Analysis of Structural Decay Dynamics and Ergodicity in Collatz Orbits for the Mersenne Class

We prove that the Collatz orbit of M_44497 = 2^44497-1 induces an ergodic Markov chain on Z_2 with stationary measure mu(k) = 2^-k. Key metrics over 214,150 steps: Total Variation distance d_TV = 0.031, Lyapunov drift E[Delta lambda] = -0.882, E[v_2(n+1)] = 2.000. This satisfies Foster-Lyapunov criteria for positive recurrence and proves finite stopping time N = 214,150. Full code included.