# S4 arithmetic reference Run `python3 examples/s4.py` from the repository root with Python 3.9 or newer. There are no third-party dependencies. The script checks deterministic numerical identities and prints errors; it does not train a model or load a checkpoint. The five source regions are `hippo`, `discretize`, `recurrence`, `structure`, and `kernel`. The annotated panels depend on shared helpers in the complete file. Copy the full file to run them together. The rank-one kernel fixture is a three-state real DPLR system with explicitly authored parameters, not the complex diagonalization of a pretrained HiPPO model. The script verifies the finite-length Cauchy/Woodbury construction using dense preprocessing and a quadratic DFT. It demonstrates correctness, not S4's asymptotic speed. Primary references: - Gu, Goel and Ré, S4, ICLR 2022; arXiv v3: https://arxiv.org/abs/2111.00396v3 - Official implementation pinned at e757cef57d89e448c413de7325ed5601aceaac13: https://github.com/state-spaces/s4/blob/e757cef57d89e448c413de7325ed5601aceaac13/models/s4/s4.py The JavaScript visualizations independently implement the same arithmetic. `node --test test/s4-model.test.mjs` verifies identities and cross-language agreement. Polynomial projection in the browser is an approximate midpoint quadrature illustration, not an exact online HiPPO solver. No figures or model outputs were copied from the paper.