From dd14582dcfd3b6e3e7b0e28f66bb0f2994f106c4 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Fri, 29 May 2026 08:43:19 -0500
Subject: Add multilayer capacity distribution matching

---
 scripts/README.md | 27 +++++++++++++++++++++++++++
 1 file changed, 27 insertions(+)

(limited to 'scripts/README.md')

diff --git a/scripts/README.md b/scripts/README.md
index e30b309..efd4208 100644
--- a/scripts/README.md
+++ b/scripts/README.md
@@ -85,6 +85,33 @@ Outputs are written under `outputs/capacity_empirical_validation/`:
 - `multilayer_capacity.csv`
 - diagnostic plots when `--plot` is set.
 
+## Multilayer Observed Capacity Distribution
+
+Run:
+
+```bash
+python scripts/multilayer_capacity_distribution.py --dimensions 64 256 1024 4096 --layers 1 2 4 8 16 --samples 100000 --batch-size 8192 --seed 456 --plot
+```
+
+This validates the full null distribution of observed capacity surprisal:
+
+\[
+S_l=-\log P(Q_l'\ge Q_l).
+\]
+
+Theory predicts:
+
+\[
+S_l\sim \mathrm{Exp}(1),
+\qquad
+\sum_{l=1}^L S_l\sim \mathrm{Gamma}(L,1).
+\]
+
+The script samples random direction cosines through the exact chi-square
+representation \(Q=X/(X+Y)\), with \(X\sim\chi^2_1\) and
+\(Y\sim\chi^2_{D-1}\). Outputs are written under
+`outputs/multilayer_capacity_distribution/`.
+
 ## Minimax Initialization Bound
 
 Run:
-- 
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