The Pythagorean Gap: A Mode-Decomposition Indicator for Financial Crises

We introduce the Pythagorean Gap, a single scalar metric that measures the dispersion of risk-adjusted returns across spectral modes of a return distribution. When asset returns are decomposed into orthogonal modes via PCA, the tangency portfolio's Sharpe ratio satisfies a Pythagorean identity: \(SR^2_{\text{tangency}} = \sum_k SR^2_k\). The Gap — defined as \(G_t = \sum_k SR^2_k - SR^2_{\text{ref}}\), where \(SR_{\text{ref}}\) is the Sharpe ratio of an equal-variance-weighted reference portfolio — measures how far mode-level Sharpe ratios have diverged from uniformity. We verify the algebraic framework in Lean 4 (machine-verified, zero sorry) and test the indicator on 18 years of S&P 500 sector ETF data (2007–2025). The Pythagorean Gap provided early warning signals in 5 of 6 major market crises; in the out-of-sample test for COVID-19, it generated 17 early-signal days with the first signal 23 trading days before the crash of March 2020. The Gap has near-zero correlation with the VIX (\(r = -0.06\)), indicating that it captures fundamentally different information: not the level of fear, but the structural breakdown of diversification. Unlike variance-based PCA indicators such as the absorption ratio (Kritzman et al., 2011), the Gap operates on Sharpe ratios, connecting crisis detection directly to portfolio optimality.

Keywords: crisis detection, diversification, mode decomposition, PCA, Pythagorean theorem, Sharpe ratio, systemic risk

JEL: G01, G11, G12, C58