Quartic Equimodular Curves and Spectral Reductions in Prism-Derived Chromatic Polynomials

We study limiting chromatic-root loci for several recursively constructed prism-derived graph families by means of finite transfer-matrix reductions. For each family, the chromatic polynomial admits a spectral expansion P(Gn,z)=∑jαj(z)λj(z)n, so the Beraha–Kahane–Weiss theorem reduces the asymptotic analysis to dominant spectral competition and residual amplitude cancellation. For the antiprism family An, the reduced transfer operator has discriminant Δ(z)=9−4z. The unit-modulus conditions for the two nontrivial spectral branches pull back to the irreducible quartic x4−8x3+2x2y2+21x2−8xy2−18x+y4+5y2=0. We prove that this entire quartic is the equimodular accumulation locus for An, with the node at z=3 separating the two dominance types. The same quartic arises for the circulant family Cn(1,2) through the same reduced spectral normal form. We also prove that An has a nonfixed isolated accumulation point at the fifth Beraha number B5=(3+5)/2, and that Cn(1,2) has the same isolated point along its even subsequence after the residual branches +1 and −1 are combined by parity. In contrast, for Cn(1,3), the identity λ+(z)λ−(z)=−1 collapses the reduced-sector equimodular locus to the vertical segment ℜ(z)=2, |ℑ(z)|≤2. For the generalized Petersen family G(n,2), we obtain an irreducible cubic equation for one invariant sector, indicating the first obstruction to the quadratic pullback method while leaving the full branch expansion open.