# Systematic Low Bias in the Operator Predictor

We accept the current early-kernel operator predictor as the working version:

\[
\hat K_t^{rule}
=
K_0^{rule}
+
t\frac{K_s^{rule}-K_0^{rule}}{s}.
\]

It gives accurate trajectory-level gap distributions across depth, width, and
finite horizon stress tests. The remaining issue is systematic: the predicted
FA/BP gap is usually slightly lower than the empirical gap.

## Error Decomposition

The gap prediction is

\[
\hat\Delta
=
\hat L^{FA}-\hat L^{BP}.
\]

So a negative gap bias can come from either:

- \(\hat L^{FA}\) too low;
- \(\hat L^{BP}\) too high;
- both.

Across the stress grid, the dominant source is \(\hat L^{BP}\) too high. That
means the linear-velocity predictor makes BP slightly too weak, rather than
making FA too strong.

| setting | gap bias | BP loss bias | FA loss bias |
|---|---:|---:|---:|
| \(d=1,w=64,T=50,s=20\) | -0.000233 | 0.000207 | -0.000026 |
| \(d=2,w=32,T=50,s=20\) | -0.005375 | 0.005715 | 0.000340 |
| \(d=2,w=64,T=25,s=10\) | -0.002668 | 0.002973 | 0.000305 |
| \(d=2,w=64,T=50,s=20\) | -0.001491 | 0.001946 | 0.000455 |
| \(d=2,w=64,T=100,s=40\) | -0.002169 | 0.002949 | 0.000779 |
| \(d=2,w=96,T=50,s=20\) | -0.001551 | 0.002465 | 0.000914 |
| \(d=3,w=64,T=50,s=20\) | -0.004916 | 0.005885 | 0.000969 |

Aggregated over all stress-grid rows:

\[
\operatorname{bias}(\hat\Delta^{linear})
\approx
-0.00213,
\]

with

\[
\operatorname{bias}(\hat L^{BP})
\approx
0.00263,
\qquad
\operatorname{bias}(\hat L^{FA})
\approx
0.00050.
\]

Therefore the systematic low gap comes mainly from over-predicting BP loss.

## Not a Normalization Bug

The normalization is unlikely to be the cause.

Reasons:

1. The same residual recursion
   \[
   r_{t+1}=(I-\eta K_t/N)r_t
   \]
   was validated in the \(T=5\) local experiment without fitted scaling.
2. The measured time-varying \(K(t)\) diagnostic at \(T=50\) matched empirical
   losses to \(10^{-4}\)-\(10^{-3}\) absolute error.
3. A wrong \(1/N\), \(1/2\), or MSE normalization would not selectively make
   the linear BP loss too high while leaving measured \(K(t)\) accurate.

## Step Matching

The step indexing also looks correct.

The code computes \(K_s\) after \(s\) SGD updates and uses

\[
\hat K_t = K_0 + \frac{t}{s}(K_s-K_0)
\]

for update index \(t=0,\dots,T-1\). Thus \(\hat K_s=K_s\). An off-by-one change
such as \(s-1\) would make the slope larger and would worsen the observed
negative bias. A denominator \(s+1\) would reduce the bias numerically, but that
would be an ad hoc correction rather than the correct indexing.

## Kernel-Curvature Probe

We probed actual kernel drift by measuring

\[
\alpha_t
=
\frac{
\langle K_t-K_0,K_s-K_0\rangle_F
}{
\|K_s-K_0\|_F^2
}.
\]

The linear-velocity model predicts

\[
\alpha_t^{linear}=t/s.
\]

In representative runs, after \(t>s\), the true \(\alpha_t\) is consistently
below \(t/s\). Example: \(d=2,w=64,N=128,T=50,s=20\).

| rule | \(t\) | actual \(\alpha_t\) | linear \(t/s\) |
|---|---:|---:|---:|
| BP | 30 | 1.199 | 1.500 |
| BP | 40 | 1.347 | 2.000 |
| BP | 50 | 1.459 | 2.500 |
| FA | 30 | 1.073 | 1.500 |
| FA | 40 | 1.123 | 2.000 |
| FA | 50 | 1.166 | 2.500 |

The same pattern appears in narrow and deeper networks.

This means \(K_t-K_0\) is not linear in time. The early drift saturates. The
linear predictor over-extrapolates the kernel drift after the early window.

## Directional-Gain Effect

Loss decrease depends on the residual-direction gain

\[
g_t
=
\frac{r_t^\top K_t r_t}{\|r_t\|^2}.
\]

For \(d=2,w=64,N=128,T=50,s=20\), at \(t=50\):

| rule | actual gain | linear gain | linear minus actual |
|---|---:|---:|---:|
| BP | 161.8 | 150.5 | -11.3 |
| FA | 179.3 | 176.6 | -2.7 |

The linear predictor underestimates the BP directional gain more strongly than
the FA directional gain. This directly explains the negative gap bias:

\[
\hat L^{BP} > L^{BP}_{emp}
\quad\Rightarrow\quad
\hat L^{FA}-\hat L^{BP}
<
L^{FA}_{emp}-L^{BP}_{emp}.
\]

## Interpretation

The missed term is kernel-path curvature:

\[
K_t
=
K_0
+
t\dot K_0
+
\frac12t^2\ddot K
+
\cdots.
\]

The early secant model captures \(\dot K\) well enough to predict the gap
shape, but it ignores that the drift velocity decays. This decay is stronger
for BP in the residual-relevant direction, so the predicted BP loss is too high.

This is why:

- fixed \(K(0)\) is biased high for the gap: it makes BP too strong;
- early re-tangent is still biased high: freezing \(K_s\) still makes BP too
  strong relative to the future path;
- linear velocity is biased low: it makes BP too weak by over-extrapolating
  drift saturation.

The empirical gap is bracketed by the conservative frozen-\(K_s\) predictor and
the aggressive linear-velocity predictor.

## Working Position

For the current paper version, use:

1. linear-velocity predictor as the main distribution predictor;
2. early re-tangent as a conservative upper bracket;
3. report the systematic low bias as kernel-curvature residual, not as fitted
   normalization error.

Do not introduce a fitted scalar correction.

If a tighter future version is needed, the clean next step is a no-fit
curvature model using two early checkpoints, e.g. \(K_{s/2}\) and \(K_s\), to
estimate a decaying velocity. That would remain predictive, but it is not
needed for the current acceptable version.