Moonshine: An Autonomous Mathematical Research Agent Centered on Conjecture Generation

2026-06-09

AI Summary

This work proposes the Neural Jacobian Conjecture (NJC), which transposes the classical Jacobian conjecture—concerning whether local non-degeneracy implies global injectivity—into the setting of single-hidden-layer affine-ridge Sigmoid neural networks. By integrating the interactive reasoning capabilities of large language models (GPT-5.5-pro, DeepSeek-V4-pro, and ChatGPT) with geometric topology and symbolic reasoning, the study establishes an autonomous research pipeline that spans problem transposition, conceptual abstraction, conjecture formulation, and automated verification. For networks of width \(N = n+1\), the authors obtain two independent complete proofs alongside a geometric-topological proof, providing preliminary validation of the NJC’s plausibility and offering a scalable framework for AI-driven mathematical discovery.

Abstract

Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single proposition as its endpoint, Moonshine builds an extensible theoretical framework through conjecture generation, bridge building, and obstacle identification. This article uses Moonshine's exploration of the Jacobian conjecture as an example. It shows how the central logic of whether local nondegeneracy can force global injectivity is transferred to one-hidden-layer affine-ridge sigmoid networks. This leads to the formulation of the \emph{Neural Jacobian Conjecture} (NJC): if such a network has strictly positive Jacobian determinant on the whole space, then it must be globally injective. By invoking GPT-5.5-pro and DeepSeek-V4-pro separately, Moonshine obtained independent complete proofs for the case \(N=n+1\). In addition, with the assistance of ChatGPT through interactive use of its web interface with GPT-5.5-pro, a geometric-topological proof was developed. These results provide preliminary evidence for the plausibility of the conjecture. The general higher-width case \(N\ge n+2\), however, remains unresolved and is left for further investigation. This work illustrates Moonshine's ability to autonomously generate meaningful mathematical problems and make rigorous progress on them.

Problem

• conjecture generation
• Jacobian conjecture
• global injectivity
• neural networks
• autonomous mathematical research

Innovation

• autonomous mathematical agent
• conjecture generation
• Neural Jacobian Conjecture
• large language models in mathematics
• global injectivity