A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map

This work analyzes the odd–even dynamics of the classical Collatz map from a 2-adic viewpoint. It proves that odd numbers with ν₂(3n + 1) = r form arithmetic progressions of relative density 1/2^{r–1}, explaining the increasing gaps observed in Collatz trajectories. Using this 2-adic information, all odd numbers sharing the same last even term are grouped into 4-adic families, showing that every family converges to 𝔽₂ = {1, 5, 21, 85,…}. The cycle 4 → 2 → 1 emerges as a unique global attractor. The framework also connects explicitly with the Structure Theorem for (d,g,h)-maps by Kontorovich and Sinai (2006), where the decreasing 2-adic density plays the role of the negative drift in their probabilistic model.

This second version provides a fully revised English text, with improved terminology, academic style adjustments, and an updated Appendix B concerning the transition from classes 4n+34n+34n+3 to 4n+14n+14n+1.
Appendix A has been expanded to clarify the correspondence between the deterministic 2-adic model and the probabilistic structure theorem of Kontorovich and Sinai, establishing a formal equivalence between both frameworks.
The mathematical content and main propositions remain unchanged.