# Downstream Random-Capacity Results

## Main Run

Output directory:

`outputs/downstream_capacity_random_main_fast`

Task:

\[
x_i\sim \mathcal N(0,I_{16}),
\qquad
y_i\sim \mathcal N(0,I_4),
\qquad
N=128.
\]

Task dimension:

\[
N n_L=512.
\]

Model:

\[
16\to n\to n\to 4.
\]

Theory:

\[
P(n)=16n+n^2+4n,
\]

\[
K_{\mathrm{FA}}(n)=(n^2-1)+(4n-1),
\]

\[
M_{\mathrm{BP}}=P-Nn_L,
\qquad
M_{\mathrm{FA}}=P-K_{\mathrm{FA}}-Nn_L.
\]

Prediction:

- BP threshold at width \(16\).
- FA threshold at width \(32\).

Trajectory count:

- BP: \(4\) seeds per width.
- FA: \(4\times 12=48\) trajectories per width.
- Widths: \(8,12,16,24,32,48,64,96\).

## Summary

| width | BP train | FA train | gap | BP margin | FA margin |
|---:|---:|---:|---:|---:|---:|
| 8 | 0.7088 | 1.064 | 0.3555 | -288 | -382 |
| 12 | 0.2252 | 0.6141 | 0.3889 | -128 | -318 |
| 16 | 0.00743 | 0.2948 | 0.2873 | 64 | -254 |
| 24 | 0.000129 | 0.0346 | 0.0345 | 544 | -126 |
| 32 | 0.000007 | 0.0108 | 0.0108 | 1152 | 2 |
| 48 | 0.000007 | 0.00392 | 0.00391 | 2752 | 258 |
| 64 | 0.000046 | 0.00189 | 0.00185 | 4864 | 514 |
| 96 | 0.000250 | 0.00231 | 0.00206 | 10624 | 1026 |

## Main Figures

- `outputs/downstream_capacity_random_main_fast/train_mse_trajectory_distribution.png`
- `outputs/downstream_capacity_random_main_fast/train_gap_trajectory_distribution_vs_fa_margin.png`
- `outputs/downstream_capacity_random_main_fast/bp_fa_train_mse_vs_width.png`
- `outputs/downstream_capacity_random_main_fast/train_gap_vs_fa_capacity_margin.png`

The cleanest figure is `train_gap_trajectory_distribution_vs_fa_margin.png`: the FA/BP train gap is large when \(M_{\mathrm{FA}}<0\), drops sharply near \(M_{\mathrm{FA}}=0\), and remains small for positive FA margin.