A Rigidity-Based Resolution of the Collatz Dynamics via 2-Adic Anchoring

This paper presents a structural resolution of the Collatz conjecture by reframing the dynamics as a rigid ultrametric system governed by irreversible symbolic compression and arithmetic non-degeneracy. The classical Collatz map is reformulated as the odd Collatz map and extended to the compact space of odd 2-adic integers, where recurrence can be analyzed at arbitrary precision. The central contribution is the identification of a previously hidden obstruction: the possibility of infinite near-identity affine drift under symbolically coherent recurrence. The paper introduces an anchoring functional to quantify irreversible information fixation under symbolic refinement and proves that the arithmetic structure of Collatz coefficients forbids such drift via explicit 2-adic valuation bounds. Combining symbolic dynamics, ultrametric geometry, affine block analysis, and valuation theory, the work establishes a global rigidity theorem: arbitrarily fine symbolically coherent recurrence necessarily collapses into exact recurrence and hence periodicity. This eliminates all aperiodic trajectories and yields the classical Collatz conjecture as a corollary. Beyond Collatz, the framework suggests a general rigidity principle applicable to a broad class of recursive and arithmetic dynamical systems characterized by symbolic compression and ultrametric structure. -Wishing everyone a very Happy Christmas! -JSR