# More Phase-Transition Trajectories

We ran two larger trajectory sweeps around the capacity transition to separate
sampling noise from finite-time undertraining.

## Runs

```text
outputs/phase_transition_zoom_T10000_1024traj
outputs/phase_transition_zoom_T30000_256traj
```

The setup is unchanged:

```text
task: random-label regression
architecture: 16 -> width -> width -> 4
optimizer: full-batch SGD
learning rate: 0.01
train samples: 128
```

The `T=10000` run uses:

```text
widths: 16, 20, 24, 28, 32, 36, 40, 48
init seeds: 2
feedback seeds per init: 64
FA trajectories per width: 128
total FA trajectories: 1024
```

The `T=30000` run focuses on the transition:

```text
widths: 24, 28, 32, 36
init seeds: 2
feedback seeds per init: 32
FA trajectories per width: 64
total FA trajectories: 256
```

## Main Plots

```text
outputs/phase_transition_zoom_comparison/trajectory_gap_clouds_T10000_vs_T30000.png
outputs/phase_transition_zoom_comparison/trajectory_gap_quantiles_T10000_vs_T30000.png
```

Per-run plots:

```text
outputs/phase_transition_zoom_T10000_1024traj/phase_transition_capacity_exhaustion.png
outputs/phase_transition_zoom_T30000_256traj/phase_transition_capacity_exhaustion.png
```

## Results

At `T=10000`, the train-gap curve is very smooth and monotone:

| width | FA margin | FA trajectories | train gap mean | train gap std |
|---:|---:|---:|---:|---:|
| 16 | -254 | 128 | 0.420780 | 0.075399 |
| 20 | -190 | 128 | 0.278329 | 0.058114 |
| 24 | -126 | 128 | 0.171090 | 0.043111 |
| 28 | -62 | 128 | 0.099732 | 0.033015 |
| 32 | 2 | 128 | 0.050912 | 0.016008 |
| 36 | 66 | 128 | 0.030891 | 0.010649 |
| 40 | 130 | 128 | 0.014464 | 0.004946 |
| 48 | 258 | 128 | 0.004212 | 0.001836 |

This smooth ramp is not caused by too few feedback seeds. Each point has 128 FA
trajectories and the standard deviations are much smaller than the mean trend.

At `T=30000`, the near-zero and positive-margin gaps collapse:

| width | FA margin | FA trajectories | train gap mean | train gap std |
|---:|---:|---:|---:|---:|
| 24 | -126 | 64 | 0.047899 | 0.026889 |
| 28 | -62 | 64 | 0.015020 | 0.009012 |
| 32 | 2 | 64 | 0.003275 | 0.003577 |
| 36 | 66 | 64 | 0.000932 | 0.001324 |

So the earlier lack of a sharp kink was mostly finite-time undertraining near
the transition. With longer training, positive-margin networks close the BP/FA
train gap, while negative-margin networks retain a nonzero gap.

## Interpretation

The hard FA capacity margin is a conservative structural boundary. It is not an
exact discontinuous transition point.

The empirical picture is:

```text
T=10000: smooth finite-time ramp
T=30000: compressed transition; positive margin almost zero, negative margin nonzero
```

This supports the phase-transition contribution, but the right language is a
soft capacity transition rather than a hard kink at exactly margin zero.

## Consequence

For the paper figure, the clean visual is the two-panel comparison:

```text
trajectory_gap_clouds_T10000_vs_T30000.png
```

It shows:

1. More trajectories do not remove the smooth `T=10000` ramp.
2. Longer training does remove the apparent positive-margin gap.
3. The remaining long-time gap is concentrated on the negative-margin side.

This is the strongest current evidence that the original smooth curve mixed two
effects:

```text
finite-time optimization gap + structural capacity gap
```