Collatz Dynamics I: Skeleton Bound and the Elimination of Non-Trivial Cycles

This paper establishes the non-existence of non-trivial cycles in the Collatz map through a purely algebraic framework. The central device is the Skeleton Condition, a structural inequality that provides a strict upper bound on the correction term in the fundamental cycle equation. When juxtaposed with the necessary growth condition 2^(S(k)) > 3^k, the Skeleton Condition yields an unavoidable structural contradiction.

In contrast to prior approaches—whether based on Diophantine approximation, asymptotic estimates, or large-scale computation—this argument is entirely algebraic and free of circular assumptions. The result closes the cycle subproblem of the Collatz conjecture with definitive rigor.

This work constitutes the first installment of the three-paper Collatz Dynamics program. Together with the companion article Collatz Dynamics II: Drift–Compression Dynamics and Global Convergence, it delineates a unified structural pathway toward a complete deterministic resolution of the Collatz conjecture.