Sharp asymptotics for the Lovász number of random circulant graphs

We study the LOVÁSZ number of dense random circulant graphs generated by independent Bernoulli bits indexed by cyclic distances. Although only $O(n)$ independent random inputs determine $O(n^2)$ globally dependent edge indicators, we prove sharp expectation asymptotics along all odd orders, including composite orders, with leading constant one. This resolves a conjecture posed for random circulant graphs in 2025. The proof combines an exact Fourier reduction and complementary duality with a positive interior smoothing that reveals a nonlinear random fixed-point equation. To analyze this equation under the strong dependence of the distance-bit model, we construct canonical tree fields, establish growing-degree Gaussian approximations through polynomial comparison and Wick/Hermite expansions, and transfer the resulting approximation back to the original sign model. A deterministic residual estimate together with an explicit exceptional-event integration yields the sharp expectation asymptotics.