# Learning Where Outcomes Change:Credit-Addressable Reasoning for Multimodal Geometry

arXiv: https://arxiv.org/abs/2608.30457 · HTML: https://mlsanity.com/paper/2608.30457/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 22 of its 39 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 22 of 22)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [MathVerse: Does Your Multi-modal LLM Truly See the Diagrams in Visual Math Problems?](https://mlsanity.com/paper/2403.14624/) (2024)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [VL-Rethinker: Incentivizing Self-Reflection of Vision-Language Models with Reinforcement Learning](https://mlsanity.com/paper/2504.08837/) (2025)
- [GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning](https://mlsanity.com/paper/2105.14517/) (2021)
- [UniGeo: Unifying Geometry Logical Reasoning via Reformulating Mathematical Expression](https://mlsanity.com/paper/2212.02746/) (2022)
- [A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram](https://mlsanity.com/paper/2302.11097/) (2023)
- [VisOnlyQA: Large Vision Language Models Still Struggle with Visual Perception of Geometric Information](https://mlsanity.com/paper/2412.00947/) (2024)
- [VisuoThink: Empowering LVLM Reasoning with Multimodal Tree Search](https://mlsanity.com/paper/2504.09130/) (2025)
- [MathCanvas: Intrinsic Visual Chain-of-Thought for Multimodal Mathematical Reasoning](https://mlsanity.com/paper/2510.14958/) (2025)
- [CodePlot-CoT: Mathematical Visual Reasoning by Thinking with Code-Driven Images](https://mlsanity.com/paper/2510.11718/) (2025)
- [Visually-Guided Policy Optimization for Multimodal Reasoning](https://mlsanity.com/paper/2604.09349/) (2026)
- [GPO: Learning from Critical Steps to Improve LLM Reasoning](https://mlsanity.com/paper/2509.16456/) (2025)
- [GeometryZero: Advancing Geometry Solving via Group Contrastive Policy Optimization](https://mlsanity.com/paper/2506.07160/) (2025)
- [GeoTikzBridge: Advancing Multimodal Code Generation for Geometric Perception and Reasoning](https://mlsanity.com/paper/2603.22687/) (2026)
- [CFPO: Counterfactual Policy Optimization for Multimodal Reasoning](https://mlsanity.com/paper/2606.23206/) (2026)
- [Geoparsing: Diagram Parsing for Plane and Solid Geometry with a Unified Formal Language](https://mlsanity.com/paper/2604.11600/) (2026)
- [GeoSym127K: Scalable Symbolically-verifiable Synthesis for Multimodal Geometric Reasoning](https://mlsanity.com/paper/2605.16371/) (2026)

## Abstract (arXiv)

Multimodal geometry reasoning requires VLMs to extract precise visual relations and preserve them through multi-step deduction. Existing free-form traces obscure the decisions that determine the answer, and trajectory-level reinforcement learning distributes a single terminal signal across the entire response. We introduce credit-addressable reasoning, in which the semantic units exposed during inference also define where learning compares alternatives and assigns credit. We instantiate this principle with Code-CoT, which retains the diagram, represents visual relations as line-addressable executable code, and organizes reasoning into typed events, and CE-GRPO, which selects event boundaries using structural priors and type-normalized entropy, samples complete continuations from shared prefixes, and converts outcome differences into localized advantages. Across nine geometry benchmarks, CE-GRPO achieves an average accuracy of 76.04, outperforming Qwen3-VL-8B and trajectory-level GRPO by 8.09 and 3.43 points, respectively. Its relative advantage increases with the number of intermediate events, demonstrating the value of representation--optimization co-design for long, dependency-heavy multimodal reasoning.