Automaton Termination and Language Restriction: From Galois Theory to Collatz Dynamics

We propose a unifying viewpoint: classical unsolvability results can be interpreted as failures of termination for automata operating under restricted transition languages. This framework is developed in two parallel settings. First, Galois-theoretic solvability by radicals is reinterpreted as termination of a branch automaton constrained to a radical-only language. Second, the odd-only Collatz dynamics is formalized as a labeled automaton governed by a Δ-update language. The paper isolates a general termination template separating structural properties (determinism, injectivity/irreversibility) from a single remaining arithmetic requirement (forced decrease). In the Collatz setting, this reduces the completion of termination to one explicitly stated quantitative collision principle. This work does not claim a proof of the Collatz conjecture; its contribution is the precise isolation of the remaining obstruction.