Figure It Out: Improve the Frontier of Reasoning with Executable Visual States

TL;DR

The paper introduces FIGR, a method that enhances reasoning by integrating executable visual construction with reinforcement learning, improving performance on complex mathematical benchmarks beyond text-only models.

Contribution

FIGR is the first approach to incorporate executable visual diagrams into multi-turn reasoning, externalizing hypotheses to improve understanding of structural constraints.

Findings

Outperforms text-only baselines on eight mathematical benchmarks.

Achieves 13.12% improvement on AIME 2025.

Achieves 11.00% improvement on BeyondAIME.

Abstract

Complex reasoning problems often involve implicit spatial and geometric relationships that are not explicitly encoded in text. While recent reasoning models perform well across many domains, purely text-based reasoning struggles to capture structural constraints in complex settings. In this paper, we introduce FIGR, which integrates executable visual construction into multi-turn reasoning via end-to-end reinforcement learning. Rather than relying solely on textual chains of thought, FIGR externalizes intermediate hypotheses by generating executable code that constructs diagrams within the reasoning loop. An adaptive reward mechanism selectively regulates when visual construction is invoked, enabling more consistent reasoning over latent global properties that are difficult to infer from text alone. Experiments on eight challenging mathematical benchmarks demonstrate that FIGR outperforms strong text-only chain-of-thought baselines, improving the base model by 13.12% on AIME 2025 and 11.00% on BeyondAIME. These results highlight the effectiveness of precise, controllable figure construction of FIGR in enhancing complex reasoning ability.