Riemannian Covariance Estimation and Regime Detection for Portfolio Management: Benchmark Results

OmniSciences Technical Report TR-2026-004 We present benchmark results for Riemannian geometric methods applied to portfolio covariance estimation, conditioning, and regime detection. Key results (615+ tests, 5 independent Monte Carlo universes): 63% of Euclidean sample covariance estimates fail positive-definiteness at realistic sample sizes — Riemannian: 0% failure (p<0.0001) Geodesic shrinkage produces dramatically better-conditioned estimates Regime detection via geodesic distance detects correlation structure shifts Honest disclosure: Portfolio performance metrics (Sharpe, drawdown) are NOT statistically significant across universes Patent-pending (U.S. Provisional Application No. 64/007,419).