Double signs of Hamiltonian circles in doubly signed complete graphs

We study Hamiltonian circles in the doubly signed complete graph ${\mathrm{\Sigma }}_n=(K_n,\sigma ,{\mathbb{F}}_2^2)$. A circle’s double sign is defined as the sum of its edge labels. I establish conditions under which Hamiltonian circles realize all four possible double signs and prove that this occurs when the set of triangle double signs contains at least three distinct values. The proof is based on an analysis of triangle bases of the binary cycle space, structural properties of $K_4$ subgraphs, and explicit Hamiltonian constructions.