Teacher Knows It Best: Spontaneous Symmetry Breaking and Tipping Points in Networked Langevin Dynamics AI Sycophancy

We formulate a statistical physics framework to model a networked stochastic dynamical system exhibiting bistability, driven by additive noise and social conformity. We apply this model to understand and mitigate AI-induced delusional spiraling-a phenomenon where algorithmic sycophancy from Large Language Models continuously reinforces inaccurate beliefs within a socially interacting society. By partitioning the network into a majority of regular agents and a minority of "aware" nodes (Teachers) placed at topological hubs, we use a degree-weighted mean-field approximation to reduce high-dimensional coupled Langevin equations into a single macroscopic drift equation. We provide a closed-form analytical derivation for the deterministic critical tipping time through a saddle-node bifurcation. We validate this analytical boundary using finite-size scaling and demonstrate a universal data collapse across diverse network topologies. Finally, we optimize an intervention strategy under a strict budget constraint that balances the topological footprint against driving velocity. We prove mathematically that under certain conditions, a highly concentrated, rapid intervention targeting massive hubs strictly outperforms a distributed, slow approach to rescue the network.