3. **Reputation decoupling (anonymity, platform churn) lowers $\beta$ and $r$.** Lower $\beta$ paradoxically *raises* endemic deception $D^* = (r/\beta)(1-1/\mathcal R_0)$; lower $r$ slows recovery and, via the Jacobian, **destabilizes the spiral** — pushing the system from damped oscillation toward a **limit cycle** (sustained boom–bust fraud waves that never settle).