# Pole-safe rational neural layer MVP

`pole_safe_experiment.py` implements the guaranteed maximal-order construction for an order-two pole:

`psi(z, x) = (z - beta)^2 * h_theta(x) / (||h_theta(x)|| + eps)`

and evaluates

`Q(z) psi(z,x)`, where `Q(z) = Q_-2/(z-beta)^2 + Q_-1/(z-beta) + Q_0`.

It compares the pole-safe branch with an unconstrained branch and an unconstrained branch using gradient clipping. All branches use the same 1,344-parameter MLP, data, optimizer, and number of steps.

Run:

```bash
/home/maxwelhelp/main/bin/python3 pole_safe_experiment.py
```

The run writes `results.json` and prints the same JSON. CUDA is used when available and falls back to CPU on CUDA runtime failure.

## Observed result

The direct math check gave an unsafe log-log growth slope of approximately `-2.0`, as expected for the order-two pole, and a safe slope approximately `0.0`. At distance `1e-5` from the pole, the unsafe norm was about `9.9e7` times its value at distance `1e-1`, while the safe norm changed by only about 0.7%; the safe product's error from `Q_-2 v` was `1.2e-5`.

In the toy regression, validation MSE was approximately:

- unconstrained: `7.996e11`
- unconstrained + gradient clipping: `1.337e11`
- pole-safe: `2.27e-2`

Maximum output norms were approximately `1.84e7`, `2.05e7`, and `1.49`, respectively. Thus the claimed boundedness/stability phenomenon and a strong synthetic optimization signal were observed.

This does not establish a result on a real SSM or sequence benchmark. The operator, pole, and data are synthetic; only one fixed seed and one order-two pole were tested, and the safe parameterization changes the effective feature dependence on frequency by design.