Method: the Jacobian lens and the J-space

Archive — the original J-space replication study

This is what the repository started as: replicating the J-space / global-workspace readout claim on open weights. That readout claim did not survive our controls — an informative null, preserved here with full rigor — and the project pivoted to the operator/operand finding on the main page. These pages are kept as-is.

This project reproduces the method of Anthropic's Verbalizable Representations Form a Global Workspace in Language Models (Gurnee, Sofroniew, Lindsey et al., 2026) on open-weights Qwen3 models, using the authors' reference implementation anthropics/jacobian-lens.

The Jacobian lens (J-lens)

A lens reads out what an intermediate residual-stream activation is "poised to make the model say". The logit lens does this by decoding a mid-layer residual \(h_\ell\) directly with the model's unembedding \(W_U\). The Jacobian lens first transports \(h_\ell\) into the final-layer basis with the average input–output Jacobian of the network:

\[ \text{lens}_\ell(h) = \operatorname{softmax}\big(W_U\,\operatorname{norm}(J_\ell\, h)\big), \qquad J_\ell = \mathbb{E}\!\left[\frac{\partial h_{\text{final}}}{\partial h_\ell}\right]. \]

The expectation is taken over a corpus of pretraining-like prompts, over source positions, and over all current-and-future target positions. Intuitively, for every vocabulary token the J-lens finds the internal activity pattern that makes the model more likely to emit that token at some point in the future — it measures causal influence on the output, not mere presence.

Computing \(J_\ell\) requires backward passes through the model: for each of the \(d_{\text{model}}\) output dimensions a one-hot cotangent is injected at every valid target position and backpropagated to layer \(\ell\). This is why the whole pipeline needs autograd on the GPU (not just inference) — see the setup page.

From the logit lens to the Jacobian lens

It helps to build the J-lens up from the simplest readout.

The logit lens (nostalgebraist, 2020) decodes a mid-layer residual in place with the model's own output head:

\[ \text{logit-lens}(h_\ell) = \operatorname{softmax}\big(W_U\,\operatorname{norm}(h_\ell)\big). \]

It asks "if the model stopped at layer \(\ell\) and read out now, what would it say?" The catch is a basis mismatch: the residual stream at layer \(\ell\) is not yet in the coordinate system the unembedding expects — the remaining layers still rotate and rescale it. So the logit lens is reasonable in late layers (basis already close to final) but noisy and systematically under-reads content in the early and middle layers — exactly where the workspace lives.

The Jacobian lens fixes the basis by transporting \(h_\ell\) with the model's own average input–output Jacobian before decoding:

\[ \text{J-lens}(h_\ell) = \operatorname{softmax}\big(W_U\,\operatorname{norm}(J_\ell\, h_\ell)\big). \]

Because \(J_\ell\) is the model's first-order sensitivity — how a nudge to \(h_\ell\) actually propagates to the output, averaged over real text — three things follow:

• It reads in the right basis. Instead of assuming the mid-layer basis equals the final one, it carries \(h_\ell\) forward with the network's own derivative.
• It measures influence, not presence. A direction that exists in \(h_\ell\) but that the downstream network ignores contributes nothing to \(J_\ell h_\ell\); the lens filters for what actually reaches the output.
• It is derived, not trained. Unlike the tuned lens (a learned affine probe fit to predict the next token), \(J_\ell\) comes from the model itself, so it cannot hallucinate content that is not causally there.

lens	transport into final basis	trained?	main failure mode
logit lens	none (\(J_\ell = I\))	no	basis mismatch → under-reads mid layers
tuned lens	learned affine \(A_\ell h + b_\ell\)	yes (fit to next token)	can hallucinate / overfit
Jacobian lens	model's average Jacobian \(J_\ell\)	no (derived)	needs backward passes to build

The logit lens is exactly the special case \(J_\ell = I\) — turn off the transport and the J-lens is the logit lens. In this codebase that is a single flag, lens.apply(..., use_jacobian=False), so every "J-lens vs logit-lens" comparison in the results is controlled: same activations, same unembedding, the only difference is whether \(h_\ell\) is carried through the model's Jacobian first. If the J-lens beats the logit lens, the improvement is due to the transport and nothing else.

The J-space

The J-space is a small, sparse subspace of the residual stream spanned by the J-lens vectors that are active at a given moment (typical sparsity \(k \le 25\)). Its defining properties, which we measure across scale:

• it accounts for a small fraction of activation variance (never more than ~10%, varying by layer);
• far more model components read from and write to it than to ordinary directions (up to ~100× in parts of the network);
• it is organised by depth into a sensory regime (early layers), a workspace regime (the middle band, where abstract content persists), and a motor regime (late layers, aligned with the output).

What we measure

Per-layer metrics (scripts/metrics.py), reindexed to a 0–100 depth scale:

metric	what it captures
jlens_top1/5, logitlens_top1	next-token accuracy of the readout; J-lens vs logit-lens baseline
read_var_frac	fraction of residual variance in the J-lens' top-\(k\) read directions (the "<10%" claim)
eff_dim	participation ratio of \(J_\ell\)'s singular values (effective dimensionality of the workspace)
kurtosis	excess kurtosis of the lens logits (a few concepts dominate → workspace signature)
top1_autocorr	persistence of the top-1 readout across adjacent positions

Lens quality (scripts/lens_eval.py): on the paper's shipped two-hop set, pass@\(k\) = the fraction of hidden bridge entities (e.g. Brazil in "the country where the Amazon ends") that appear at lens rank \(\le k\) at their best layer.

Causal importance (scripts/causal_swap.py): using the shipped probe-swap set, we move the residual component along a bridge entity's J-lens direction onto a different entity's direction, across a layer band, and check whether the model's answer flips accordingly — versus a matched-norm random control. If the workspace band flips (while the control and the early/late bands do not), the J-space is causally steering the computation.

The scale question

The paper studies Claude. Our contribution is to ask, on open weights, when these properties appear: we run the identical pipeline on Qwen3 1.7B, 8B, and 32B and compare the metrics above across model size — does the J-space exist already at 1.7B, or does it sharpen (lower variance fraction, cleaner workspace band, stronger causal flips) as models scale?