A Complete Prime-Cofactor Face for Circulant Determinants of Order 112

For a positive integer n, let S(C_n) be the set of determinants of integraln by n circulant matrices. This preprint gives an exact computer-assistedclassification of the prime-cofactor face comparing 49q with 64q at order112. In the ray group of Q(zeta_7) of modulus 16(1-zeta_7), isomorphic toC8^2 x C4^2 x C2^2, an explicit Galois-stable affine set H8 of eight classesdefines a Chebotarev set P8. For every rational prime q, both signs of 49q occur while neither sign of64q occurs if and only if q=7 or q belongs to P8. In fact, neither sign of64q occurs for any prime q. The separator-prime set has natural and Dirichletdensity 1/3072; its least prime is 7 and its least unramified prime is 320839.Explicit 112-coefficient polynomials certify determinants 343 and49*320839. The accompanying exact-arithmetic archive certifies the completeunit packets, frozen ray basis, four HNF/SNF contraction systems, all remainingnormalization atoms, both ramified primes, Chebotarev density, leastness scan,and the explicit determinant witnesses. This result classifies the fixed 49q/64q face; it does not classify the fullset S(C_112).