Moxia: A Trust-First Neuro-Symbolic Execution Architecture for Self-Explaining Mathematical Reasoning

Abstract

We present Moxia (formerly AXIOM), a trust-first neuro-symbolic architecture for self-explaining mathematical reasoning over natural-language input. Its language model is strictly a canonicalizer: it rewrites informal problem text into a narrow schema consumed by a deterministic Computer-Algebra-System (CAS) pipeline, which derives and verifies the answer or abstains as a first-class output. Routing follows a 1:1:1 alignment of problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 4,783 routes shipped, 71% of which answer without invoking the language model, and zero LOST_CORRECT regressions as a standing release gate. Because the answer is derived rather than generated, so is its explanation: every handler emits a step trace of the computation it performed, rendered as prose by a layer covering all 4,785 task files that cannot narrate a step the handler did not take. Derivations export to Lean 4 as well: 479 task files (10%) emit a theorem from the problem's declared data, 445 accepted by the Lean kernel with Mathlib; that gate covers a fixture corpus, so live output is generated, not machine-checked. We report two numbers and never fuse them. On the full 7-category MATH test split, designed against, Moxia answers 90.2% (4,510/5,000) with one confident-wrong answer (99.98% trust on parseable). On held-out MATH-500, never designed against, it answers 89.2% (446/500) with zero confident-wrong answers. The 1.0 pp gap is the substantive result: a registry that had merely memorized problem shapes would collapse on held-out data, and this one does not. The rule-only path answers the 20,000-record lm-eval arithmetic benchmark at 100%, 1 ms per record. What we emphasize is not an accuracy figure but the forward dynamic: every logged abstain is a candidate correct after one ship cycle, since new tasks compose without regressing the registry.