Axiom: A Householder-Parameterized Pure Unitary RNN for Long-Range Sequence Modeling

We present **Axiom**, a recurrent neural network whose hidden-to-hidden transition is parameterized as a product of Householder reflections, forming a strict unitary matrix. Unlike LSTM and GRU, Axiom contains no forget gate — the unitary transition matrix guarantees lossless information preservation across arbitrary sequence lengths by mathematical construction. On standard long-range memory benchmarks, Axiom achieves 76.5–99.9% accuracy on the delayed copy task using 8,584 parameters, while LSTM (111,368 parameters, 13× more) scores 12.5–13.5% — random chance — across all delays on both GPU and TPU v6e. On the Adding Problem (Hochreiter & Schmidhuber, 1997), Axiom achieves MSE 0.00046 at length 200 versus…