${\mathcal{S}}^1$: An Interpretable State Space Model via an End-to-End Closed-Form Operator

The linear dynamics of state-space models are – in principle – mathematically solvable, yet the nonlinear readouts required for expressivity render the full computation inaccessible to mathematical analysis. We introduce ${\mathcal{S}}^1$, a state-space model derived from a system of complex nonlinear oscillator networks in which the entire computation – both dynamics and readout – admits an exact, end-to-end mathematical description. The key is a nonlinear coordinate transformation that simultaneously linearises the recurrent dynamics and uniquely determines the readout nonlinearity, which in turn enables an exact analytical expression for the end-to-end computation done by the network. A broad class of connectivity patterns gives rise to traveling wave modes in the recurrent layer, and classification emerges from nonlinear interactions between wave modes, which we illustrate in an exact manner. We validate ${\mathcal{S}}^1$ on EEG classification and sequential MNIST, providing a first-principles account of classification that no existing sequence model can offer, while also achieving competitive performance on benchmarks. The same mathematical framework also enables principled control: specific eigenmodes can be selectively amplified or suppressed to causally redirect model classifications. These results demonstrate a new SSM architecture that achieves competitive performance while adding precise interpretability and control.