State Propagation Also Satisfies: A Complex-Valued State-Space Model for Deterministic State Tracking

Despite the dominance of massive language models, leading paradigms like Transformers and Mamba fundamentally falter at continuous deterministic state tracking, suffering from catastrophic out-of-distribution (OOD) collapse when generalizing to extended sequences. To shatter this bottleneck, we present the Complex State Propagator (CSP), a radically minimalist recurrent paradigm that operates strictly on the complex phase manifold without intermediate output projections, amplitude modulations, or per-step non-linearities. Crucially, we unveil an unprecedented architectural marvel: the phase signal survives extreme depth with absolute zero informational attenuation. By implementing an exact four-quadrant Coordinate-to-Phase (C-to-2) transformation governed by the (atan2(y, x)) activation, CSP forces the continuous optimization landscape to seamlessly align with discrete cyclic groups. Remarkably, with a mere 3-layer hidden topology trained on short inputs (length 16),CSP demonstrates absolute mathematical purity, achieving a nearly 100% validation accuracy and F1-score when generalized to a 4× prolonged OOD length of 64 on the canonical Mod-3 tracking task. Our work establishes complex-valued, pure-phase propagation not merely as a compact alternative, but as a dominant frontier that beats heavy networks at a fraction of their size.