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  "claims": [
    {
      "claim": 1,
      "literal_claim": "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)."
    },
    {
      "claim": 2,
      "literal_claim": "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)."
    },
    {
      "claim": 3,
      "literal_claim": "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)."
    },
    {
      "claim": 4,
      "literal_claim": "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)."
    },
    {
      "claim": 5,
      "literal_claim": "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)."
    }
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    "claim5_risk_improves": true
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  "paper_id": "G4ve69pimc"
}