# Scripts

## Static Alignment Beta Law

Run:

```bash
python scripts/static_alignment_beta.py --rows 16 --cols 16 --samples 20000 --seed 7 --plot
```

This samples independent matrix pairs \(A,B\), computes:

\[
Q =
\frac{\langle A,B\rangle_F^2}{\|A\|_F^2\|B\|_F^2},
\]

and compares the empirical distribution with:

\[
\mathrm{Beta}\left(\frac12,\frac{D-1}{2}\right),
\qquad
D=\texttt{rows}\times\texttt{cols}.
\]

Outputs are written under `outputs/static_alignment_beta/`, which is ignored by Git.

## Capacity Scaling

Run:

```bash
python scripts/capacity_scaling.py --plot
```

This computes:

\[
C_l(q)
=
-\log\Pr(Q_l\ge q)
\]

for equal-width feedback blocks with \(D=n^2\), then sums over the number of feedback-aligned layers:

\[
C_{\mathrm{all}}=\sum_l C_l(q_l).
\]

The default run compares two regimes:

- `fixed`: \(q=0.01\), where \(C_{\mathrm{all}}\) grows like \(\Theta(Ln^2)\).
- `chance`: \(q=1/D\), where \(C_{\mathrm{all}}\) grows mostly with \(L\).

Outputs are written under `outputs/capacity_scaling/`.

## Minimax Initialization Bound

Run:

```bash
python scripts/minimax_initialization.py --dimension 32 --feedback-samples 20000 --target-samples 10000 --seed 11 --subspace-dim 4 --plot
```

This estimates the feedback second-moment matrix:

\[
M_\mu=\mathbb E_\mu[\hat b\hat b^\top]
\]

for several initialization distributions. The worst-case expected squared alignment is:

\[
\inf_{\|a\|=1}
\mathbb E_\mu[(a^\top \hat b)^2]
=
\lambda_{\min}(M_\mu).
\]

The prior-free minimax theorem says:

\[
\sup_\mu \lambda_{\min}(M_\mu)=\frac1D,
\]

with equality for isotropic feedback. Outputs are written under `outputs/minimax_initialization/`.