Polar Geometry of Time-Odd EEG Dynamics

Abstract

Abstract Objective. Longitudinal EEG recordings vary across sessions because of electrode reapplication, referencing, recording conditions, and physiological state. We asked whether multichannel EEG contains a time-odd geometric structure that remains subject-discriminative across repeated recordings and extended inter-session intervals. Approach. We constructed a local state-derivative operator T_ij = corr(m_i, Delta x_j) and isolated its exact skew component, K = 1/2(T - T^T). Polar decomposition, K = Q_K M_K, separated interaction magnitude from normalized orientation. We applied the same decomposition to signed imaginary coherency, Omega = Im(G), providing a frequency-domain test based on a separate estimation procedure. We evaluated cross-session subject matching in four public longitudinal EEG datasets, including the Dortmund cohort of 206 participants with recordings separated by approximately five years. Alpha-beta fusion used fixed equal weights without calibration or learned weighting. Main results. The empirical operator T was already strongly dominated by its skew component, indicating that K isolates rather than creates the observed time-odd structure. Polar orientation improved longitudinal discriminability for K to Q_K in every principal comparison. For the approximately one-month RestCog interval, area under the receiver-operating-characteristic curve (AUC) increased from 0.9222 to 0.9597 and rank-1 identification (CMC@1) from 0.6333 to 0.8167. In Dortmund, AUC increased from 0.8901 to 0.9625 and CMC@1 from 0.4757 to 0.6214. Fixed alpha-beta fusion further increased Dortmund Q_K performance to CMC@1 = 0.8350, AUC = 0.9855, and equal-error rate (EER) = 5.90%. The spectral operator reproduced the raw-to-polar AUC advantage for Omega to Q_Omega in both alpha and beta across all four datasets. Matched-metric controls preserved the polar AUC advantage in all 40 raw-to-polar comparisons. Significance. Across two mathematically linked skew EEG operators obtained using different estimation procedures, removing interaction-magnitude weighting while retaining normalized orientation improved cross-session discriminability. The gain reflects improved genuine-impostor geometry rather than necessarily greater absolute similarity between repeated recordings. Polar orientation provides a compact longitudinal descriptor requiring no performance-tuned internal parameter once preprocessing, temporal support, and frequency band are fixed.