{p-adic Padovan Dynamics: Unified Theory of Companion Matrix Orders Modulo Prime Powers}

The p-adic Padovan Dynamics Theory unifies three Zenodo works on the Padovan companion matrix $P$ order mod $p^e$. We prove $\operatorname{ord}_{p^e}(P) = \operatorname{ord}_p(P) \cdot p^{e-1 + v_p(\det(P^{\operatorname{ord}_p(P)}-I_3))}$, via Hensel lifting and GL(3,Z/p^e Z) structure. Examples: p=5 (20→100), p=7 (42→294). Implies perfect observability of dynamical trajectories and exponential growth, generalizing to linear recurrences.