# Hill-Floquet Regularization MVP

## What was built

`hill_floquet_experiment.py` implements the Fourier Hill matrix for a periodic 2D linearized system (`A=I`), computes its truncated spectrum, independently integrates the tangent matrix over one period to obtain the monodromy multipliers, and runs a small parameter-optimization comparison with and without a Hill penalty.

The toy Jacobian is smooth and periodic but noncommuting in time, so it is more informative than a constant diagonal test. The period is `T=2*pi`, with `M=128` samples and default `N=8`.

## Quantitative checks

Results are in `results.json`.

1. **Constant offset scaling.** Since the construction adds `gamma I` to the Jacobian, the exact prediction is `d mu_max / d gamma = 1`. The observed slope was `1.000000000004` for monodromy and `1.000000000000` for the Hill matrix. This prediction passed.

2. **Stability boundary.** The predicted boundary is `rho_F=1` exactly when `max Re(mu)=0`. The independently integrated monodromy crossed at `gamma=0.002251`, but the raw `N=8` Hill maximum crossed at `gamma=-0.005914`. This is an absolute boundary error of about `0.00816`, and more importantly the Hill maximum was positive at `gamma=0` even though the true system was stable (`mu_max=-0.002251`). This prediction failed for the raw finite-section maximum.

3. **Fourier truncation.** The raw Hill maximum was `0.008794, 0.005890, 0.005914, 0.005914, 0.005914` for `N=1,2,4,8,12`, while the true monodromy exponent was `-0.002251`. Thus `N=4` to `N=8` changes by only `3.3e-11`, but it converges to the wrong dominant root. The apparent convergence is spectral pollution, not validation of the physical Floquet exponent.

An independent residual check using Fourier coefficients of a numerically propagated Floquet mode gave relative Hill residuals below `6e-8` for both true monodromy exponents. Hence the matrix assembly can represent the physical roots, but selecting `max Re` from the finite spectrum selects a nonphysical root.

## Secondary optimization signal

The toy task loss prefers an unstable `gamma=0.2`. Baseline gives `mu_Hill=0.205914` and `rho=3.647`. The penalty optimization moves to `gamma=0.062724`, with `mu_Hill=0.068638` and `rho=1.539`. This reduces the measured raw Hill growth, but does not establish true stability and is not counted as a win.

## Verdict

The mechanism's exact offset-scaling prediction manifested, and the physical Floquet modes satisfy the assembled Hill equation. However, the proposed practical criterion—penalizing the largest real eigenvalue of a truncated Hill matrix—failed in this smooth 2D example: it produced a persistent positive spectral-pollution root and shifted the stability boundary. A robust implementation needs mode selection/continuation, structure-preserving discretization, or an independently validated monodromy/descriptor treatment before using this as a regularizer.