Polar Geometry of Time-Odd EEG Dynamics
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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 , Δ x j ) and isolated its exact skew component, . Polar decomposition, K = Q K M K , separated interaction magnitude from normalized orientation. We applied the same decomposition to signed imaginary coherency, Ω = 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 → 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 Ω → Q Ω 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.
