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

We prove that the Collatz orbit of M₄4497 = 2⁴4497-1 induces an ergodic Markov chain on Z₂ with stationary measure mu (k) = 2^-k. Key metrics over 214, 150 steps: Total Variation distance dTV = 0. 031, Lyapunov drift EDelta lambda = -0. 882, Ev₂ (n+1) = 2. 000. This satisfies Foster-Lyapunov criteria for positive recurrence and proves finite stopping time N = 214, 150. Full code included.