Add large empirical capacity validation

2 files changed, 52 insertions, 0 deletions

diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md
index 896961b..dcf797a 100644
--- a/notes/02_experiment_notes.md
+++ b/notes/02_experiment_notes.md
@@ -260,6 +260,57 @@ Result:
 
 This cleanly separates the fixed-threshold regime, where total cost scales like \(Ln^2\), from the chance-level regime, where per-layer cost is nearly width-independent.
 
+## Large Empirical Capacity Validation Run Log
+
+Script:
+
+```bash
+python scripts/capacity_empirical_validation.py --dimensions 64 128 256 512 1024 2048 4096 --samples 100000 --batch-size 2048 --seed 123 --plot
+```
+
+Setup:
+
+- dimensions \(D\): `64, 128, 256, 512, 1024, 2048, 4096`
+- samples per dimension: `100000`
+- total \(Q\) samples: `700000`
+- sampler: fixed-target Gaussian direction, using rotational invariance
+- chance thresholds: \(q=c/D\), \(c\in\{0.5,1,2,5,10\}\)
+- fixed thresholds: \(q\in\{0.001,0.002,0.005,0.01\}\)
+- multilayer all-event validation: \(c\in\{0.5,1\}\), layers \(L\in\{1,2,4,8\}\)
+
+Distribution calibration:
+
+| \(D\) | empirical mean | theory mean | KS statistic | empirical q99 | theory q99 |
+|---:|---:|---:|---:|---:|---:|
+| `64` | `0.0156793` | `0.0156250` | `0.0034949` | `0.1007135` | `0.1007080` |
+| `128` | `0.0078528` | `0.0078125` | `0.0019669` | `0.0511649` | `0.0510968` |
+| `256` | `0.0039289` | `0.0039062` | `0.0029576` | `0.0259898` | `0.0257333` |
+| `512` | `0.0019553` | `0.0019531` | `0.0016418` | `0.0129654` | `0.0129127` |
+| `1024` | `0.0009706` | `0.0009766` | `0.0031219` | `0.0063694` | `0.0064679` |
+| `2048` | `0.0004882` | `0.0004883` | `0.0031095` | `0.0032339` | `0.0032368` |
+| `4096` | `0.0002444` | `0.0002441` | `0.0024262` | `0.0016322` | `0.0016191` |
+
+Capacity-tail calibration:
+
+- chance-threshold rows: `35/35` had expected hits \(\ge 20\)
+- max absolute smoothed cost error over all chance rows: `0.2023` nats
+- mean absolute smoothed cost error over all chance rows: `0.01915` nats
+- for chance rows with expected hits \(\ge 500\): max error `0.02771` nats, mean error `0.00607` nats
+
+Multilayer product-capacity calibration:
+
+- rows with expected hits \(\ge 20\): `49`
+- max absolute smoothed total-cost error: `0.45788` nats
+- mean absolute smoothed total-cost error: `0.03880` nats
+- for rows with expected hits \(\ge 500\): max error `0.04319` nats, mean error `0.00826` nats
+
+Interpretation:
+
+- The beta-law distribution calibration is very tight across two orders of magnitude in dimension.
+- Empirical \(C(q)=-\log P(Q\ge q)\) matches the theoretical incomplete-beta cost when the expected tail count is large enough.
+- Multilayer capacity accumulation matches the product law until the all-event becomes too rare for the finite sample budget.
+- Fixed-threshold high-dimensional tails quickly become too rare for direct Monte Carlo, which is itself consistent with the exponential/geometric volume-collapse interpretation.
+
 ## Minimax Initialization Run Log
 
 Script:
diff --git a/notes/03_paper_outline.md b/notes/03_paper_outline.md
index 65a5c47..7e2e5d4 100644
--- a/notes/03_paper_outline.md
+++ b/notes/03_paper_outline.md
@@ -164,6 +164,7 @@ Scaling:
 
 - depth and width sweeps.
 - threshold regimes.
+- large empirical tail calibration for \(C(q)=-\log P(Q\ge q)\).
 
 Trajectory: