The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers which inflation propensity remains so unpredictable it could be used to generate reliable proof of work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here we establish an ad hoc equivalent of modular arithmetic for Collatz sequences, based on five arithmetic rules we prove apply on the entire Collatz dynamical system and which iteration exactly define the full basin of attraction leading to any odd number. We further simulate these rules to gain insight on their quiver geometry and computational properties, and observe they allow to linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules allow to linearize Collatz convergence, one in ZFC and one in Peano arithmetic.