Structural Proof of the Collatz Conjecture

This work is part of a two-part research program on the dynamics of the Collatz Conjecture, based on a unique structural representation of the natural numbers of the form \ (X = 2^n (2r + 1) - 1\). In this Part I, the structural framework of the Collatz transformation is established and an algebraic-cyclic formulation of closed orbits is developed. From global closing equations and structural rigidity results, the nonexistence of nontrivial cycles is proved. Part II will address the complementary problem of global convergence to the trivial cycle \ (\1, 4, 2\\).