Add initialization coverage distribution matching

2 files changed, 71 insertions, 0 deletions

diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md
index 2fad18f..c7e7168 100644
--- a/notes/02_experiment_notes.md
+++ b/notes/02_experiment_notes.md
@@ -385,6 +385,64 @@ Random-target means remain close to \(1/D\) for all distributions, but worst-cas
 
 This empirically illustrates the prior-free minimax theorem: without target or weight prior information, anisotropic feedback cannot improve the worst-case angular bound.
 
+## Initialization Coverage Distribution Matching Run Log
+
+Script:
+
+```bash
+python scripts/initialization_distribution_matching.py --dimension 128 --target-samples 100000 --feedback-samples 100000 --batch-size 8192 --seed 2026 --subspace-dim 8 --anisotropy 64 --plot
+```
+
+Theory:
+
+For a feedback initialization distribution \(\mu\):
+
+\[
+M_\mu=\mathbb E[\hat b\hat b^\top].
+\]
+
+For a random target direction \(a\):
+
+\[
+A(a)=a^\top M_\mu a.
+\]
+
+If \(\lambda_i\) are eigenvalues of \(M_\mu\), then:
+
+\[
+A(a)
+=
+\frac{\sum_i \lambda_i G_i}{\sum_i G_i},
+\qquad
+G_i\sim\chi^2_1.
+\]
+
+Setup:
+
+- dimension \(D=128\)
+- target samples: `100000`
+- feedback samples for empirical \(M_\mu\): `100000`
+- schemes: `isotropic`, `rademacher`, `subspace`, `axis`, `geometric_axis`
+- subspace dimension: `8`
+- geometric axis anisotropy: `64`
+
+Results:
+
+| scheme | population \(\lambda_{\min}\) | empirical \(\lambda_{\min}\) | predicted mean | empirical mean | predicted std | empirical std | KS |
+|---|---:|---:|---:|---:|---:|---:|---:|
+| `isotropic` | `0.0078125` | `0.0072928` | `0.0078125` | `0.0078124` | `~0` | `0.0000346` | `nan` |
+| `rademacher` | `0.0078125` | `0.0072982` | `0.0078125` | `0.0078125` | `~0` | `0.0000342` | `nan` |
+| `subspace` | `0` | `0` | `0.0077967` | `0.0078031` | `0.0037405` | `0.0037335` | `0.00299` |
+| `axis` | `0` | `0` | `0.0078396` | `0.0078514` | `0.0109834` | `0.0109230` | `0.00351` |
+| `geometric_axis` | `0.0005111` | `0.0004100` | `0.0078116` | `0.0078155` | `0.0010441` | `0.0010425` | `0.00470` |
+
+Interpretation:
+
+- All schemes have mean coverage near \(1/D\), as expected from averaging over uniformly random targets.
+- Isotropic and rademacher have population point-mass coverage at \(1/D\); empirical spread comes from finite-sample \(M_\mu\) estimation noise, so KS against a point mass is not meaningful.
+- Non-isotropic schemes match the predicted target-coverage distribution tightly, with max nondegenerate KS `0.00470`.
+- The distribution shape, not just the mean, exposes the tradeoff: subspace and axis schemes keep the same average but create zero-coverage worst-case directions.
+
 ## Functional Capacity Overlap Run Log
 
 Script:
diff --git a/notes/03_paper_outline.md b/notes/03_paper_outline.md
index 328871f..6055722 100644
--- a/notes/03_paper_outline.md
+++ b/notes/03_paper_outline.md
@@ -128,6 +128,19 @@ Prior-aware corollary:
 \operatorname{tr}(\Sigma_A M_\mu).
 \]
 
+Distributional validation target:
+
+\[
+A(a)=a^\top M_\mu a
+=
+\frac{\sum_i \lambda_i G_i}{\sum_i G_i},
+\qquad
+G_i\sim\chi^2_1.
+\]
+
+Use this to compare predicted target-coverage distributions for isotropic,
+subspace, axis, and anisotropic feedback initializations.
+
 ## 7. Trajectory Bridge
 
 Introduce local approximation: