arXiv:2606.00671v2 Announce Type: replace
Abstract: We present AXIOM, a trust-first neuro-symbolic architecture for natural-language mathematical reasoning. 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 between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 4,783 such 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 tasks that cannot narrate a step the handler did not take. We report two numbers and never fuse them. On the full 7-category MATH test split, designed against, AXIOM answers 90.2% correctly (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% correctly (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% in 1 ms per record; the system has served ~30,000 public queries. 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. The discipline behind it (math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, abstain as first-class output) transfers to trustworthy neuro-symbolic systems beyond mathematics.
Abstract: We present AXIOM, a trust-first neuro-symbolic architecture for natural-language mathematical reasoning. 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 between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 4,783 such 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 tasks that cannot narrate a step the handler did not take. We report two numbers and never fuse them. On the full 7-category MATH test split, designed against, AXIOM answers 90.2% correctly (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% correctly (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% in 1 ms per record; the system has served ~30,000 public queries. 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. The discipline behind it (math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, abstain as first-class output) transfers to trustworthy neuro-symbolic systems beyond mathematics.
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