The Mathematics of Quintile Portfolios: Rank Sufficiency, the Winner's Curse Deflator, and the Optimal Selected Fraction

The quintile portfolio ranks a cross section by a score and holds the top fifth. This paper develops a theory of the heuristic. The equal-weighted top-k portfolio is the projection of a score vector onto equal-weighted k-subsets in every ₚ metric ($p>1$) and every separable Bregman divergence; under diagonal covariance, sorting by μᵢ/σᵢ² is the corresponding mean--variance rule. Under a budget constraint, an equal-weighted selection S is exactly mean--variance optimal if and only if μ=aΣS+b for $a>0$. Rank-only allocation annihilates strictly increasing transformations; estimation losses enter through boundary inversions and are quadratic under small Gaussian noise. Selection on a noisy Gaussian score asymptotically realizes ² of its apparent excess. For a dollar-neutral Gaussian top-minus-bottom spread, the Sharpe-optimal fraction solves φ(zq)=2qzq, giving q^=0.2703 independently of cross-sectional size, residual correlation, information coefficient, and scale. Its transfer coefficient is $0.8999$, versus $0.8853$ at $q=0.20$. Score-independent factor loadings preserve the optimum under an expected-variance criterion. With unequal tails, the heavier side controls local behavior: a tail index below two favors concentration, while indices above two on both sides exclude full concentration but leave the optimum distribution-dependent. Student-t calculations give q^=0.207 for t₄ and $0.251$ for t₁₀. For long-only portfolios, a market-mode variance floor eliminates any fixed interior optimum; the optimal name count grows only logarithmically with cross-sectional size. Under independent Gaussian score dynamics, expected turnover decreases with breadth. A 2017--2025 CRSP momentum audit does not identify a universal fraction. In 2021--2025, q^=0.2703 has Sharpe $0.507$ versus $0.567$ for a fifth, with paired block-bootstrap interval $[-0.164,0.083]$. A persistence-calibrated Gaussian-null simulation gives a central $68%$ range $[0.15,0.40]$ for the sixty-month sample maximizer. The turnover law fits common persistence near the measured $0.902$ but shows residual drift across fractions. The analysis separates proved claims, conditional models, and descriptive evidence. Lean~4 kernel-checks the discrete exchange and cancellation core and records open obligations.