The Atlas Learns to Move: A Dynamic Arithmetic Atlas for Collatz and Beyond

Arithmetic as Atlas represents integers relative to chosen landmark families. A Road supplies the landmarks; Memory records a number's position between them. This paper extends that static framework to arithmetic dynamics. Given a transformation, the Dynamic Arithmetic Atlas records how an inhabitant moves between landmark chambers and how its normalized Memory changes. The accelerated Collatz map is used as the first developed case. On the dyadic Road, halving preserves normalized Memory exactly, while an odd step decomposes into a leading-position rise and a trailing-valuation descent. This separation yields a three-letter migration grammar, exact chamber populations, a prohibition against three consecutive rises, residue-defined one-halving corridors, explicit Mersenne migration words, an exit-valuation law, a distinction between accumulated debt and later recovery, and reverse inheritance ladders with a limiting Memory coordinate. The results do not prove the Collatz conjecture, and several ingredients are classical. The contribution is a unified representational architecture that turns orbit data into a geography of territory, migration, corridor, exit, recovery, and inheritance. The final sections state a general Dynamic Atlas method for other arithmetic systems and consider the human-AI collaboration through which the framework developed.