MECA: A Mechanism-Centered Agent for Constructing Well-Specified and Valuable Mathematical Conjectures

2026-07-30

AI Summary

This work addresses the limitation of existing automated conjecture generation methods, which often produce vague or ill-defined propositions that offer little guidance for formal proof. To overcome this, the authors propose MECA, a multi-agent framework that uniquely centers mathematical mechanisms—such as inequalities and invariants—to guide conjecture formulation. In MECA, an explorer agent generates and tests mechanism-driven conjectures, while a critic agent evaluates their rigor and research value; the two agents iteratively refine hypotheses and conclusions through collaboration and integrate an automated theorem prover for validation. Experimental results demonstrate that MECA not only reconstructs known conjectures effectively but also generates 100 novel, well-formed conjectures in semi-open problem settings that exhibit strong research potential and remain challenging for current theorem provers.

Abstract

Automatically constructing well-specified and valuable mathematical conjectures remains a central challenge in AI-assisted mathematical discovery. Many existing open problems and conjectures are often too broad, underspecified, or difficult to connect to plausible proof or refutation strategies. We view a mathematical mechanism as a structure or reasoning principle that connects the assumptions of a candidate problem to its target conclusion, such as an inequality, invariant, decomposition, or reduction to an intermediate claim. We present MECA (MEchanism-centered Conjecture Agent), a multi-agent framework that constructs conjectures by jointly developing candidate statements and their supporting mechanisms. Explorer agents propose mechanisms, test how they apply, and revise the candidate conjecture accordingly, while critic agents assess their mathematical validity and research value. Their feedback guides changes to the assumptions, scope, and conclusion. Through this process, MECA transforms broad research directions into precise conjectures with substantive mathematical support while retaining a clearly identified unresolved core. We evaluate MECA in two complementary settings. First, we compare it with a generate-and-revise baseline on reconstructing preselected target-paper conclusions from target-conditioned but article-blind source materials. Second, we construct 100 semi-open problems from literature-derived seeds and existing open problems and evaluate them through independent proof and refutation attempts by automated provers. Our results indicate that mechanism-centered refinement produces well-specified and research-worthy conjectures that remain challenging for current automated provers.

Problem

• mathematical conjectures
• AI-assisted discovery
• well-specified problems
• research value
• automated reasoning

Innovation

• mechanism-centered reasoning
• mathematical conjecture generation
• multi-agent framework
• automated mathematical discovery
• conjecture refinement