Unit Liars in a Cubic Frobenius Test: An Exact Formula and an N^(-3/2) Semiprime Bound

Let q be a prime such that 4q − 27 = a², put f_q(X) = X³ − qX − q, and consider the cubic algebra A_N = (Z/NZ)[X]/(f_q). 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.

Reference implementation: [cubic-frobenius-unit-test](https://github.com/mariotrevi/cubic-frobenius-unit-test), a public single-threaded C/GMP implementation of the cubic Frobenius unit test.

This is a preliminary preprint and has not been peer reviewed.