# 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/`.

## Functional Capacity Overlap

Run:

```bash
python scripts/functional_capacity_overlap.py --parameters 96 --task-rank 24 --constraint-ranks 0 24 48 72 84 96 --trials 100 --seed 5 --plot
```

This samples a task-sensitive subspace \(S\) of dimension \(d\) and an alignment
constraint subspace \(E\) of dimension \(k\) inside \(\mathbb R^P\). It validates:

\[
\Delta d_{\mathrm{hard}}
=
\max(0,k-(P-d))
\]

and:

\[
\mathbb E[\operatorname{tr}(P_E P_S)]
=
\frac{kd}{P}.
\]

Outputs are written under `outputs/functional_capacity_overlap/`.

## Synthetic MLP FA/BP Trajectories

Run:

```bash
python scripts/trajectory_mlp_fa.py --samples 128 --hidden-widths 24 24 --steps 80 --lr 0.02 --eval-every 10 --feedback-runs 3 --data-seed 3 --init-seed 4 --feedback-seed-start 50 --plot
```

This trains one BP baseline and several FA runs from the same initial weights on
a synthetic regression task. At each checkpoint, the script records:

- training loss;
- full-model cosine between the BP gradient and the FA surrogate gradient at the FA weights;
- hidden-layer-only cosine between the BP and FA gradients, excluding the output layer where gradients are identical;
- layerwise \(Q_l=\cos^2(W_{l+1}^{\top},B_l)\).

Outputs are written under `outputs/trajectory_mlp_fa/`:

- `summary.csv`
- `trajectories.csv`
- `layer_metrics.csv`
- diagnostic plots when `--plot` is set.