Jacobian Lens and Global Workspace Interpretability

Why This Matters

Reading an intermediate residual activation

Logit lens

1. h_l intermediate activation
2. norm no geometry correction
3. W_U unembedding
4. tokens ranked readout

J-lens

1. h_l intermediate activation
2. J_l mean Jacobian correction
3. norm + W_U final-layer coordinates
4. tokens ranked readout

Sensory range: L0–L38

Workspace-like range: L38–L92

Motor range: L92–L100

The logit lens reads an intermediate residual-stream activation by sending it directly through the model's final normalization and unembedding. This is easy to compute, but it assumes that a feature has the same coordinates at an early layer and at the output. That assumption becomes less plausible as many layers transform the representation.

Gurnee et al. (2026) introduce the Jacobian lens, or J-lens, to correct for this representational drift. The method estimates how a perturbation at layer $\ell$ changes final-layer activations on average across prompts, source positions, and current or later target positions. The resulting token-indexed directions provide a vocabulary readout for intermediate activations.

The paper then asks a larger empirical question: do the representations found by this lens act like a limited shared workspace for verbal report, flexible inference, and internal reasoning? Its evidence includes coordinate swaps, concept injections, ablations, cross-task transfer, and controls against non-J-space directions. These are intervention-based experimental findings on the tested models. They are not mathematical consequences of the J-lens definition.

The Jacobian-Lens Construction

Let $h_{\ell,t} \in \mathbb{R}^d$ be the residual-stream activation at layer $\ell$ and token position $t$. A perturbation to this activation can affect the final residual state $h_{L,t'}$ at every later position $t' \geq t$. The paper averages the corresponding Jacobians over current and later target positions:

$$J_\ell = \mathbb{E}_{\text{prompt},\,t,\,t'\geq t} \left[ \frac{\partial h_{L,t'}}{\partial h_{\ell,t}} \right].$$

The expectation uses 1,000 prompts sampled from a pretraining-like distribution. For an activation $h_\ell$, the vocabulary readout is

$$\operatorname{lens}(h_\ell) = \operatorname{softmax} \left( W_U\,\operatorname{norm}(J_\ell h_\ell) \right),$$

where $W_U$ is the model's unembedding matrix. The rows of $W_UJ_\ell$ are called J-lens vectors. Setting $J_\ell=I$ recovers the direct logit-lens approximation.

Proposition

Mean-Jacobian First-Order Response

Statement

For a perturbation $\varepsilon\delta$ at $h_{\ell,t}$, let $F_c(h_{\ell,t})=h_{L,t'}$ denote the downstream residual map in sampled context $c=(\text{prompt},t,t')$. Then

$$\mathbb{E}_c\left[F_c(h_{\ell,t}+\varepsilon\delta)-F_c(h_{\ell,t})\right] = \varepsilon J_\ell\delta+o(\varepsilon),$$

with $J_\ell=\mathbb{E}_c[DF_c(h_{\ell,t})]$. For any fixed linear readout $A$, applying $A$ gives the corresponding average first-order readout change, $\varepsilon AJ_\ell\delta+o(\varepsilon)$.

Intuition

Each prompt induces its own local linear map from an intermediate activation to a later activation. Averaging these maps retains effects that recur across contexts while reducing prompt-specific geometry. The result is a fixed layer-wise map that can be reused as a readout.

Proof Sketch

Apply the multivariate first-order Taylor expansion to each context: $F_c(h+\varepsilon\delta)-F_c(h)=\varepsilon DF_c(h)\delta+r_c(\varepsilon)$. Take expectations. The assumptions permit the expectation to pass through the linear term and make $\mathbb{E}[r_c(\varepsilon)]=o(\varepsilon)$. Linearity of $A$ gives the readout statement. This argument establishes a local first-order identity. It does not prove that applying $J_\ell$ to a full activation exactly reconstructs the model's nonlinear computation.

Why It Matters

The proposition isolates the rigorous part of the method. The J-lens is based on the derivative of the actual downstream computation, rather than on an assumption that intermediate and final representations already share one coordinate system.

Failure Mode

The approximation can fail for large interventions, sharp nonlinear changes, or contexts far from the averaging distribution. Averaging also discards context-specific Jacobian structure. Normalization and softmax add further nonlinearity. The actual logit derivative contains the derivative of the final normalization and cannot in general be reduced to $W_UJ_\ell\delta$. A high-ranked lens token is a diagnostic readout, not an exact decomposition of the model's state.

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From J-Lens Vectors to the J-Space

At one layer, the vocabulary supplies more J-lens vectors than the residual stream has dimensions. They form an overcomplete frame, not a basis. A vector therefore has many possible expansions in J-lens coordinates.

The paper imposes sparse nonnegative structure. If $V_\ell$ has J-lens vectors as columns, the $k$-sparse J-space is

$$\mathcal{J}_{\ell,k} = \left\{ V_\ell a: a\geq 0,\;\lVert a\rVert_0\leq k \right\}.$$

Geometrically, this is a union of cones, one cone for each active set of at most $k$ token directions. The study usually takes $k\leq25$ and estimates a sparse nonnegative reconstruction with gradient pursuit. The reconstructed part is the activation's J-space component; the residual is its non-J-space component. Because the dictionary is overcomplete and the feasible set is nonconvex, neither the coordinates nor a nearest reconstruction must be unique.

What the Experiments Establish

Unless a paragraph names another model, the results below use the paper's default model, Claude Sonnet 4.5.

Verbal report: Figures 6–8

The authors compare interventions along pure J-lens vectors, the J-space part of independently estimated concept vectors, and the remaining non-J-space part. Across concepts and workspace layers, the J-space component contains a median of only 6–7% of concept-vector variance. Even so, pure J-lens swaps move the target concept into the output top five on 88% of trials. J-space concept components reach 59%, while equal-norm non-J-space components reach 5%. Clamping the relevant J-lens coordinates removes the remaining non-J-space effect.

The 88% number has a narrow meaning: target-to-top-five success in this swap protocol. It is not an 88% accuracy result, a universal report-flip rate, or a claim about all model internals.

Internal reasoning: Figures 12–17

In the paper's clearest single example, the model completes "the number of legs on the animal that spins webs is" with 8. The J-lens surfaces the unspoken intermediate spider. Swapping the spider coordinate for ant changes the top output from 8 to 6.

On 50 two-hop factual prompts, raw J-lens coordinate swaps put the target answer at top one on 54% of trials for Haiku 4.5 and 70% for both Sonnet 4.5 and Opus 4.5. A separate 90-prompt probe decomposition gives a more pointed control: swapping only the probe's J-space component succeeds on 61% of trials, compared with 28% for the non-J-space component. When the relevant J-space coordinates are clamped, that 28% falls to 6%.

These results support a causal role for the manipulated coordinates in the tested tasks. They do not show that every reasoning step uses the J-space or that the named token direction is a complete account of the representation.

Flexible use and selectivity: Figures 18–24

The flexible-use experiment applies one source-to-target concept swap across multiple downstream functions. Across 192 trials, the target-appropriate answer reaches top one in 76 trials at standard strength and 101 trials at double strength. Success varies by concept category and is predicted by how strongly the source concept is loaded before intervention.

Selectivity tests separate flexible inference from more automatic processing. A language-coordinate swap changes explicit language reports and questions that require computing with the language, but leaves fluent continuation and local anomaly detection largely unchanged. Heavy ablation of the ten strongest J-lens directions drives controlled multi-hop accuracy near zero while leaving several classification, comparison, and extractive tasks near baseline.

Layer range and capacity: Figures 27–30

The study reports all layer numbers on a normalized 0–100 scale. Several measurements identify an approximate workspace-like range from L38 to L92; these endpoints are descriptive, not universal cutoffs. Before that band, coherent token readouts are weak; near the output, readouts become tied to the imminent token, which the paper calls the motor range. Within the middle band, median sparse occupancy plateaus around 25 J-lens vectors. The excess activation variance explained by the J-lens reconstruction over a same-size random-direction control never exceeds 10% in the reported measurement.

The authors caution that weak early readouts may partly reflect a limitation of the J-lens itself, and that the L92 transition into the output-linked motor range was selected somewhat post hoc.

This is evidence for a small, privileged component, not evidence that 90% of the activation is irrelevant. The non-J-space component carries most of the variance and supports substantial processing.

Common Confusions

Watch Out

An intervention result is not a mathematical theorem

Changing a model's output after a controlled activation intervention is strong evidence that the manipulated direction plays a causal role in that protocol. It is not the same kind of claim as a convergence proof. The percentages on this page are sample results for particular models, prompts, layer bands, and intervention rules. They can fail to transfer when those conditions change.

Watch Out

J-space is not the model's full reasoning state

At median occupancy, the reported J-lens reconstruction explains at most 10% excess activation variance over a same-size random-direction control. Some automatic computations continue when J-space coordinates are swapped or ablated. The method also reads token-aligned concepts more readily than multi-token relations, bindings, or features without a compact vocabulary name.

Watch Out

Workspace-like function is not evidence of subjective experience

The experiments test functional properties associated with global workspace accounts: report, directed modulation, flexible use, selectivity, and a form of broadcast across depth. The authors do not claim that a transformer reproduces the brain architecture proposed by global workspace theory. The results do not settle whether a language model has subjective experience.

Exercises

ExerciseAdvanced

Problem

Ignore normalization and softmax in a two-dimensional synthetic example. Let $J$ have rows $(1,0.5)$ and $(0,2)$, let $h=(2,-1)^\top$, and let $W_U$ have rows $(1,0)$, $(0,1)$, and $(-1,-1)$. Compute the three J-lens logits $W_UJh$ and rank the tokens. Then set $J=I$ to obtain the direct logit-lens logits. What changed?

Hint

Reveal Solution

ExerciseResearch

Problem

In the 90-prompt probe experiment, the J-space component swap succeeds on 61% of trials. The non-J-space component succeeds on 28%, but falls to 6% when the relevant J-space coordinates are clamped to their clean values. What inference does this control support, and what does it not prove?

Hint

Reveal Solution

Scope and Limitations

The J-lens is restricted to concepts aligned with individual vocabulary tokens. Its readout is a flat list, so it does not directly represent how concepts are bound into relations. Some layer-position readouts remain hard to interpret. The mean Jacobian is also distribution-dependent: a corpus shift can change the average map, and context-specific effects may cancel under averaging.

The main experiments use proprietary model activations and reported protocols. The public article provides detailed methods, controls, and interactive figures, but this page has not independently replicated the numerical results. Treat the results as a well-documented experimental report awaiting broader replication, not as settled model-independent law.

References

• Gurnee, W., Sofroniew, N., Pearce, A., et al. "Verbalizable Representations Form a Global Workspace in Language Models." Transformer Circuits Thread, published July 6, 2026. Primary article.
• Gurnee et al. (2026), Methods, "The Jacobian Lens" and Figure 4 for $J_\ell$, the readout equation, and lens-coordinate patching.
• Gurnee et al. (2026), "The J-space supports verbal report," Figures 6–8 for concept swaps, injection, the 88% top-five result, and component controls.
• Gurnee et al. (2026), "The J-space mediates internal reasoning," Figures 12–17 for the spider-to-ant example, two-hop trials, probe decomposition, and arithmetic intermediates.
• Gurnee et al. (2026), "The J-space supports flexible generalization," Figures 18–19 for cross-function concept swaps and the 76/192 and 101/192 results.
• Gurnee et al. (2026), "The J-space selectively mediates flexible but not automatic cognition," Figures 20–24 for task selectivity and J-space ablation.
• Gurnee et al. (2026), "The J-space's structure supports its function," Figures 27–30 for the normalized L38–L92 band, occupancy, and activation-variance measurements.
• Gurnee et al. (2026), Discussion, "Limitations and open questions" for token-vocabulary restrictions, missing relational structure, and inconsistent readouts.
• Belrose, N., Furman, Z., Smith, L., et al. "Eliciting Latent Predictions from Transformers with the Tuned Lens." arXiv:2303.08112. Comparison point for learned layer-wise vocabulary translators.
• Marks, S., Rager, C., Michaud, E. J., Belinkov, Y., Bau, D., Mueller, A. "Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models." arXiv:2403.19647. Feature-level causal circuit analysis.