Complex-Time Neural Networks: Geometric Temporal Access for Long-Range Reasoning

Most neural architectures model time as a one-dimensional real-valued variable, constraining temporal reasoning to sequential propagation along a single axis. We introduce Complex-Time Neural Networks (CTNN), a new class of architectures in which temporal coordinates are elements of the complex plane T = t + iτ ∈ ℂ, where Re(T) preserves chronological ordering and Im(T) encodes an orthogonal experiential dimension. Within this geometry, Im(T) < 0 defines a memory domain enabling retrospective retrieval, Im(T) = 0 corresponds to present-moment computation, and Im(T) > 0 defines an imagination domain for prospective projection. We prove the Expressive Separation Theorem (Theorem 1), establishing that, within the temporally coupled function class GTCP and under explicit Assumptions A1–A4 (in particular the bounded projection Assumption A3), CTNN accesses temporally coupled functions at O(1) cost with respect to temporal distance Δ1, Δ2, while real-time architectures incur Ω(Δ1 + Δ2) sequential steps. For layered compositions, this yields an exponential composition gap within GTCP under A1–A4. These advantages hold under the stated assumptions and may not directly generalize to broader function classes or large-scale settings where A3 cannot be maintained. Therefore, Theorem 1 provides a formal separation result for GTCP, while CTNN more broadly defines a geometric framework for temporal computation. As the first concrete instantiation of this framework, we develop Complex-Time Convolutional Neural Networks (CTCNN). CTCNN achieves state-of-the-art performance on Something-Something V2 (70.2 ± 0.4%, +1.1% over VideoMAE v2, p < 0.01), strong performance on Kinetics-400 (78.4 ± 0.3%), and substantial gains on Long Range Arena Path-X (87.3% vs. 79.6%, +7.7%), using 3.4× fewer parameters than VideoMAE v2. Learnable angular parameters α and β provide computationally interpretable parameters related to memory-access span and prospection breadth, with values varying systematically across task families.