From 2446bb758f6ec7456c1279e6b8ea5ebda9530cc3 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Thu, 28 May 2026 22:57:15 -0500
Subject: Add functional capacity overlap simulation

---
 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 94f9fff..f24c117 100644
--- a/scripts/README.md
+++ b/scripts/README.md
@@ -84,3 +84,30 @@ The prior-free minimax theorem says:
 \]
 
 with equality for isotropic feedback. Outputs are written under `outputs/minimax_initialization/`.
+
+## Functional Capacity Overlap
+
+Run:
+
+```bash
+python scripts/functional_capacity_overlap.py --parameters 96 --task-rank 24 --constraint-ranks 0 24 48 72 84 96 --trials 100 --seed 5 --plot
+```
+
+This samples a task-sensitive subspace \(S\) of dimension \(d\) and an alignment
+constraint subspace \(E\) of dimension \(k\) inside \(\mathbb R^P\). It validates:
+
+\[
+\Delta d_{\mathrm{hard}}
+=
+\max(0,k-(P-d))
+\]
+
+and:
+
+\[
+\mathbb E[\operatorname{tr}(P_E P_S)]
+=
+\frac{kd}{P}.
+\]
+
+Outputs are written under `outputs/functional_capacity_overlap/`.
-- 
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