  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. Here we establish an ad hoc equivalent of modular arithmetic for
Collatz sequences to automatically demonstrate the convergence of infinite
quivers of numbers, based on five arithmetic rules we prove apply on the entire
Collatz dynamic and which we further simulate to gain insight on their graph
geometry and computational properties. We then formally demonstrate these rules
define an automaton that is playing a Hydra game on the graph of undecided
numbers we also prove is embedded in 24N-7, proving that in ZFC the Collatz
conjecture is true, before giving a promising direction to also prove it in
Peano arithmetic.
