What question did this study set out to answer?

The aim is to develop a reliable neuro-symbolic architecture for mathematical reasoning that can validate its outputs.

May 31, 2026Open Access

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

Key Points

The aim is to develop a reliable neuro-symbolic architecture for mathematical reasoning that can validate its outputs.
AXIOM combines a large language model with deterministic CAS handlers using a 1:1:1 task-routing discipline.
Empirical validation was performed on the MATH benchmark across four domains with a total of 2747 records.
The architecture prioritizes reliability, zero regressions, and problem-solving velocity.
Overall, the architecture achieved 94.36% total correctness across all categories.
Specific domain results included: Algebra 98.15%, Counting & Probability 91.35%, Number Theory 95.37%, and Precalculus 87.73%.
The architecture maintained zero regressions across 250+ consecutive deployments.

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). 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. Empirical validation on the MATH benchmark test split (4 active domains, 2747 records total): Algebra: 1165 / 1187 (98. 15%), 100% trust on parseableCounting & Probability: 433 / 474 (91. 35%), 100% trust on parseableNumber Theory: 515 / 540 (95. 37%), 100% trust on parseablePrecalculus: 479 / 546 (87. 73%), 100% trust on parseableTOTAL: 2592 / 2747 (94. 36%), 100% trust on parseable All four MATH categories simultaneously above the 70/90/70 per-domain floor (parseable ≥ 70%, trust on parseable ≥ 90%, total correct ≥ 70%) - the architecture's verifiable bottom-up thesis empirically validated. The task registry has grown to >3000 task triples with zero LOSTCORRECT regressions across 250+ consecutive ship commits. ## What's in this deposit- axiomworkshop. pdf — the paper (compiled PDF) Live interactive demo: https: //huggingface. co/spaces/Squagghy/axiom-solver This is v1. 3: updated benchmark numerics. Architecture and methodology unchanged from v1. 0. Future versions will be deposited under the same concept-DOI as the architecture evolves.

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