ACED: Reference-Anchored Aggregation of Natural-Language and Executable Evidence for Mathematical Reasoning

Natural-language self-consistency and executable programs provide complementary test-time evidence for mathematical reasoning, yet either channel may support the same semantic error. Thus we propose ACED, a reference-anchored framework that treats the self-consistency answer as the default and executable outputs as supporting evidence rather than interchangeable votes. ACED converts source-specific answer counts into smoothed supports, combines them through a weighted logarithmic pool, and overrides the reference only when a calibrated relative-evidence margin is exceeded. The same rule supports both candidate selection and admission of executable-only answers. Across five math word-problem benchmarks, ACED matches or outperforms the corresponding baselines. On GSM8K, ACED-CANDIDATE recovers 89.7% of the attainable candidate-set gain with only three damaging overrides, while ACED-ALL reaches 86.9% accuracy versus 82.6% for direct Program-of-Thought aggregation. These results show that preserving evidence provenance enables more reliable integration of natural-language and executable reasoning.