{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/p^eZ. 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/p^eZ). 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