ALGEBRAIC STRUCTURES IN THE COLLATZ CONJECTURE AND ITS STOPPING TIME ACCORDING TO THE DESCENT EXPRESION

In this work, we present a structural analysis of orbits and parity sequences within the Collatz Conjecture through the formulation of Diophantine equations and an exponential descent operator. We analytically demonstrate that discrete trajectories are deterministically segmented into infinite disjoint families parameterized under arithmetic progressions congruent modulo 2^j . Furthermore, we study the finiteness of solutions for fixed alternating patterns using induction and formalize the general equations for arbitrary sequences. A sieve algorithm is implemented and validated via C++ high-performance software, achieving a coverage density of 98.78% for the domain below 3 \times 10^8 in approximately 20 minutes on consumer-grade hardware. This framework provides a structural foundation that may have implications for cryptographic applications based on iterative dynamical systems.