On the Distribution of Orders of Companion Matrices over Finite Fields

We study the arithmetic dynamics of the Padovan companion matrix over modular rings Z/pᵉZ. We investigate the behavior of the multiplicative order under p-adic lifting and formulate a structural relation between base order and higher prime power extensions. The analysis connects linear recurrences, matrix dynamics, and p-adic valuation theory, highlighting a lifting phenomenon governed by the algebraic structure of GL (3, Z/pᵉZ). Numerical experiments support the proposed order growth law. No claim of full generality or complete proof is made. Citation: Franesi, P. (2026). p-adic Padovan Dynamics and Companion Matrix Orders Modulo Prime Powers. Zenodo. https: //doi. org/10. 5281/zenodo. 19679124