From LLM-Generated Conjectures to Lean Formalizations: Automated Polynomial Inequality Proving via Sum-of-Squares Certificates

2026-05-14 · Source: Artificial Intelligence · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

A neuro-symbolic framework named NSPI has been developed to automate the proving of polynomial inequalities, a significant challenge in mathematical reasoning due to complex algebraic structures and large search spaces. NSPI integrates Large Language Models (LLMs) with symbolic computation to overcome the scalability limitations of purely symbolic methods and the restricted scope of LLM-guided approaches. The framework operates by having an LLM propose an approximate polynomial Sum-Of-Squares (SOS) decomposition, which is then refined through symbolic computation to achieve an exact polynomial SOS representation. This exact representation directly proves the inequality and is subsequently certified in Lean, establishing an end-to-end pipeline from heuristic discovery to machine-checked proof. NSPI demonstrates effectiveness and scalability on benchmarks involving polynomials with up to 10 variables.

Key takeaway

For AI scientists and research scientists working on automated mathematical reasoning, NSPI offers a scalable approach to polynomial inequality proving. You should consider integrating neuro-symbolic frameworks that leverage LLMs for heuristic discovery and symbolic computation for rigorous proof generation and formal verification. This method can enhance the scalability and reliability of automated theorem proving systems, especially for complex problems with many variables.

Key insights

NSPI combines LLMs and symbolic computation to automate polynomial inequality proving, generating machine-checked Lean proofs.

Principles

• Combine LLMs with symbolic methods.
• Refine approximate LLM outputs symbolically.

Method

An LLM generates an approximate Sum-Of-Squares decomposition, which symbolic computation refines into an exact polynomial SOS representation. This exact representation proves the inequality and is then certified in Lean.

In practice

• Use LLMs for initial conjecture generation.
• Apply symbolic methods for proof refinement.
• Certify proofs using formal verification tools.

Topics

• Automated Polynomial Inequality Proving
• Neuro-Symbolic Frameworks
• Sum-of-Squares Certificates
• Large Language Models
• Formal Verification

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Artificial Intelligence.