2-adic Scenarios in Collatz Dynamics: Preliminaries on Geometry and Odd-Step Jump Heights

This work proposes a novel geometric reading of the classical Collatz problem by rigorously separating the underlying arithmetic space from the iterated dynamics that acts upon it. Instead of starting from the iteration of the Collatz function, we first introduce a fixed, pre‑dynamic geometric organization of the natural numbers: the 2-adic table‑scenario, based on the unique decomposition of every integer as an odd part times a power of two. This scenario provides a global coordinate system that is independent of any particular trajectory. A second, complementary scenario is then defined on the set of odd numbers alone. It classifies each odd integer according to the 2-adic height of its successor under the Collatz map—that is, according to how many consecutive divisions by two are possible after applying the “3n+1” step. This height determines, prior to any iteration, the potential for vertical contraction that each odd step can induce. Within this double‑scenario framework, the usual Collatz iteration appears as a dynamical play performed on a fixed geometric stage. The approach does not attempt to prove the Collatz conjecture, but rather aims to clarify the structural properties of the arithmetic space that precede the dynamics. In particular, it leads to the notion of a “stable corner”—a finite, invariant region of the scenario that acts as a final scene. The classical cycle (1‑4‑2‑1) is reinterpreted as the unique stable corner accessible from the rest of the 2-adic landscape. This reformulation shifts the emphasis from tracking individual orbits to a global question of uniqueness of an invariant region within a geometrically defined arithmetic space.