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

Abstract

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 $U_i(S_i\cdot S_{i+1})$ and $V_i(S_i\cdot S_{i+2})$ is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler $O(n)$ open chain with nearest-neighbor interaction $V_i({\mathbf{\sigma }}_i\cdot {\mathbf{\sigma }}_{i+1})$ and axial single-spin potential $U_i({\sigma }_i^z)$ for every integer $n\ge 1$ and every system size $L\ge 1$. The homogeneous linear specialization maps the foundational frustrated $J_1$-$J_2$ 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 $J_1$-$J_2$ standard Potts open chain for every $q\ge 2$ and every $L\ge 1$. 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.

17 pages (extended from 12 pages), 3 figures, 2 tables; added the link and message-level citations to the human-AI conversation transcript [38]