On the Convergence of the 3x + 1 Sequences

The Collatz conjecture posits that iterating the function f(n) = n/2 for even n and f(n) = 3n + 1 for odd n eventually reaches 1 for any positive integer n. In this paper, we analyze the accelerated odd-to-odd Collatz operator T(n) = (3n+1)/2^k. We derive explicit algebraic representations for m-step iterations, formulate a probabilistic measure on the exponent sequence (k_j), and employ linear forms in logarithms to bound non-trivial cycles. Consequently, we establish the long-term density convergence of the sequences toward the fundamental cycle (4, 2, 1).