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 $\ell_p$ metric ($p>1$) and every separable Bregman divergence; under diagonal covariance, sorting by $\mu_i/\sigma_i^2$ is the corresponding mean--variance rule. Under a budget constraint, an equal-weighted selection $S$ is exactly mean--variance optimal if and only if $\mu=a\Sigma\one_S+b\one$ 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 $\IC^2$ of its apparent excess. For a dollar-neutral Gaussian top-minus-bottom spread, the Sharpe-optimal fraction solves $\varphi(z_q)=2qz_q$, giving $q^{\ast}=0.2703$ 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 $q^{\ast}=0.207$ for $t_4$ and $0.251$ for $t_{10}$. 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, $q^{\ast}=0.2703$ 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.