Can language models and calculators solve math without lying?

AXIOM splits mathematical reasoning between a language model (rewrites problems into structured schemas) and a computer algebra system (solves deterministically and verifies). Unlike black-box neural approaches, it produces zero false confidences: answers are either correct, correct-but-missed, or abstained. Deployed in production with a registry of 3,100+ routing rules, the system improves monotonically—each missed case can become handled without breaking prior ones. The framework shows how to build trustworthy neuro-symbolic systems where auditability and safety matter as much as speed.