Non-divergence of Positive Collatz Orbits

This paper establishes a structural non-divergence result for the Collatz dynamics through a detailed analysis of the odd return map R (n) =3n+12^{₂ (3n+1) }, which maps odd integers to odd integers by removing all powers of two from 3n+1. The analysis is conducted entirely within the classical 3n+1 framework. No modified dynamical rule is introduced; all arguments are derived from the standard Collatz iteration. The proof develops several structural components of the return dynamics: • Resolution tree structure. Residues modulo 2ᵐ organize into a deterministic resolution hierarchy. At each level exactly two residues remain unresolved while all others resolve into balanced return classes. • Exact class balance. For contracting returns from residues n 1 4, the transition probabilities between the two return classes 1 8 and 5 8 are exactly balanced. • Exceptional tower sparsity. The unresolved residues form a rigid two-branch tower governed by the identity 3 (k) +5=2ᵏ. These tower residues occur with exponentially decreasing density and cannot contribute linear drift. • Depth separation and fresh-bit involutions. The return dynamics exhibit a strict separation between prefix-dependent behavior and fresh-bit decisions determined by higher binary digits. This yields exact equidistribution among return subclasses. • Entropy counting bound. Combining class balance with the fresh-bit involutions produces a combinatorial bound on the number of expanding trajectories. Together these structural results imply that the cumulative logarithmic drift of any Collatz orbit cannot remain positive indefinitely. Consequently, no positive integer orbit diverges to infinity under the classical Collatz iteration. The paper does not assume stochastic models or probabilistic heuristics; all arguments are deterministic and rely only on explicit arithmetic properties of the return dynamics.