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# Results
All numbers below are from the faithful J-lens (`src/jlens_v3.py`), which
computes exactly the paper's quantity: rows of `W_U * J_l` where
`J_l = E[ d h_final / d h_l ]` (average residual-to-residual Jacobian, read out
through the unembedding). Verification: at the last layer, J must be the
identity, and the check returns cosine similarity **1.0000**.
## 1. Both-ways comparison: old proxy vs faithful lens (trained 10.65M char model)
Pearson r between token frequency and J-lens norm, per layer. n = 65 tokens
(full char vocab). The correlation survives the faithful implementation at
every layer.
```
Layer proxy r faithful r
L0 -0.661 -0.643
L1 -0.673 -0.668
L2 -0.653 -0.672
L3 -0.648 -0.685
L4 -0.562 -0.637
L5 -0.665 -0.606
```
Top tokens by faithful norm are consistently rare characters (`?`, `z`, `q`,
`$`); bottom are common ones (space, `e`, `t`, `i`).
## 2. Frequency-matched synthetic pair (`src/synthetic_pair.py`)
Two new characters at identical 0.1% unigram frequency in Shakespeare:
`@` appears only after the trigger "the " (predictable in context);
`#` appears at uniform random positions. Faithful J-lens norm per seed, range
over layers 0-5:
```
seed @ norm (predictable) # norm (noise) ratio freq corr r
0 0.0232 - 0.0246 0.0152 - 0.0154 1.51-1.60 -0.59..-0.68
1 0.0224 - 0.0237 0.0148 - 0.0151 1.50-1.60 -0.59..-0.66
2 0.0215 - 0.0233 0.0154 - 0.0163 1.35-1.51 -0.62..-0.66
```
Reading: at equal frequency, the structured token scores ~1.4-1.5x higher.
The frequency anti-correlation holds, but the lens also carries genuine
conditional-predictability signal.
## 3. Loss-reweighting causal test (`src/loss_reweight.py`)
Three models per seed, identical init + minibatch order: 'q' targets weighted
x2 in the loss, plain control, and a same-total-loss control upweighting random
non-'q' targets. Question: does raising effective frequency causally reduce
'q's faithful J-lens norm? PENDING — run completes within hours of this file
being written; the summary table is printed by
`python3 src/loss_reweight.py --step summary --layers 2,3,4`.
## 4. Historical / do-not-copy
- Original proxy finding (r = -0.65, `jlens_v2`): superseded by the faithful
implementation; kept only for the both-ways comparison.
- GPT-2 correlation (r = -0.18, `gpt2_jlens.py`): UNDER-POWERED (96 token
positions, n=100 sampled tokens) and computed a different quantity
(norm-per-batch vs norm-of-mean). Directionally consistent but not
publishable evidence on its own.
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