Structural and Stochastic Analysis of the Set G_i in Collatz Dynamics: Toward the Exclusion of Disjoint Components via the Gebahi Ribbon Sequence

This paper formalizes the algebraic properties of the set G_i of odd preimages within the Collatz conjecture framework. We introduce the concept of the Gebahi Ribbon, defined as the infinite ordered set formed by the exact succession of terms extracted by the saturation sequence (p_n) = (1; 5; 3; 13; 17; 11; 7; 9; ...). By exploiting the modular congruence cycles modulo 3 and modulo 9 induced within its sub-branches, we establish a rigorous transition toward a 2-adic probabilistic model. Through an ergodic Markov chain, we quantify the global logarithmic growth expectation of the inverse tree at +0.5817 bit, which is validated numerically by a stable statistical convergence toward an equiregular stationary distribution pi = (1/2, 1/2). Finally, by modeling the inverse exploration as a multitype Galton-Watson branching process within the ring of 2-adic integers Z_2, we prove the primitivity and irreducibility of its gain matrix, ensuring the complete saturation of its dyadic residue classes. Confronted with Terence Tao’s asymptotic density theorem (2019), this local saturation mathematically excludes the existence of any disjoint connected component, thereby structurally validating the Collatz conjecture.