted-class value for every n = 8q ≥ 32, disproving the corresponding signed-circulant minimum conjecture on an infinite arithmetic progression. All decisive comparisons and classification witnesses are exact; floating-point linear algebra is used only for discovery and stress testing.