# A Human-AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension

arXiv: https://arxiv.org/abs/2609.00322 · HTML: https://mlsanity.com/paper/2609.00322/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 60 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Accelerating scientific discovery with Co-Scientist](https://mlsanity.com/paper/2502.18864/) (2025)
- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [SciSciGPT: Advancing Human-AI Collaboration in the Science of Science](https://mlsanity.com/paper/2504.05559/) (2025)
- [When Should Users Check? Modeling Confirmation Frequency in Multi-Step Agentic AI Tasks](https://mlsanity.com/paper/2510.05307/) (2025)
- [A New Workflow for Materials Discovery Bridging the Gap Between Experimental Databases and Graph Neural Networks](https://mlsanity.com/paper/2602.00756/) (2026)

## Abstract (arXiv)

Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field O(n)-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions Ui(Si·Si+1) and Vi(Si·Si+2) is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler O(n) open chain with nearest-neighbor interaction Vi(boldsymbolσi·boldsymbolσi+1) and axial single-spin potential Ui(σiz) for every integer n≥1 and every system size L≥1. The homogeneous linear specialization maps the foundational frustrated J1-J2 model onto the canonical J-h field model---with n=1,2,3 being the Ising, XY, and Heisenberg classical spin models, respectively; the theorem resolved a longstanding challenge for n=3 published in 1990. Its proof was done with an AI-synthesized recursive Householder moving frame and understood via a human-recognized hidden reciprocity. An analogous theorem holds when the continuous O(n) spins are replaced by the q-state Potts spins, implying a closed-form exact solution of the J1-J2 standard Potts open chain for every q≥2 and every L≥1. The emergence of these theorems from a human-AI co-development framework suggests that sustained AI involvement throughout a systematic research program may incubate autonomous scientific breakthroughs and make aspects of the discovery process experimentally testable.