Add functional capacity overlap simulation

2 files changed, 33 insertions, 0 deletions

diff --git a/notes/01_theory_notes.md b/notes/01_theory_notes.md
index d524000..7a54374 100644
--- a/notes/01_theory_notes.md
+++ b/notes/01_theory_notes.md
@@ -357,6 +357,14 @@ d-d_{\mathrm{hard}}
 \max(0,k-(P-d)).
 \]
 
+Interpretation:
+
+- \(P-d\) is the local redundant parameter dimension.
+- If \(k\le P-d\), generic alignment constraints can be absorbed by redundant directions without reducing hard local function rank.
+- If \(k>P-d\), every additional generic constraint reduces hard local function rank one-for-one.
+
+This gives the proposed FA/BP functional-gap onset: parameter-volume cost can grow before the model loses hard function rank, but once the alignment burden exhausts redundancy, FA should separate from BP more sharply.
+
 Soft overlap model. Let \(E\) be the alignment constraint subspace and \(S\) be the task-sensitive subspace:
 
 \[
diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md
index 05b108b..889d1bd 100644
--- a/notes/02_experiment_notes.md
+++ b/notes/02_experiment_notes.md
@@ -284,3 +284,28 @@ Random-target means remain close to \(1/D\) for all distributions, but worst-cas
 - subspace and axis initializations leave entire orthogonal directions uncovered.
 
 This empirically illustrates the prior-free minimax theorem: without target or weight prior information, anisotropic feedback cannot improve the worst-case angular bound.
+
+## Functional Capacity Overlap Run Log
+
+Script:
+
+```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
+```
+
+Setup:
+
+- parameter dimension \(P=96\)
+- task-sensitive rank \(d=24\)
+- redundant dimension \(P-d=72\)
+
+Result:
+
+- \(k=0\): hard loss `0`, theory `0`; soft overlap `0`, theory `0`
+- \(k=24\): hard loss `0`, theory `0`; soft overlap `6.0315`, theory `6`
+- \(k=48\): hard loss `0`, theory `0`; soft overlap `12.0018`, theory `12`
+- \(k=72\): hard loss `0`, theory `0`; soft overlap `18.0018`, theory `18`
+- \(k=84\): hard loss `12`, theory `12`; soft overlap `21.0029`, theory `21`
+- \(k=96\): hard loss `24`, theory `24`; soft overlap `24`, theory `24`
+
+This validates the redundancy-exhaustion interpretation: hard functional rank remains intact until alignment constraints exceed the redundant dimension \(P-d\), while soft overlap grows linearly as \(kd/P\).