path: root/blog/jacobian-lens-frequency.qmd

---
title: "What the Jacobian Lens Actually Measures"
date: "2026-08-02"
---

### A SMALL REPLICATION OF ANTHROPIC'S J-LENS, THE TOKEN-FREQUENCY CONFOUND WE FOUND, AND THE BUG WE ALMOST PUBLISHED

*This is a story about trying to look inside a language model. We found something
Anthropic didn't mention in their paper — and then we found that we'd made a
mistake, fixed it, and the thing was still there. That second part is the
stronger result.*

---

**The short version.** We reimplemented Anthropic's Jacobian lens faithfully
(verified against their released code) and found that the J-space ranking is
strongly confounded by token frequency: rare tokens score high, common tokens
score low (r ≈ -0.6 to -0.7 at every layer, p ~ 10^-9 or less). Anthropic never
controls for frequency — not in the paper, not in the released code. Digging
into *why* gave us the most interesting result: the frequency signal splits
into two separable parts. Half lives in the static geometry of the model's
word-scoring matrix — baked into the lens by definition, so any user inherits
it. A smaller, layer-dependent part lives in what the layers themselves do, and
vanishes at the final layer. A frequency-matched synthetic pair shows the lens
also carries genuine structure signal (confirmed by a clean-boundary control at
~1.3x), and a causal test found the demotion effect is small under loss
reweighting. We are **not** claiming the J-space doesn't exist. We're claiming
that any "privileged subspace" interpretation needs a frequency control first.
The full story — numbers, mistakes, and all — is below.

---

## 1. THE MACHINE THAT GUESSES WORDS

A language model is, at its heart, a machine that guesses the next word. Show it
"the cat sat on the" and it produces a list of probabilities for what comes next:
"mat" high, "chair" high, "banana" low. Everything it "knows" is wrapped up in
that guessing.

The interesting question is: *where* does the guessing happen? A modern model
has dozens of layers, each transforming the sentence a little. Somewhere in
those layers, the model is deciding that "cat" is an animal, that "sat" is past
tense, that a location is coming. We would like to watch that happen. The
problem is that the inside of a transformer is a soup of high-dimensional
vectors, and no one has a map.

For a long time, people used the "logit lens": at each layer, take the
representation, and ask "if the model had to guess *right now*, what would it
guess?" The trouble is that representations change coordinate systems as they
travel through the layers, so early layers give you nonsense. It's like trying
to read a letter that's been translated into a language you don't know — at the
start of the chain, the translation is too rough.

## 2. ANTHROPIC'S IDEA: THE JACOBIAN LENS

In 2026, Anthropic published a paper — "Verbalizable Representations Form a
Global Workspace in Language Models" — introducing a smarter version: the
*Jacobian lens*. Instead of asking "what would the model guess right now?", it
asks a sharper question: *"if I nudge this representation a tiny bit, how much
does the final guess move?"*

That's what a Jacobian is: a table of "how much does each output move when each
input moves." The lens computes, for every layer, the average nudge-effect of
that layer's representation on every word in the vocabulary, averaged over a
thousand different contexts. Words whose representations are strongly "poised"
to be spoken — ready to be said, should the occasion arise — get big numbers.
Anthropic calls this collection of word-vectors the **J-space**, and they claim
it's a kind of "global workspace": a small, privileged subset of the model's
internal state that can be reported on, modulated, and used for reasoning. They
even note the resemblance to theories of consciousness, carefully, the way you
would mention a bear while making clear you are not feeding it.

The headline claim that caught our eye: **the J-space has limited capacity —
only 10 to 50 concepts are "active" at once.** A tiny privileged workspace
inside a big model. That's a strong claim. Strong claims deserve strong tests.

## 3. THE ITCH

The moment we read the paper, something felt off. Here's the thing about token
frequencies: in any language, a handful of words ("the", "of", "and") appear
all the time, and thousands of words appear almost never. In the model's
vocabulary of 50,257 tokens, the rarest are nearly invisible.

Now, the J-lens vector for a word is a gradient — it measures how much the
model's computation tunes toward that word. And there's a mechanical quirk of
gradients through softmax: the *less* likely a word is, the *larger* the raw
gradient term can be. A gradient of log-probability contains a term that looks
like (1 - p), where p is the word's probability. Rare words have small p, so
(1 - p) is close to 1. Common words have large p, so (1 - p) is small. If the
lens is ranking words by the size of this gradient, the ranking is partly
pre-written by the frequency distribution before the model even learns
anything.

(A note on that intuition: it applies directly to our first, simpler
implementation, which differentiated through the softmax. With the faithful
lens the mechanism is different — it turns out to live partly in the geometry
of the word-scoring matrix itself. Section 6 has the full decomposition.)

In other words: **a "privileged workspace" might just be a frequency effect
wearing a fancy hat.**

## 4. OUR FIRST ATTEMPT — AND THE BUG THREE REVIEWERS FOUND

We set out to test this on a small model we could train ourselves: a
10.65-million-parameter character-level transformer (Karpathy's nanoGPT),
trained on Shakespeare. Small enough to run on a 4GB GPU in a few hours. Big
enough to have real layers.

Our first implementation looked reasonable. We hooked into each layer, computed
the gradient of log-probability for every character, averaged over contexts,
and — sure enough — found a strong correlation: rare characters had big
J-lens norms, common characters had small ones (r ≈ -0.65). We were excited.
We were also wrong.

Before publishing anything, we did something slightly unusual: we asked three
independent AI reviewers to try to tear the work apart. We gave them our code
and our results and asked them to find the flaws. All three, independently,
found the same one:

**Our implementation was not computing Anthropic's Jacobian lens.**

Anthropic's lens computes the average Jacobian from a layer to the *final
representation* — the residual stream — and *then* reads it out through the
model's word-scoring matrix. Our code instead differentiated through the
softmax directly. That folds a frequency-dependent calibration factor — the
(1 - p) term — into the thing being averaged. Our beautiful correlation might
have been an artifact of our own measurement.

This is the part of the story we like best, because it's the part that's easy
to skip: we had built a measurement that *looked* like the paper's and wasn't.
The reviewers caught it, we fixed it, and the honest result got stronger.

## 5. THE RIGHT WAY

We rebuilt the lens to match the paper's definition exactly. The faithful
computation is:

> For each layer ℓ, compute the average Jacobian from that layer to the final
> residual stream, over all source positions, all future positions, and many
> prompts. The J-lens vector for a word is that matrix read through the
> model's own unembedding rows.

We verified our implementation the way you verify a ruler: at the last layer,
the Jacobian from a layer to itself is the identity matrix, so the faithful
J-lens vectors *must* equal the model's word-scoring rows. Our check returned
cosine similarity 1.0000 — exactly. The ruler is correct.

(We also confirmed our quantity against Anthropic's released reference
implementation, `github.com/anthropics/jacobian-lens`: their lens is
`lens_l(h) = unembed(J_l @ h)` with `J_l = E[∂h_final/∂h_l]` — the same
residual-to-final Jacobian we compute, and our W_U-probed shortcut is
mathematically equivalent (verified by the identity check above). Their
estimator has two differences of detail: it excludes the first 16 positions
(attention sinks) and the last position from the average, and it averages over
source positions rather than (source, future) pairs. We re-ran our analysis
with their exact estimator choices: the frequency correlation is essentially
identical at every layer (max delta 0.008, see results.md section 1b), so the
result is robust to those choices.)

One technical note before moving on, because it matters for the
interpretation: we capture the residual stream *before* the model's final
layer norm. That matches the paper's definition — the Jacobian stops at the
final residual stream and the J-lens vectors are the rows of W_U·J_ℓ, with
normalization applied only when *reading* the lens, i.e.
softmax(W_U·norm(J_ℓ·h_ℓ)). Under that definition the last-layer identity
check is exact by construction.

## 6. WHAT WE FOUND: FREQUENCY IS EVERYWHERE

On the real trained model, all six layers, both the old (buggy) proxy and the
faithful lens, correlated with token frequency like this:

```
  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
```

The correlation survived the faithful implementation — slightly *stronger*, if
anything. The rare characters ('?', 'z', 'q', '$') sit at the top of the
J-space ranking; the common ones (space, 'e', 't', 'i') sit at the bottom. On
the paper's own quantity, the J-lens ranking is frequency-confounded. The
statistics are not subtle: Pearson r ≈ -0.61 to -0.69 (p ~ 10^-9 to 10^-13),
and Spearman rank correlation is even stronger (-0.69 to -0.85), so the
result is not an artifact of a few extreme common tokens.

### THE MOST INTERESTING THING WE FOUND: WHERE THE CORRELATION COMES FROM

The faithful lens vector for token k is W_U[k]·J_ℓ — the row of the
unembedding matrix times the layer Jacobian. The correlation can come from
either factor, and the two behave very differently. This decomposition is, we
think, the actual novel mechanistic contribution of this project: **the
frequency confound is not one thing.**

1. **Static geometry — the ruler.** The unembedding row norms ||W_U[k]||
   themselves anti-correlate with frequency (r = -0.61, Spearman -0.81).
   Rare tokens get bigger rows in the word-scoring matrix. Since the lens
   reads through W_U by definition, any user of the lens — including
   Anthropic's capacity analysis — inherits this bias automatically. A
   frequency control would have to live in the geometry, not in the prompts.

   Where does that geometry come from? Not from initialization: fresh models
   show no frequency correlation in their row norms (r ≈ +0.00 to -0.19
   across three seeds). After training it is -0.61. The model *learns* to
   push rare tokens' rows outward as it learns the corpus, and nothing in the
   objective or the readout ever corrects it. The lens inherits learned
   geometry, not a fixed one.
2. **Layer dynamics — the layers.** Regress out the W_U component and an
   anti-correlation still survives in layers 0-4 (partial r ≈ -0.24 to
   -0.36) — something about what the layers themselves do keeps boosting
   rare tokens — and it vanishes at the last layer (+0.06). The mechanism
   of that layer-dependent part is still under investigation.

So the lens carries a frequency signal from both the ruler it reads with and
from what the layers do — and the two are separable. That is the finding we
would most want someone to test at scale.

A fact-check before we go further. We were about to claim "Anthropic does not
control for frequency anywhere," and that is the kind of claim that should be
checked, not asserted. We checked it four ways: our own scan of the paper's
text, two independent adversarial reviewers who read the full paper including
the appendix, and — after a reader pointed us to it — Anthropic's own released
companion code (`github.com/anthropics/jacobian-lens`, Apache-2.0). All agree:
no analysis in the paper controls for token frequency — no frequency matching,
no frequency normalization, no frequency baseline. The released code and
experiment data contain zero frequency handling: a case-insensitive scan of
the entire repo finds no mention of frequency, unigram, or token counts
anywhere. The one related detail is an appendix note about a separate baseline
method (the "template lens"), where they filter "high-frequency noise tokens"
and explicitly call that "not a principled approach." To be precise: that note
concerns the template lens, not the main J-lens — it is not evidence that they
observed this confound in the J-lens itself. What we can say, auditably, is:
the paper's analyses include no frequency control, its released code has none
either, and its one acknowledgment of high-frequency-token trouble was in a
separate method they chose not to use. Any "privileged subspace"
interpretation needs a frequency control first.

**Related work.** We are not alone in noticing the raw lens is distorted by
token statistics. An independent research-engineer analysis of the same paper
(willkn, GreaterWrong, 24 Jul 2026), working on GPT-2-medium (355M — thirty
times our model), found that the raw fitted Jacobian "misweights structural
tokens (grammar, punctuation) over semantic content" — dominant spectral
channels carry ~10x the residual pathway's gain, and structural tokens are the
high-frequency tokens. They also found the Jacobian essentially full-rank
(562-858 dimensions for 90% of spectral variance), matching our toy-scale rank
result. Their fix — a single-parameter shrinkage regularizer (J + λI) that
restores next-token faithfulness — is worth testing against our frequency
correlation; Anthropic's released fitting code applies no such regularization.
Whether shrinkage removes the correlation is an experiment we have not run
yet; it is a natural next step.

## 7. BUT NOT *ONLY* FREQUENCY

Now the twist. Correlation is not causation, so we ran a cleaner test. We made
a new corpus with two brand-new characters, both at *exactly* the same
frequency (0.1%):

- `@` — appears only after the trigger "the ". The model can predict it in
  context. It is *poised to be said*.
- `#` — appears at random positions. Nothing predicts it.

Same frequency. Different structure. If the J-lens were purely a frequency
meter, the two tokens would get identical norms. Here is what three separate
training runs showed:

```
  seed   @ norm (predictable)   # norm (noise)   ratio
  0      0.0232 - 0.0246        0.0152 - 0.0154   1.51 - 1.60
  1      0.0224 - 0.0237        0.0148 - 0.0151   1.50 - 1.60
  2      0.0215 - 0.0233        0.0154 - 0.0163   1.35 - 1.51
```
*(Norm ranges are across layers 0-5 within each seed; ratios are per-layer
@/#. The middle-layer summary follows.)*

The predictable token scores **~1.4-1.5x higher** than the noise token at
identical frequency, in every layer of every seed. Middle-layer ratio across
the three seeds: 1.47 ± 0.09, bootstrap 95% CI [1.37, 1.53] — entirely above
1. So the lens is not a pure frequency meter: at equal frequency, the two
tokens differ in norm.

One caveat, found by a reviewer: the noise token '#' was inserted at random
character positions, which slices through the middle of a word 58% of the time
(th#e, ki#ng — letter on both sides), while '@' always sits at a clean word
boundary after "the ". That confounds predictability with n-gram corruption —
so we ran the control that isolates them: '#' inserted at random *word
boundaries* (0% word-slicing, still unpredictable), same 0.0998% frequency,
three fresh seeds.

The control is done, and it is the honest kind of result — partly confirming,
partly correcting:

```
  placement of '#'      middle-layer ratio @/#   bootstrap 95% CI
  random (58% slicing)  1.47 ± 0.09              [1.37, 1.53]
  clean boundary (0%)    1.31 ± 0.08              [1.26, 1.40]
```

The corruption confound was real: it inflated the estimate by about 12%. But
it was not the whole story. At identical frequency, with clean boundaries and
nothing sliced, the predictable token still scores ~1.3x higher than the
unpredictable one, and the CI stays entirely above 1 in every seed. The
conditional-predictability signal — the thing "verbalizable" should mean —
survives the control, modestly smaller than our first estimate.

## 8. THE CAUSAL TEST: WHAT ACTUALLY HAPPENED

The last experiment was the one designed to make the frequency story causal.
Train three models per seed from the *identical* starting weights and the
*identical* minibatch order — the only difference is the loss: one model gives
the letter 'q' twice the learning pressure (2x CE weight on 'q' targets, which
raises its effective frequency without corrupting the text), one is a plain
control, and one upweights the same number of random *other* letters (to check
that "any reweighting" isn't the thing doing the work). Three seeds, three
models each. If doubling 'q's effective frequency causally shrinks its J-lens
norm below both controls, the frequency story is causal, not just
correlational. The faithful lens norm of 'q' (layers 2-4, mean per seed):

```
  seed | q(2x)  control  ctrl_random | q/control  q/ctrl_random
  0    | 0.0163  0.0174   0.0152    |   0.934       1.069
  1    | 0.0150  0.0171   0.0161    |   0.881       0.932
  2    | 0.0161  0.0158   0.0169    |   1.019       0.952
```

Cross-seed: q/control mean = 0.944 (bootstrap 95% CI [0.881, 1.019]),
q/ctrl_random mean = 0.985 (CI [0.932, 1.069]).

What this shows, honestly:
1. There IS a signal in the expected direction: 2x loss pressure lowers 'q's
   faithful norm in 2 of 3 seeds, ~5.6% on average below the plain control.
2. It is small — and the sharper truth is that this design was never powered
   to see it. Observed sd 0.07 on the ratio against a mean deficit of 0.056:
   a two-sided 80%-power test needs ~13 seeds per arm, and we ran 3. The
   experiment could answer "is the effect large?" (no — CI crosses 1.0, one
   seed goes the other way, and the effect vanishes against the
   random-upweight control) but not "is there any effect at all?". The right
   version is ~13 seeds per arm, or a manipulation that moves frequency more
   than 2x.
3. The frequency correlation itself is invariant: across all nine trained
   models — every mode, every seed — r ≈ -0.63 to -0.69. Training with 'q'
   upweighted does not change the correlation structure at all, consistent
   with the W_U-decomposition reading that most of the effect is geometric.
4. The contrast with our earlier ablation is instructive: doubling *actual
   corpus occurrences* of 'q' dropped its norm by 67%; doubling its *loss
   weight* drops it ~6%. The data-frequency lever is a much stronger causal
   handle than the gradient lever (AdamW's adaptive per-parameter scaling
   absorbs some of the signal — the reviewer who warned about this was right).

Net: the frequency confound is strongly correlational and geometrically
stable; the causal lever we could afford to test is weak. This is the honest
state of the causal evidence. (Absolute 'q' norms differ across experiments —
base model 0.011 vs these 0.015-0.018 — so only within-experiment
comparisons are meaningful.)

## 9. WHAT WE ARE NOT SAYING

Let us be very careful here, because it would be easy to overclaim.

- We are **not** saying the J-space doesn't exist. We haven't tested
  Anthropic's actual capacity claim (which is about *occupancy* — how often
  J-lens directions are used per position — not about the rank of the word
  vectors).
- We are **not** saying they have no controls at all. Their occupancy
  analysis compares against random-direction baselines, and their probes
  subtract mean concept directions. Those are real experimental controls —
  but none of them is a token-frequency control, which is the specific thing
  our claim is about.
- We are **not** saying the lens is useless. The synthetic-pair result shows it
  carries real structure signal.
- We are **not** saying "it's just linear algebra." Our toy models don't show
  the compression Anthropic sees in large models; that's a limitation of toy
  models, not evidence against large ones.

What we **are** saying is narrower and, we think, more durable: on the paper's
own measurement, J-lens *rankings* are strongly confounded by token frequency
at every scale we can test, and frequency is a variable any J-lens analysis
should control for. Whether the confound survives at Anthropic's scale is an
empirical question — one we are taking to bigger models next.

## 10. WHAT'S NEXT

Toy scale answers the methodological question. Scale answers the real one. We
want to run the faithful lens on a real language model (V = 50K, d = 768 — the
regime where Anthropic's claims live) with proper statistical power, and to run
the occupancy test their capacity claim is actually about. That's the next
post.

The learned-geometry finding makes one piece of that cheap and decisive: if
the W_U row-norm anti-correlation is a general property of softmax-output
models trained on Zipfian data, it should appear in GPT-2's unembedding matrix
directly — no Jacobian computation at all. One line of code at scale, and it
either generalizes or it doesn't.

## 11. HOW TO REPRODUCE EVERYTHING

All code, data-prep scripts, experiment scripts, tests, and this analysis live
in the repository: <https://git.jayrup.me/c/jspace-nanogpt.git/>. Summary of
results in `results.md`. Reproduction steps in the README. The only
requirements are a Linux machine with Docker, a CUDA GPU (any modern card; we
used a 4GB Quadro K2200), and the `pytorch/pytorch:2.4.1-cuda11.8` image.

Run the test suite:
```
sh scripts/test.sh
```

Rebuild the main experiment from scratch:
```
# 1. train the character-level model on Shakespeare (10.65M params)
# 2. compute the faithful J-lens + old proxy, all layers:
python3 src/jlens_v3.py --checkpoint out-shakespeare-char/ckpt.pt \
    --data_dir data/shakespeare_char --layers 0,1,2,3,4,5
# 3. synthetic frequency-matched pair:
python3 src/synthetic_pair.py --step prep
python3 src/synthetic_pair.py --step train --seed 0
python3 src/synthetic_pair.py --step jlens --seed 0
python3 src/synthetic_pair.py --step summary
# 4. loss-reweighting causal test:
python3 src/loss_reweight.py --step train --mode q --seed 0
python3 src/loss_reweight.py --step summary
```

---

*Written in the spirit of the rule we keep trying to follow: the first
principle is that you must not fool yourself — and you are the easiest person
to fool.*