In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator

2025-07-22

AI Summary

This paper presents four open conjectures in graph theory generated autonomously by the AI system TxGraffiti. The system integrates symbolic computation, analysis of graph invariants, and heuristics encoded by mathematicians, refined over years of human–AI collaborative iteration. Each conjecture is concise, formulated exclusively in terms of natural graph invariants, and empirically verified on hundreds of graphs; none has yet been proven or disproven. Methodologically, TxGraffiti transcends conventional AI-as-assistant paradigms by producing mathematically meaningful, standalone conjectures—marking the first instance where machine-generated conjectures are treated as original creative outputs in their own right. The contribution thus lies both in advancing graph theory with novel, falsifiable hypotheses and in establishing a collaborative interface that stimulates joint exploration between human mathematicians and AI. Moreover, the work prompts foundational reflection on the capacity of machines to participate meaningfully in creative mathematical reasoning.

Technology Category

• Humans and AI: Other Foundations of Human Computation & AI
• Machine Learning: Graph-based Machine Learning
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

We present four open conjectures in graph theory generated by the automated conjecturing system exttt{TxGraffiti}. Each conjecture is concise, grounded in natural graph invariants, and empirically validated across hundreds of graphs. Despite extensive effort, these statements remain unresolved--defying both proof and counterexample. They are not only mathematical challenges but creative expressions--born of symbolic pattern recognition and mathematician-defined heuristics, refined through years of human dialogue, and now offered back to the community as collaborative artifacts. These conjectures invite not only formal proof, but also reflection on how machines can evoke wonder, spark curiosity, and contribute to the raw material of discovery. By highlighting these problems, we aim to inspire both human mathematicians and AI systems to engage with them--not only to solve them, but to reflect on what it means when machines participate meaningfully in the creative process of mathematical thought.

Problem

• Open conjectures in graph theory generated by AI
• Conjectures defy proof and counterexample despite validation
• Machines contribute creatively to mathematical discovery process

Innovation

• Automated conjecturing system generates graph theory conjectures
• Symbolic pattern recognition refines conjectures via human dialogue
• Empirical validation across hundreds of graphs ensures robustness