Almost-sure non-divergence of positive Collatz orbits: a 2-adic density-zero result

This paper studies non-divergence of positive Collatz orbits through the odd return map R (n) =3n+12^{₂ (3n+1) }, which maps odd integers to odd integers by applying the Collatz step 3n+1 followed by all required halvings. The paper establishes that the set of positive integers whose Collatz orbits exhibit divergent-type behavior has 2-adic density zero. In other words, at every finite level of 2-adic precision, the fraction of residue classes whose dynamics remain compatible with divergence decays exponentially. The argument combines four deterministic structural ingredients: Computational verification below 2^68, using the known complete convergence check in that range. A growing isolation gap for the binary recurrence variety T=\n: 3n+5=2ᵏ\, obtained via Baker-type finiteness results for tower re-entry. Exact class balance in the return-graph resolution tree, showing that contracting returns split evenly between the 1 8 and 5 8 classes at every resolved level. Depth separation and bit-level involutions, showing that successive class decisions depend on distinct binary positions and generate exact combinatorial symmetry in the admissible orbit patterns. These ingredients yield a 2-adic entropy bound: the Haar measure of the set of 2-adic integers whose return dynamics sustain the class-1 frequency required for divergence decays exponentially in the number of resolved returns. The paper proves that the set of potentially divergent positive integers has 2-adic density zero: almost every positive integer, in the 2-adic sense, does not have a divergent Collatz orbit. The paper does not claim a full proof that no individual positive integer diverges. Instead, it isolates the remaining gap between this density-zero result and full non-divergence, and explains how the unresolved obstruction lies in passing from an almost-sure statement in 2-adic measure to a pointwise statement for every positive integer. All arguments are deterministic and remain within the classical 3n+1 dynamics; no modified dynamical rule is introduced.