Explicit Equimodular Curves for Prism-Graph Chromatic Polynomials and the Beraha-Kahane-Weiss Limit Set

For the prism (cyclic ladder) graphs $G_n=C_n\text{？}\square \text{？}{P}_2$ , the chromatic polynomial admits a four-branch transfer-matrix expansion $P\left(G_n,z\right)={\displaystyle \sum }_{j=0}^3\text{？}{\alpha }_j\left(z\right){\lambda }_j{\left(z\right)}^n$ , ${\lambda }_0\left(z\right)=z^2？3z+3$ , ${\lambda }_1\left(z\right)=1？z$ , ${\lambda }_2\left(z\right)=3？z$ , ${\lambda }_3\left(z\right)=1$ , with explicit polynomial amplitudes ${\alpha }_j\left(z\right)$ . By the Beraha-Kahane-Weiss mechanism, accumulation of chromatic roots as $n\to \infty$ is confined to loci where two or more