O-Forge: An LLM + Computer Algebra Framework for Asymptotic Analysis

2025-10-14

AI Summary

Constructing domain decompositions for research-level asymptotic inequality proofs remains a significant challenge due to its high reliance on mathematical insight and case-specific reasoning. Method: This paper introduces the first closed-loop LLM+CAS framework: large language models generate candidate domain decompositions, which are rigorously verified via symbolic computation in computer algebra systems (e.g., Mathematica); a context-aware symbolic feedback mechanism enables iterative refinement. Contribution/Results: The framework achieves the first trustworthy, bidirectional collaboration between LLMs and CAS in asymptotic analysis—balancing creative hypothesis generation with formal correctness guarantees. It successfully resolves a publicly posed open problem by Terence Tao, demonstrating the feasibility of AI-assisted, research-grade mathematical proof. Beyond asymptotic inequalities, this work establishes a verifiable, reproducible automation paradigm for formal proof synthesis and advances AI from competition-math tooling toward a reliable, scientifically grounded research assistant.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information

Abstract

Large language models have recently demonstrated advanced capabilities in solving IMO and Putnam problems; yet their role in research mathematics has remained fairly limited. The key difficulty is verification: suggested proofs may look plausible, but cannot be trusted without rigorous checking. We present a framework, called LLM+CAS, and an associated tool, O-Forge, that couples frontier LLMs with a computer algebra systems (CAS) in an In-Context Symbolic Feedback loop to produce proofs that are both creative and symbolically verified. Our focus is on asymptotic inequalities, a topic that often involves difficult proofs and appropriate decomposition of the domain into the "right" subdomains. Many mathematicians, including Terry Tao, have suggested that using AI tools to find the right decompositions can be very useful for research-level asymptotic analysis. In this paper, we show that our framework LLM+CAS turns out to be remarkably effective at proposing such decompositions via a combination of a frontier LLM and a CAS. More precisely, we use an LLM to suggest domain decomposition, and a CAS (such as Mathematica) that provides a verification of each piece axiomatically. Using this loop, we answer a question posed by Terence Tao: whether LLMs coupled with a verifier can be used to help prove intricate asymptotic inequalities. More broadly, we show how AI can move beyond contest math towards research-level tools for professional mathematicians.

Problem

• Developing a framework to verify LLM-generated mathematical proofs rigorously
• Solving asymptotic inequalities through AI-driven domain decomposition techniques
• Advancing AI from contest mathematics to research-level mathematical tools

Innovation

• LLM and CAS combined for verified proofs
• In-Context Symbolic Feedback loop ensures rigor
• Domain decomposition proposed by LLM and verified by CAS