Square-Root Price Impact Is Necessary for Endogenous Manipulation Cycles in Learning-Agent Markets

We study a minimal agent-based market in which a single evolutionary-optimized institutional agent interacts with 20{,}000 herding retail traders. The agent spontaneously discovers a multi-cycle predatory strategy, producing 8--11 complete cycles over 2000 trading days with total portfolio return of $+51\%$ (best of 20 seeds; mean $+37.7\%$). Mean-field reduction maps the system onto a nonlinear oscillator that undergoes two distinct bifurcations: a continuous Hopf transition as institutional capital exceeds a critical threshold $C_c$, with oscillation amplitude $A \propto (C-C_c)^\alpha$ where $\alpha$ is consistent with the standard prediction of $1/2$; and a discontinuous fold transition in the herding-scale parameter space. The limit cycle persists even at $\beta = 0$: position-tracking feedback coupled with square-root price impact creates a self-sustained nonlinear oscillator requiring no retail herding. Square-root impact is shown to be necessary: linear impact eliminates the Hopf bifurcation entirely and renders the retail market unconditionally stable. Manipulation cycles thus emerge as the optimal-control solution of a nonlinear dynamical system, and a structural analogy to Maxwell's demon frames the agent as an information-processing controller that reduces the entropy rate of the price process.