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  "subfield": "asset_pricing",
  "subfield_secondary": "portfolio choice / covariance estimation (theoretical correction)",
  "primary_method": "theoretical_model_analytical",
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   "estimand": "not applicable (analytical correction of a proof)",
   "strategy": "not applicable; corrected KKT-based derivation of the adjusted covariance matrix and constrained MLE equivalence",
   "key_assumptions": [
    "returns i.i.d. jointly normal (for Proposition 2)",
    "adjusted covariance matrix nonsingular",
    "sample covariance matrix positive semidefinite"
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   "causal_language_strength": 1,
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   {
    "claim": "The original equation omitted a term, so the original proofs hold only with no-short-sale constraints and not with upper bounds on weights.",
    "evidence_type": "equation_or_proof",
    "evidence_ref": "Corrected equation (10) vs. original equation (8)"
   },
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    "claim": "With the corrected adjustment, the shrunk covariance matrix is symmetric and PSD, and the constrained minimum variance portfolio is one of its GMV portfolios.",
    "evidence_type": "equation_or_proof",
    "evidence_ref": "Proposition 1 and proof"
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    "claim": "Under normality, the adjusted matrix is the constrained MLE whose implied GMV portfolio satisfies the weight constraints, and conversely.",
    "evidence_type": "equation_or_proof",
    "evidence_ref": "Proposition 2 and proof; Morrison (2005); Zwiernik, Uhler, and Richards (2017)"
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    "evidence_type": "equation_or_proof",
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