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

Abstract

AXIOM is a trust-first neuro-symbolic math reasoning architecture combining a frozen large language model (Qwen 2.5 7B Instruct, used as rewriter only) with deterministic SymPy CAS handlers organized via a 1:1:1 task-routing discipline (trigger regex + narrow prompt + closed-form handler).4,783 such routes ship, 71% of which reach an answer without invoking the language model at all.Because the answer is derived rather than generated, the derivation itself is available: every handler emits a step trace of the computation it actually performed, and a synthesizer renders that trace as prose. The layer covers 4,785/4,785 tasks and can only describe steps the handler took, which is what distinguishes it from a chain-of-thought narration that may or may not describe the path to the answer.The architecture commits to three ordered priorities (the "North Star"):(1) reliability - the system either solves correctly or abstains with "UNKNOWN";(2) zero regressions across ship cycles;(3) velocity within 3s/problem.Two measurements are reported and never combined.DESIGNED-AGAINST - the full MATH test split, all 7 domains, 5000 records. Task triples were built from MATH problem shapes, so this measures how much of the benchmark's shape space the registry covers:Intermediate Algebra: 880 / 903 (97.45%)Number Theory: 496 / 540 (91.85%)Algebra: 1078 / 1187 (90.82%)Counting & Probability: 428 / 474 (90.30%)Precalculus: 493 / 546 (90.29%)Prealgebra: 735 / 871 (84.39%)Geometry: 400 / 479 (83.51%)TOTAL: 4510 / 5000 (90.20%), one confident-wrong answer (99.98% trust on parseable)HELD OUT - MATH-500, never designed against and never used to choose what to build: 446 / 500 (89.2%), with zero confident-wrong answers (100.00% trust on parseable).The 1.0 pp gap between the two is the substantive result. A registry of thousands of narrow routes is exactly the artifact one would expect to memorize benchmark shapes, and a shape-memorizing system collapses on held-out data. This one does not: the routes encode mathematical templates, so perturbing the quantities in a covered problem makes the handler recompute rather than recall.The rule-only path answers the 20,000-record lm-eval arithmetic benchmark at 100% with a median latency of 1 ms and no LLM call.## What's in this deposit- axiom_workshop.pdf - the paper (compiled PDF)Live interactive demo: https://huggingface.co/spaces/Squagghy/axiom-solver This is v2.0.0: coverage extended from 4 to all 7 MATH domains, a faithful derivation layer added, and held-out MATH-500 reported alongside the designed-against split.Deposited under the same concept-DOI as previous versions.