Testing for Cointegration in Cliord Algebra: Cross-Level Bivectors and Non-Parametric Inference in Cl(2d,0)

We propose a non-parametric test for cointegration based on bivector means in the Cliord algebra Cl(2d,0), where d is the number of integrated series. The test embeds a d-dimensional I(1) system in a 2d-dimensional state space (levels and rst dierences) and computes the geometric product of consecutive state vectors. The resulting bivector decomposes into three interpretable blocks: cross-level bivectors that detect cointegration, cross-predictability bivectors that identify the direction of error correction, and velocity co-movement bivectors that capture short-run dy- namics. Our main result establishes that the cross-level bivector in the e1e2 plane is strictly non-zero under cointegration and converges to zero under independent random walks, extending the spectral identity of Vázquez Broquá and Sudjianto (2026) from the univariate to the multivariate setting. As a secondary result, we show that the cross-predictability bivectors e14 and e23 provide a non-parametric identication of the error-correction loading vector α, recovering the direction of adjustment without estimating the VECM. The test uses permutation inference on increments, requires no hyperparameters, and produces diagnostic channel proles with no analogue in existing cointegration tests. Applied to US Treasury yields (TB3MS, GS5, GS10) the same data used by Sudjianto and Narain (2026) for geometric estimation with rotors the test conrms cointegration, identies bilat- eral error correction with the short rate adjusting faster, and demonstrates a natural pipeline: GDF pre-testing, geometric cointegration testing, rotor-based estimation.