Filtrations and Orbit Dynamics: A Unified Mean Rho-Length Theorem

For three families of polynomial-style dynamics on finite fields, power maps x ↦ xd on Fp* (prime or composite d) and linear maps v ↦ Mv on Fpn, the mean rho-length over tail points admits a single closed form arising from one Abel summation over a depth filtration. We isolate the combinatorial template (Lemma 2.2 plus the Endomorphism Template Theorem 2.4) and give explicit closed forms for the level counts in each setting in terms of the d-adic signature of p−1 (power maps) and the Jordan partition of the nilpotent part of the Fitting decomposition (linear maps). As a corollary, prime-d power-map trees and single-Jordan-block trees are structurally identical under the substitution (d,h) ↦ (p,n). The formula is verified on 86,453 matrices and 2,592 composite-d power maps in exact rational arithmetic with zero discrepancy.