# Dense Long-Training Phase Transition

The previous `T=30000` transition plot only had four width points. That was not
enough to decide whether the long-time curve has a kink or a soft ramp.

We ran a denser long-training sweep:

```text
outputs/phase_transition_dense_T30000_352traj
```

Setup:

```text
task: random-label regression
architecture: 16 -> width -> width -> 4
optimizer: full-batch SGD
learning rate: 0.01
train samples: 128
steps: 30000
widths: 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40
init seeds: 1
feedback seeds: 32
FA trajectories: 352 total
```

Main plot:

```text
outputs/phase_transition_dense_T30000_352traj/dense_T30000_gap_logscale.png
```

## Result

| width | FA margin | FA trajectories | train gap mean | train gap std |
|---:|---:|---:|---:|---:|
| 20 | -190 | 32 | 0.146803 | 0.042009 |
| 22 | -158 | 32 | 0.100526 | 0.034803 |
| 24 | -126 | 32 | 0.046464 | 0.024474 |
| 26 | -94 | 32 | 0.029671 | 0.023591 |
| 28 | -62 | 32 | 0.014195 | 0.009368 |
| 30 | -30 | 32 | 0.007466 | 0.004624 |
| 32 | 2 | 32 | 0.003729 | 0.003787 |
| 34 | 34 | 32 | 0.002079 | 0.002232 |
| 36 | 66 | 32 | 0.000791 | 0.000605 |
| 38 | 98 | 32 | 0.000553 | 0.000545 |
| 40 | 130 | 32 | 0.000162 | 0.000134 |

The dense sweep does not show a sharp kink. It shows a smooth, approximately
log-linear soft ramp in the long-training FA/BP train gap.

## Interpretation

The earlier `T=10000` curve was partly finite-time undertraining, because
longer training collapses the positive-margin gap from roughly `0.05` to
roughly `0.004` near margin zero.

However, after adding dense `T=30000` points, the long-training curve is still
not a hard step. The correct conclusion is:

```text
hard FA margin predicts a capacity-controlled regime variable,
not a discontinuous empirical transition.
```

The empirical transition is soft:

```text
more negative FA margin -> larger train gap;
positive FA margin -> small but nonzero finite-time/soft-alignment tail;
no sharp kink at margin zero.
```

## Consequence for the Paper

Do not write:

```text
FA and BP are identical until redundant parameters are exhausted, then a sharp
gap appears.
```

Use:

```text
The hard margin gives a conservative redundancy-exhaustion boundary. Empirically,
the FA/BP train-gap distribution changes smoothly with the margin; longer
training removes most positive-margin gap, while negative margins retain a
capacity-controlled gap.
```

This is still a useful capacity result, but the contribution should be framed as
a scaling law plus soft capacity transition, not a hard phase transition.