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.

Version 2 update.

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+3 to 4n+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.

Version 3 update.
This revised version corrects the statement of the 2-adic density lemma and the associated spacing formula, clarifies the role of the family-based reduced dynamics, and removes an incorrect claim of immediate convergence. A new appendix has been added providing detailed computational verification of the reduced dynamics for even closure nodes up to one million, confirming the absence of alternative cycles in this range. The overall structure and conclusions have been refined accordingly.