One Pole, Two Failures: The Marchenko–Pastur 1/(1-γ) Singularity in Overparameterized Regression and in Markowitz Portfolios

Two phenomena that are usually discussed in separate literatures share one mathematical origin. In machine learning, the test error of minimum-norm least squares spikes when the number of parameters p crosses the number of samples n ("double descent"). In quantitative finance, the realized risk of a Markowitz portfolio built from a sample covariance matrix blows up as the number of assets N approaches the length of the history T. We show that both are governed by the same object, the mean of an inverse Wishart matrix, whose large-dimensional limit is the value ∫ λ^-1 dμ_MP(λ) = 1/(1-γ) of the Marchenko–Pastur law. We give exact finite-sample expressions, not only asymptotics: for Gaussian designs the expected excess risk of minimum-norm least squares is σ^2 p/(n-p-1) below the interpolation threshold and r^2(1-n/p) + σ^2 n/(p-n-1) above it, and for Gaussian returns the in-sample and out-of-sample variances of the sample minimum-variance portfolio, relative to the true minimum, have expectations (T-N+1)/T and (T-1)/(T-N). Every formula is checked against Monte Carlo simulation, and the reproducing script is included. We then map the standard remedies onto each other: optimally tuned ridge regression removes the double-descent peak, and eigenvalue clipping at the Marchenko–Pastur edge cuts the out-of-sample risk inflation at N/T = 0.8 from 5.0× to 1.2×. Finally we show numerically that the exact results fail for heavy-tailed (Student-t) returns, where the in-sample risk is underestimated even more severely than the Gaussian formula predicts. The contribution is a unified, self-contained, and verified treatment. Most of the individual ingredients are classical, and we attribute them accordingly.