From c47a74792e6ee78181b087ead595465140b77825 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Fri, 29 May 2026 04:53:39 -0500
Subject: Add large empirical capacity validation

---
 scripts/README.md | 31 +++++++++++++++++++++++++++++++
 1 file changed, 31 insertions(+)

(limited to 'scripts/README.md')

diff --git a/scripts/README.md b/scripts/README.md
index 37dc68b..e30b309 100644
--- a/scripts/README.md
+++ b/scripts/README.md
@@ -54,6 +54,37 @@ The default run compares two regimes:
 
 Outputs are written under `outputs/capacity_scaling/`.
 
+## Large Empirical Capacity Validation
+
+Run:
+
+```bash
+python scripts/capacity_empirical_validation.py --dimensions 64 128 256 512 1024 2048 4096 --samples 100000 --batch-size 2048 --seed 123 --plot
+```
+
+This uses rotational invariance to fix the target direction and sample random
+feedback directions:
+
+\[
+Q=\frac{z_1^2}{\|z\|^2},
+\qquad
+z\sim \mathcal N(0,I_D).
+\]
+
+It validates:
+
+- beta-law distribution calibration across dimensions;
+- empirical tail probabilities for \(q=c/D\) and fixed \(q\);
+- empirical capacity cost \(C(q)=-\log P(Q\ge q)\);
+- multilayer product scaling for all-alignment events.
+
+Outputs are written under `outputs/capacity_empirical_validation/`:
+
+- `distribution_summary.csv`
+- `capacity_tails.csv`
+- `multilayer_capacity.csv`
+- diagnostic plots when `--plot` is set.
+
 ## Minimax Initialization Bound
 
 Run:
-- 
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