TS-CAN: Time Series Forecasting via the Clifford Geometric Product

Deep time series forecasters have advanced rapidly by redesigning where representations interact, whether across patches, variables, or scales. The interaction itself, however, remains largely limited to dot-product similarities or learned weighted sums. These classical primitives solely measure how strongly two representations agree. Classical time series analysis instead suggests that relating two signals requires both an in-phase component (co-movement) and a quadrature component (oriented, out-of-phase structure). In this work, we show that the geometric product of Clifford algebra, $\mathbf{uv}=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\wedge \mathbf{v}$, yields exactly this dual decomposition within a single operation. Its generalized inner product captures scalar agreement, while its antisymmetric wedge term serves as a discrete, learnable counterpart of the Lévy area from rough path theory, capturing higher-order directional structure. Building on this principled mathematical foundation, we propose TS-CAN, a forecaster whose sole token interaction is the geometric product itself. Applied causally along time and sparsely across the feature dimension, this geometric interaction is representationally dense enough to internalize nonlinear channel mixing, entirely eliminating the need for self-attention and heavy feed-forward networks (FFNs). Operating at a strict linear complexity of $\mathcal{O}(N)$, TS-CAN achieves the lowest MSE in 28 of 32 long-term forecasting settings across eight benchmarks, while performing strongly on short-term tasks. These findings establish a new accuracy-efficiency Pareto frontier and suggest that how representations interact is a fundamental, unexplored design dimension for time series modeling.