Let q be a prime such that 4q − 27 = a², put fq (X) = X³ − qX − q, and consider the cubic algebra AN = (Z/NZ) X/ (fq). The cubic character of N modulo q selects one of the two nonidentity conjugations of this cyclic cubic algebra. This preprint gives an exact product formula for the proportion ρ (N, q) of units satisfying the selected Frobenius congruence. As the principal application, for every product N = pr of two distinct odd primes and every admissible q, ρ (N, q) < N^ (-3/2). The exponent and leading constant are conditionally asymptotically sharp. An explanatory appendix develops the composite-modulus cubic algebra from polynomial quotient rings, determinant norms, finite fields, and the Chinese remainder theorem. The record includes the main preprint, the companion manuscript proving the explicit Frobenius-selection rule, and a complete source/data archive for the paper and its semiprime illustration. This is a preliminary preprint and has not been peer reviewed.
David Bernier (Thu,) studied this question.