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+# Exact e0 Distribution Validation
+
+This note records the clean distributional validation for the actual FA initial
+operator erosion theorem.
+
+## Theorem Being Validated
+
+Use a one-hidden-layer MLP with Gaussian feedback:
+
+```text
+x -> W1 -> ReLU -> W2 -> output
+```
+
+For fixed forward weights, data, gates, and residual:
+
+```text
+speed_BP = ||g_output^BP||^2 + ||g_hidden^BP||^2
+speed_FA = ||g_output^BP||^2 + <g_hidden^BP, g_hidden^FA(B)>
+e0(B) = 1 - speed_FA / speed_BP
+```
+
+Because the hidden FA gradient is linear in Gaussian feedback `B`:
+
+```text
+<g_hidden^BP, g_hidden^FA(B)> = <C(W,r), B>
+```
+
+Therefore:
+
+```text
+<C(W,r), B> ~ Normal(0, sigma_B^2 ||C(W,r)||^2)
+```
+
+and:
+
+```text
+e0(B) ~ Normal(
+  1 - ||g_output^BP||^2 / speed_BP,
+  sigma_B^2 ||C(W,r)||^2 / speed_BP^2
+)
+```
+
+This is an exact conditional distribution for one-hidden-layer Gaussian FA at
+initialization. It does not involve training horizon `T`.
+
+## Experiment
+
+Script:
+
+```text
+scripts/actual_fa_initial_erosion_distribution.py
+```
+
+Run:
+
+```text
+python scripts/actual_fa_initial_erosion_distribution.py \
+  --widths 16 32 64 128 \
+  --init-seeds 3 \
+  --feedback-samples 4096 \
+  --torch-threads 8 \
+  --outdir outputs/actual_fa_initial_erosion_distribution
+```
+
+Outputs:
+
+```text
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_theory_vs_empirical.png
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_moment_calibration.png
+outputs/actual_fa_initial_erosion_distribution/e0_distribution_rows.csv
+```
+
+## Results
+
+Representative rows:
+
+| width | init | theory mean | empirical mean | theory std | empirical std | KS | p-value |
+|---:|---:|---:|---:|---:|---:|---:|---:|
+| 16 | 0 | 0.113830 | 0.113413 | 0.027228 | 0.027401 | 0.0150 | 0.314 |
+| 16 | 1 | 0.113705 | 0.113859 | 0.027261 | 0.027278 | 0.0103 | 0.778 |
+| 32 | 0 | 0.121730 | 0.121544 | 0.018952 | 0.018693 | 0.0114 | 0.654 |
+| 32 | 2 | 0.227193 | 0.227691 | 0.031148 | 0.030779 | 0.0166 | 0.206 |
+| 64 | 0 | 0.031155 | 0.031119 | 0.003652 | 0.003583 | 0.0118 | 0.609 |
+| 128 | 0 | 0.023004 | 0.022989 | 0.001630 | 0.001610 | 0.0135 | 0.441 |
+
+All tested panels show close agreement between theory density and empirical
+random-feedback histograms.
+
+## Paper Use
+
+This is the clean validation figure for the actual FA local erosion theorem.
+
+It should replace any tangent-hierarchy figure if the goal is:
+
+```text
+theory predicts a distribution
+large random-feedback sampling produces the same distribution
+```
+
+Suggested main-text wording:
+
+```text
+For one-hidden-layer Gaussian FA, the conditional distribution of the initial
+operator erosion is exactly Gaussian. Figure X compares this predicted density
+against 4096 random feedback draws per initialization across widths; the
+empirical histograms and theoretical densities coincide without fitted
+parameters.
+```
+
+This figure belongs to:
+
+```text
+Contribution 4: actual FA initial operator erosion theorem
+Contribution 5: distributional empirical validation
+```
+
+It does not require a training-time axis.