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# 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/`.
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