Add functional capacity overlap simulation

1 files changed, 8 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:
 
 \[