Technical Summary: Hilbert-Geo

Problem Statement

Geometric problem solving is a critical multimodal reasoning task. While recent advances in Multimodal Large Language Models (MLLMs) have achieved significant success in plane geometry, they struggle considerably with solid geometry. The paper identifies three primary hurdles preventing MLLMs from mastering solid geometry:

1. 3D Spatial Complexity: Solid geometry requires precise modeling of 3D spatial relationships, occluded structures, and transformations, which often exceeds the abstract spatial logic capabilities of current models.
2. Multimodal Grounding: Interpreting 2D representations of 3D objects requires sophisticated visual-linguistic alignment. MLLMs frequently suffer from perception errors and hallucinations when inferring implicit spatial information from diagrams.
3. Quantitative Reliability: MLLMs are prone to calculation errors in quantitative tasks, undermining the reliability of their solutions.

Existing datasets and benchmarks (e.g., SolidGeo, MathVerse-Solid) are described as scattered, potentially containing data leakage, relying on LLM-generated answers prone to errors, or including low-relevance physics/math modeling problems. Furthermore, most benchmarks assess only final answers, failing to detect reasoning flaws.

Methodology: Hilbert-Geo Framework

To address these challenges, the authors propose Hilbert-Geo, the first unified formal language framework specifically designed for solid geometry. The system employs a Parse2Reason pipeline consisting of two distinct stages:

1. Solid Geometry Formal Language

The core of the framework is a formal language extension of the existing FormalGeo (plane geometry) system.

• Predicate Library: A comprehensive library containing 120 predicates, including 35 fundamental predicates, 20 entity types (e.g., Cube, Cone, Prism), 35 entity relationships (e.g., parallel/perpendicular between planes), and 30 attribute descriptors.
• Theorem Bank: An extension of the plane geometry theorem system, containing 220 theorems covering premises and conclusions for solid geometry (e.g., sphere volume formulas, line-plane perpendicularity judgments).
• Conditional Description Language (CDL): A formalized language used to represent both natural language problem descriptions and visual diagrams. CDL eliminates multimodal ambiguities by grounding textual entities in visual structures.

2. Parse2Reason Pipeline

• Multimodal Formalization Parser (M2FP): This module translates natural language text and visual diagrams into the formal CDL. It utilizes prompt engineering with expert-designed few-shot examples to guide MLLMs in parsing geometric conditions into structured predicates. The parser ensures cross-modal consistency by verifying that entities in text and image CDLs are declared in the construction CDL.
• Solid Geometry Reasoning Engine (SGRE): Once the problem is formalized into CDL, the SGRE performs relational inference and algebraic computation. It operates as a search-based engine that builds a reasoning tree:
  • It initializes a node-based tree where nodes represent geometric states.
  • It iteratively matches expandable nodes against the theorem bank, applying logical substitutions and calculations.
  • It generates strictly correct, verifiable, and human-readable reasoning processes.
  • The engine is designed to handle the NP-hard nature of the search space, though it notes that combinatorial explosion can occur in highly complex problems.

Key Contributions

The paper outlines four primary contributions:

1. Solid Geometry Formal Framework: Development of a specialized Predicate Library and Theorem Bank tailored for 3D entities and spatial relationships, extending the theoretical foundations of plane geometry formalization.
2. Multimodal Formalization Parser: A pipeline capable of parsing both natural language and visual diagrams into a unified formal language (CDL), resolving cross-modal ambiguities.
3. Geometry Reasoning Engine: A dedicated engine for performing verifiable, symbolic deduction on formalized problems, ensuring logical coherence and traceability.
4. Expert-Annotated Datasets: The curation of two high-quality datasets:
   • SolidFGeo2k: 1,908 solid geometry problems with formal annotations, reasoning paths, and answers.
   • PlaneFGeo3k: 3,022 plane geometry problems with similar rigorous annotations.
     Both datasets are manually annotated by domain experts to ensure accuracy and consistency, addressing the scarcity of high-quality reasoning resources.

Experimental Results

The authors conducted extensive experiments comparing Hilbert-Geo against leading MLLMs (including GPT-5, Gemini 2.5 Pro, Claude 3.7 Sonnet) and human performance.

• SolidFGeo2k Performance:
  • Hilbert-Geo achieved 77.3% accuracy (using Gemini-2.5-Pro for parsing), substantially outperforming the best MLLM, Gemini-2.5-Pro (54.2%), and GPT-5 (50.6%).
  • It approached human performance (81.8%) and significantly surpassed open-source models (e.g., Llama 3.3 70B at 33.6%).
• MathVerse-Solid Performance:
  • Hilbert-Geo achieved 84.1% accuracy, outperforming GPT-5 (62.9%) and Gemini 2.5 Pro (59.7%).
• PlaneFGeo3k Performance:
  • Demonstrating generality, Hilbert-Geo achieved 80.2% accuracy on plane geometry tasks, outperforming leading MLLMs (GPT-5 at 68.1%) and approaching human performance (87.9%).
• Error Analysis:
  • Analysis of MLLM failures revealed that visual perception errors and reasoning errors account for ~73-76% of failures.
  • Hilbert-Geo significantly reduced hallucinations and visual perception errors by enforcing formal constraints during the parsing phase.
  • The reasoning engine showed no calculation or reasoning errors when the input CDL was correct; failures were primarily attributed to parsing distortions or combinatorial explosion in the search space.

Significance and Claims

The paper claims that Hilbert-Geo represents a significant step forward in automated geometric reasoning by bridging the gap between neural perception and symbolic rigor. Its significance lies in:

• Addressing the 3D Gap: It is the first unified framework to tackle the unique challenges of solid geometry, moving beyond the limitations of plane geometry-focused systems.
• Reliability and Verifiability: By decoupling perception (parsing) from reasoning (symbolic deduction), the system generates strictly correct, verifiable, and human-readable solutions, mitigating the "black box" nature and hallucination risks of pure MLLMs.
• Benchmarking: The introduction of SolidFGeo2k and PlaneFGeo3k provides a critical, high-quality benchmark for evaluating geometric reasoning, moving away from datasets reliant on potentially flawed LLM-generated answers.
• Generality: The framework's success on both solid and plane geometry tasks demonstrates the robustness of the neural-symbolic approach in handling diverse geometric reasoning scenarios.

The authors conclude that while visual perception and logical reasoning deficits remain bottlenecks for current MLLMs, structured, formal approaches like Hilbert-Geo offer a viable path to overcoming these limitations in spatial logic and multimodal interpretation.