Reasoning over mathematical objects: on-policy reward modeling and test time aggregation

TL;DR

This paper introduces new training data, evaluation benchmarks, and methods for improving language models' ability to reason over mathematical objects, demonstrating enhanced performance and generalization across formats.

Contribution

It provides the Principia suite for training and benchmarking, develops on-policy judge training techniques, and shows how test-time aggregation scales reasoning capabilities.

Findings

Strong LLMs struggle on Principia benchmarks

Training recipes significantly improve reasoning performance

On-policy training enhances cross-format reasoning generalization

Abstract

The ability to precisely derive mathematical objects is a core requirement for downstream STEM applications, including mathematics, physics, and chemistry, where reasoning must culminate in formally structured expressions. Yet, current LM evaluations of mathematical and scientific reasoning rely heavily on simplified answer formats such as numerical values or multiple choice options due to the convenience of automated assessment. In this paper we provide three contributions for improving reasoning over mathematical objects: (i) we build and release training data and benchmarks for deriving mathematical objects, the Principia suite; (ii) we provide training recipes with strong LLM-judges and verifiers, where we show that on-policy judge training boosts performance; (iii) we show how on-policy training can also be used to scale test-time compute via aggregation. We find that strong LMs such as Qwen3-235B and o3 struggle on Principia, while our training recipes can bring significant improvements over different LLM backbones, while simultaneously improving results on existing numerical and MCQA tasks, demonstrating cross-format generalization of reasoning abilities.