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

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(\boldsymbolσ_i\cdot\boldsymbolσ_{i+1}) and axial single-spin potential U_i(σ_i^z) for every integer n\ge1 and every system size L\ge1. 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\ge2 and every L\ge1. 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.