{On the Distribution of Orders of Companion Matrices over Finite Fields}

We study the multiplicative order of companion matrices associated with the polynomial x^3 - x - 1 over finite fields. For primes where the polynomial is irreducible, we define a normalized invariant derived from the order of the corresponding matrix in the norm-one subgroup of F_{p^3}^*. We investigate the statistical behavior of this invariant over primes and propose a conjectural limiting distribution on (0,1), empirically consistent with an exponential law on logarithmic scale. Numerical experiments up to 10^5 primes support this behavior via standard statistical tests including Kolmogorov–Smirnov distance and model comparison criteria. The framework connects arithmetic properties of finite fields with probabilistic and dynamical structures arising from Frobenius actions. No claim of proof of the limiting distribution is made.