[
  "In the population setting, Theorem 1 characterizes excess risk as a function of the magnitude and direction of the performative effect together with spurious features (Section 4, Theorem 1).",
  "Corollary 2 shows the optimal regularization parameter in the population regime is proportional to the strength of the performative effect, with optimal risk remaining strictly positive (Section 4, Corollary 2).",
  "Theorem 3 establishes a deterministic equivalent of the performative fixed point for over-parameterized ridge regression when the number of features exceeds the number of samples (Section 5, Theorem 3).",
  "Theorem 4 shows the optimal regularization moves in the same direction as the performative effect on predictive features under low noise, but in the opposite direction under high noise, in the over-parameterized regime (Section 5, Theorem 4).",
  "Numerical experiments in Section 6 confirm that in the over-parameterized setting, performative effects can improve optimally-regularized risk when performativity reinforces existing trends, contrasting with the population-regime degradation (Section 6)."
]