Geometric Transfer Entropy: A Grade-Resolved Directional Information Flow for Dynamical Coupling of Time Series

We introduce Geometric Transfer Entropy (GTE), a measure of directional
information flow between time series that preserves the grade structure of the
Clifford-algebraic embedding underlying the Geometric Signal Dynamics (GSD)
programme. Building on the kinematic embedding and geometric product of Vázquez
Broquá (2026a), the spectral bivector identity of Vázquez Broquá (2026b), the joint
embedding for cointegration of Vázquez Broquá (2026c), and the antisymmetric
entropy HAS of Vázquez Broquá (2026d), we define GTEX→Y (τ ) as the sum of
two grade-resolved channels: a scalar channel T(0)X→Y that captures transfer of alignment/level information, and a bivector channel T(2)X→Y that captures transfer of rotational information (velocity, curvature, phase). The net flow ΦXY = GTEX→Y −GTEY →X decomposes analogously into a scalar and a rotational flow, ΦXY = Φ(0) XY + Φ(2) XY . Across seven benchmark systems, GTE recovers the classical Schreiber transfer entropy on level-coupled linear systems, detects velocity-coupled systems four times more strongly than Schreiber TE, and correctly identifies the direction
of coupling in stochastic oscillators where classical TE reverses sign. The bivariate signature (Φ(0), Φ
(2)) produces a taxonomy of dynamical coupling mechanisms and yields geometric diagnostics of cointegration, common confounding, and pure rotational transfer that classical information flow measures cannot provide.