Cross-sectional topological anomaly scores and intraday return predictability in the S&P 500: A BallMapper, decoder-conditional VAE, and Function-on-Function regression approach

Anomaly detection methods in financial time series score statistically unusual observations in observable data, not topologically misexpected persistent deviations in the latent structure of co-movement. This study constructs a stock-level topological anomaly score jointly conditioned on market-level topological structure and cross-sectional peer context, and tests whether its history carries predictive content for return curves. Intraday data for ten liquid S&P 500 constituents (April 2025--March 2026) are embedded via Takens delay embedding, graphed by BallMapper, and scored by three decoder-conditional variational autoencoder variants. Predictive content is assessed by penalised function-on-function regression and confirmed across all assets, intraday bar frequencies, and scoring variants, revealing a consistent temporal fingerprint -- gradual accumulation of return impact, a frequent early reversal of its direction, and broadly distributed predictive content weighted toward recent anomaly history. When the reversal occurs depends on market regime; how evenly the anomaly history contributes to prediction depends on bar frequency.