Multi-agent Autoformalization of Tensor Network Theory

2026-07-08

AI Summary

This work addresses the lack of formal verification for foundational results in tensor network theory—such as the fundamental theorem of matrix product states—and the challenge of preserving mathematical intent during large-scale autoformalization. To this end, it introduces the first multi-agent collaborative framework for the automatic formalization of complex physical theories. Built upon the Lean theorem prover, the framework integrates domain-specialized large language model agents, structured mathematical blueprints, and a human-in-the-loop review mechanism. It successfully formalizes the fundamental theorem of matrix product states, uncovers a novel proof pathway absent from the literature, and extends formalization to physical concepts like symmetry-protected topological phases. The project also establishes TNLean, the first library for tensor networks and quantum information in Mathlib, with all code and formalization blueprints publicly released.

Abstract

We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library \href{https://github.com/LionSR/TNLean}{TNLean}, together with a \nChapters{}-chapter \href{https://lionsr.github.io/TNLean/blueprint/}{blueprint} of the formalization effort.

Problem

• autoformalization
• tensor network theory
• mathematical intent
• formal verification
• theoretical physics

Innovation

• multi-agent autoformalization
• tensor network theory
• matrix-product states
• Lean formalization
• symmetry-protected topological phases