Check directional predictor across N

1 files changed, 78 insertions, 0 deletions

diff --git a/notes/14_early_predictor_probe.md b/notes/14_early_predictor_probe.md
index 5cfe058..43b9b6f 100644
--- a/notes/14_early_predictor_probe.md
+++ b/notes/14_early_predictor_probe.md
@@ -173,3 +173,81 @@ T\lambda_0+\frac{T(T-1)}{2}\dot\lambda_0
 \]
 
 This is more directly tied to loss dynamics than global kernel overlap.
+
+## Cross-N Directional Predictor Check
+
+We also tested a no-fit directional-gain extrapolation across
+
+\[
+N\in\{64,96,128,160,192,224,256,320\}
+\]
+
+with width fixed at \(64\), target horizon \(T=50\), 2 init seeds, and 4
+feedback seeds per init.
+
+The attempted predictor used
+
+\[
+\lambda_t\approx\lambda_0+t\dot\lambda_0,
+\qquad
+\dot\lambda_0\approx\frac{\lambda_s-\lambda_0}{s},
+\]
+
+and
+
+\[
+L_T
+\approx
+L_0
+\exp\left[
+-\frac{2\eta}{N}
+\left(
+T\lambda_0+\frac{T(T-1)}{2}\dot\lambda_0
+\right)
+\right].
+\]
+
+This failed as a finite-\(T\) gap predictor. Across \(N\), the directional-gap
+prediction had low correlation and large negative bias:
+
+| early step | fixed \(K(0)\) corr | fixed MAE | directional corr | directional MAE |
+|---:|---:|---:|---:|---:|
+| 1 | 0.979 | 0.0241 | 0.124 | 0.2838 |
+| 2 | 0.979 | 0.0241 | 0.138 | 0.2964 |
+| 5 | 0.979 | 0.0241 | 0.110 | 0.3686 |
+
+So the simple directional exponential formula should not be used.
+
+The reason is likely that a scalar gain
+
+\[
+\lambda_t=\frac{r_t^\top K_t r_t}{\|r_t\|^2}
+\]
+
+does not preserve residual direction changes. BP and FA rotate the residual in
+different ways, and the gap is a difference between two residual trajectories,
+not just a difference between two scalar decay rates.
+
+The more viable predictive object is therefore not a scalar exponential loss
+model. It is a compressed operator model, for example:
+
+\[
+K_t^{FA}
+\approx
+K_0^{FA}
++
+\alpha_t
+\left(K_0^{BP}-K_0^{FA}\right),
+\]
+
+or a low-rank/residual-subspace version. The scalar order parameter can control
+operator interpolation, but the residual update must remain vector-valued:
+
+\[
+r_{t+1}
+=
+\left(I-\frac{\eta}{N}K_t^{compressed}\right)r_t.
+\]
+
+This preserves direction changes and is closer to the measured \(K(t)\) product
+that worked.