Null-Validated Topological Signatures of Financial Market Dynamics

Financial markets exhibit temporal organization that is not fully captured by volatility measures or linear correlation structure. We study a null validated topological approach for quantifying market complexity and apply it to Bitcoin daily log returns. The analysis uses the $L^1$ norm of persistence landscapes computed from sliding-window delay embeddings. This quantity shows strong co-movement with stochastic volatility during periods of market stress, but remains intermittently elevated during low volatility regimes, indicating dynamical structure beyond fluctuation scale. Rolling correlation analysis reveals that the dependence between geometry and volatility is not stationary. Surrogate based null models provide statistical validation of these observations. Rejection of shuffle surrogates rules out explanations based on marginal distributions alone, while departures from phase randomized surrogates indicate sensitivity to nonlinear and phase dependent temporal organization beyond linear correlations. These results demonstrate that persistence landscape norms provide complementary information about market dynamics across market conditions.