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Geoint-R1: Formalizing Multimodal Geometric Reasoning with Dynamic Auxiliary Constructions

CVPR 2026 · Vision-language and multimodal › Multimodal LLMs · approach: Spatial reasoning in VLMs

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Introduces Geoint-R1, a Qwen2.5-VL-7B-based model trained with SFT then GRPO using a verification reward (correctness, auxiliary-line construction, format), plus the 1,885-problem Geoint benchmark with Lean4 auxiliary constructions for formal multimodal geometry reasoning.

Problem

Multimodal LLMs struggle to produce formally verifiable geometry solutions that dynamically identify, construct, and visualize auxiliary geometric elements from text and diagrams.

Approach

Combines SFT on Lean4-annotated auxiliary-construction solutions with GRPO whose reward explicitly credits correct auxiliary-line construction, plus difficulty-ordered curriculum sampling, and a new benchmark pairing problems with TikZ diagrams and Lean4 code.

Key results

• Geoint-R1 (7B) reaches 64.72% average accuracy, the best overall, with 57.01% on answer-based questions versus 51.53% for Gemini-1.5-pro.
• On proof-based questions GPT-4o scores highest at 77.99% vs Geoint-R1's 72.43%.
• Ablation: removing the verification reward drops average accuracy from 64.72% to 55.78%.

Surprising findings

• Removing the verification reward causes the largest drop on auxiliary-line answer accuracy (68.63% to 32.35%).

Limitations

• Proof-based performance still trails GPT-4o and Gemini-1.5-pro.
• Error analysis shows misapplied theorems, incomplete logic chains, and symbolic/computation errors.
• Benchmark is small (1,885 items), and training uses only 1,614 samples.

Experimental setup

Datasets: Geoint

Benchmarks: Geoint

Metrics: accuracy process-based proof score

Baselines: Phi-3.5-V-4B LLaVA-v1.6-7B InternLM-XComposer-2.5-7B Yi-VL-6B InternVL3-8B Qwen-VL-2.5-7B GPT-4o Gemini-1.5-pro Gemini-1.5-flash MMR1-Math-v0-7B Math-LLaVA-13B

Builds on: GRPO DeepTheorem

Closest papers by meaning

• Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning (CVPR 2026 · similarity 0.92)
• GeoBench: Rethinking Multimodal Geometric Problem-Solving via Hierarchical Evaluation (ICLR 2026 · similarity 0.91)
• GeoTikzBridge: Advancing Multimodal Code Generation for Geometric Perception and Reasoning (CVPR 2026 · similarity 0.91)
• SolidGeo: Measuring Multimodal Spatial Math Reasoning in Solid Geometry (NeurIPS 2025 · similarity 0.91)
• Synthesizing Multimodal Geometry Datasets from Scratch and Enabling Visual Alignment via Plotting Code (ICML 2026 · similarity 0.90)
• GeoX: Geometric Problem Solving Through Unified Formalized Vision-Language Pre-training (ICLR 2025 · similarity 0.89)