The Multigrade Signal Dynamics Signature

The Geometric Signal Dynamics (GSD) programme decomposes univariate time series into the algebraic grades of Cl(3,0) – scalar, bivector, pseudoscalar – with interpretable di- agnostics for persistence, rotation, and fragility. Its multivariate extensions are fragmented: cointegration uses Cl(2d,0) and discards the pseudoscalar channel, random-matrix scaling usesCl(3d,0) onlyasavectorspace,andtransferentropyremainspairwise. Weintroducethe Multigrade Signal Dynamics Signature (MSDS): the multigrade-decomposed, time-averaged ˆ iterated geometric product of the joint kinematic embedding in Cl(3n,0)∼ = Cl(3,0) ⊗n. The multigrading (k1,...,kn) inherited from the tensor decomposition turns every subspace into an interpretable channel – per-series fragility, cross-bivectors, directional cross-fragility, joint pseudoscalar – and every existing GSD primitive recovers as a specific slice. We prove (Proposition 1) that the cross-multigrade-(1,1) component of MSDS2 equals the quadrature spectrumofthecross-spectraldensity, whilethecommutingtensorvariantCl(3,0)⊗R Cl(3,0) recovers the co-spectrum: the choice of ambient algebra thus factorises directional from time-symmetric information, sharpening Conjecture 1 of Vázquez Broquá (2025a) into an explicit identity. We position MSDS as a canonical compression of the Lyons signature. For low ambient dimension n the count 23n is orders of magnitude below truncated Lyons N k=0(3n)k atmoderatedepthN; forlargernthisrawinequalityreversesbutthemultigrade structure allows interpretable aggregation that keeps the working feature count comparable to log-signature variants. The decisive contribution is interpretive: MSDS slices map to existing time-series primitives rather than to abstract tensor entries. Seven case studies on synthetic and real data, including a 25-year SPDR sector ETF panel, delineate where the framework wins (third-order tasks, noise-dominated regimes, direction-of-adjustment in cointegration) and where it matches generic feature methods at substantially lower dimen- sional cost with explicit interpretability (synthetic regime panels, mechanism-typed stress classification). Specialised parametric tools retain the lead on existence-of- cointegration tests and pure second-order spillover. MSDS closes the multivariate gap of the GSD pro- gramme; subsequent papers can build directly in its language.