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+# Phase Transition Visualization
+
+This note records the first clean phase-transition visualization for the
+capacity-exhaustion contribution.
+
+## Goal
+
+We want a figure that shows:
+
+```text
+available FA capacity decreases
+-> redundant directions are exhausted
+-> FA/BP train gap opens
+-> gap grows while BP remains capable
+```
+
+The key x-axis is the hard FA capacity margin:
+
+```text
+M_FA = P - K_FA - N*out
+```
+
+where:
+
+- `P` is the parameter count;
+- `K_FA` is the hard feedback-alignment constraint count;
+- `N*out` is the random-label task dimension.
+
+The predicted transition is at:
+
+```text
+M_FA = 0
+```
+
+## Experiment
+
+This run uses SGD, not Adam.
+
+Configuration:
+
+```text
+task: random-label regression
+architecture: 16 -> width -> width -> 4
+widths: 8, 12, 16, 24, 32, 48, 64, 96
+train samples: 128
+optimizer: full-batch SGD
+learning rate: 0.01
+steps: 3000
+init seeds: 4
+feedback seeds per init: 8
+FA trajectories: 256
+```
+
+Output directory:
+
+```text
+outputs/phase_transition_sgd_256_lr001_T3000
+```
+
+Main figures:
+
+```text
+outputs/phase_transition_sgd_256_lr001_T3000/phase_transition_capacity_exhaustion.png
+outputs/phase_transition_sgd_256_lr001_T3000/phase_transition_gap_only.png
+```
+
+## Result
+
+Summary:
+
+| width | FA margin | BP train | FA train | FA-BP train gap |
+|---:|---:|---:|---:|---:|
+| 96 | 1026 | 0.000249 | 0.009436 | 0.009187 |
+| 64 | 514 | 0.003354 | 0.053192 | 0.049838 |
+| 48 | 258 | 0.016732 | 0.146680 | 0.129947 |
+| 32 | 2 | 0.089511 | 0.386777 | 0.297266 |
+| 24 | -126 | 0.201897 | 0.600796 | 0.398899 |
+| 16 | -254 | 0.547408 | 0.993219 | 0.445812 |
+| 12 | -318 | 0.872346 | 1.204686 | 0.332340 |
+| 8 | -382 | 1.260234 | 1.443789 | 0.183555 |
+
+The gap is small in the redundant region:
+
+```text
+M_FA > 0
+```
+
+It rises sharply around the hard FA capacity boundary:
+
+```text
+M_FA ≈ 0
+```
+
+and becomes largest in the FA-deficient but BP-capable region:
+
+```text
+M_FA < 0
+and
+M_BP = P - N*out > 0
+```
+
+At the smallest widths, BP itself also becomes under-capacity:
+
+```text
+M_BP < 0
+```
+
+In that both-deficient regime, the FA-BP gap decreases because both methods
+fail to fit the random labels.
+
+## Interpretation
+
+The visual story is not simply "smaller network means larger gap." The clean
+phase-transition story is:
+
+1. high redundancy:
+   ```text
+   M_FA >> 0
+   ```
+   BP and FA both have enough effective room, so the train gap is small.
+
+2. FA redundancy exhausted:
+   ```text
+   M_FA crosses 0
+   ```
+   FA starts paying the feedback-alignment burden in task-relevant directions,
+   so the gap opens.
+
+3. FA-deficient but BP-capable:
+   ```text
+   M_FA < 0, M_BP > 0
+   ```
+   the gap grows and peaks.
+
+4. both-deficient:
+   ```text
+   M_BP < 0
+   ```
+   BP also cannot memorize, so the difference between FA and BP no longer
+   grows monotonically.
+
+This is the right phase-transition framing for the paper.
+
+## Caveat
+
+The hard margin is intentionally simple. It uses:
+
+```text
+K_FA = (width*width - 1) + (width*out - 1)
+```
+
+for the two-hidden-layer MLP. It predicts the transition location well enough
+to provide a clean regime variable, but the exact peak is shifted because real
+FA alignment is soft, not a hard rank constraint.
+
+Do not claim that hard capacity margin alone predicts the exact final train
+gap. The correct claim is:
+
+```text
+hard FA margin predicts the regime where the gap opens;
+tangent/operator dynamics or empirical trajectories determine the gap size.
+```