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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+- Original proxy finding (r = -0.65, `jlens_v2`): superseded by the faithful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.