Jacobian Scopes: A Unified Geometric Framework for Token-Level LLM Attribution

Toni J. B. Liu; Baran Zadeoğlu; Gurbir Arora; Raphaël Sarfati; Nicolas Boullé; Christopher J. Earls

cs.CLJan 23, 2026

Jacobian Scopes: A Unified Geometric Framework for Token-Level LLM Attribution

Authors: Toni J. B. Liu, Baran Zadeoğlu, Gurbir Arora, Raphaël Sarfati, Nicolas Boullé, Christopher J. Earls

Organizations: ∗Cornell University, USA · ‡Goodfire AI, USA · †Imperial College London, UK

Abstract

Large language models (LLMs) make next-token predictions based on clues present in their context, such as semantic descriptions and in-context examples. Yet, elucidating which prior tokens most strongly influence a given prediction remains challenging due to the proliferation of layers and attention heads in modern architectures. We propose Jacobian Scopes, a suite of gradient-based, token level causal attribution methods for interpreting LLM predictions. Grounded in perturbation theory and information geometry, Jacobian Scopes quantify how input tokens influence various aspects of a model's prediction, such as specific logits, the full predictive distribution, and model uncertainty (effective temperature). Through case studies spanning instruction understanding, translation, and in-context learning (ICL), we demonstrate how Jacobian Scopes reveal implicit political biases, uncover word- and phrase-level translation strategies, and shed light on recently debated mechanisms underlying in-context time-series forecasting. On token-ablation faithfulness benchmarks (LAMBADA and IWSLT2017) across six LLMs, the distribution-level Fisher and Temperature Scopes significantly outperform established gradient-based attribution methods in 10 of 12 model-dataset settings. To facilitate exploration of Jacobian Scopes on custom text, we open-source our implementations and provide a cloud-hosted interactive demo at https://huggingface.co/spaces/Typony/JacobianScopes.

Figures & tables

Figure 1: Overview of Jacobian Scopes. (a) An autoregressive LLM maps an input token sequence to a hidden state $\bm{y}$ , which is decoded into a predictive distribution over the vocabulary. (b) A small perturbation to any input token is propagated to the output via the Jacobian $\bm{J}_{t}$ , then projected onto a direction $\bm{v}$ in output space; the magnitude of this projection defines the token’s influence score, computable in a single backward pass. The three Scopes correspond to $\bm{v}=\bm{w}_{\mathrm{target}}$ (target unembedding row), $\bm{v}=\bm{u}_{1}$ (principal Fisher direction), and $\bm{v}=\hat{\bm{y}}$ (normalized hidden state). (c) The three Jacobian Scope variants each target a different feature of the model’s output: Semantic Scope asks which input tokens drive the predicted probability of a specific target word; Fisher Scope asks which tokens most change the overall shape of the predicted distribution; and Temperature Scope asks which tokens control the model’s confidence, i.e., how sharply peaked the distribution is.

Figure 2: Semantic Scope reveals the potential for political bias that may be present in models such as LLaMA-3.2. The prediction for the subject being a “liberal” is attributed to the input token “Columbia”, while the prediction for “conservative” is attributed to the token pair “the South”.

Figure 3: Fisher Scope visualizes LLaMA-3.2 3B’s strategy for translating Dante’s Inferno [ 2 ] . Left: Magnitude of influence from each source word (blue) on the predicted distribution at each target position (red). Right: Probability distribution of the top 7 words predicted at each target position. (a) LLaMA assigns high probability to “world”, the unambiguous literal translation of “mondo”. (b) When translating the phrase “priegoti ch’a la mente altrui mi rechi” (literally: “I pray thee to bring me to another’s mind”), LLaMA’s predicted distribution is highly uncertain, with each top token hinting at a different phrasing strategy.

Figure 4: Left: Temperature Scope reveals different strategies employed by LLaMA-3.2 during time-series forecasting: (a) When continuing the dynamics of a partially observed Lorenz system, LLaMA attends to regions in the input history that exhibit patterns similar to those near the cutoff. (b) When extrapolating unstructured Brownian motion, LLaMA shows diminished influence of early context. Right: Predicted probability distribution over the top 30 most probable tokens.

1B/1.5B-scale
	LLaMA-3.2 1B		Qwen2.5-1.5B		Gemma-3 1B
Method	LMB	IWS	LMB	IWS	LMB	IWS
Random	$-0.26\pm 0.01$	$-0.30\pm 0.01$	$-0.35\pm 0.01$	$-0.25\pm 0.01$	$-0.42\pm 0.01$	$-0.41\pm 0.01$
Integrated Grads	$-1.19\pm 0.01$	$-1.04\pm 0.01$	$-1.71\pm 0.01$	$-1.36\pm 0.01$	$-0.69\pm 0.01$	$-0.87\pm 0.01$
Input $\times$ Grad	$-1.28\pm 0.01$	$-1.04\pm 0.01$	$-1.72\pm 0.01$	$-1.35\pm 0.01$	$-1.56\pm 0.01$	$-1.36\pm 0.01$
Semantic Scope	$-1.30\pm 0.01$	$-1.06\pm 0.01$	$-1.68\pm 0.01$	$-1.24\pm 0.01$	$-1.64\pm 0.01$	$-1.30\pm 0.01$

Table 1: AOPC (higher magnitude is better) on LAMBADA (LMB) and IWSLT2017 DE

\to

EN (IWS) for all six models, averaged over ablation fractions

k\in\{5\%,10\%,20\%\}

via the trapezoidal rule over 1000 test passages.

Figure 5: Performance of Fisher Scope improves slightly with increasing rank

r

in the low-rank approximation ( eq. 23 ), on LLaMA-3.2 1B and LAMBADA (1000 passages). Dashed lines show rank-free methods; shaded bands are

\pm 1

standard error. Rank 1, the default Fisher Scope, captures nearly all of the attainable faithfulness with a single backward pass.

	Semantic Scope	Fisher Scope	Temperature Scope
Quantity explained	target logit $z$	full distribution $\bm{p}$	confidence $\\|\bm{y}\\|_{2}$
Influence score	$\\|\bm{w}_{\mathrm{target}}^{\intercal}\bm{J}_{t}\\|_{2}$	$\\|\bm{u}_{1}^{\intercal}\bm{J}_{t}\\|_{2}$	$\\|\hat{\bm{y}}^{\intercal}\bm{J}_{t}\\|_{2}$
Recommended use case	semantic reasoning & bias analysis	NLP tasks with non-unique predictions	in-context learning on numerical data

Table 2: Comparison of the Jacobian Scopes methods introduced.

\bm{w}_{\mathrm{target}}

denotes the unembedding vector of the target token,

\bm{u}_{1}

the principal Fisher direction, and

\hat{\bm{y}}

the normalized hidden direction.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Fisher Scope attribution examples on (a) LAMBADA and (b) IWSLT2017 DE $\to$ EN, using LLaMA-3.2 3B. Darker blue indicates higher influence; the target token is highlighted in red. Right: Probability distribution of the top 7 words predicted at the target position.

1B/1.5B-scale
	LLaMA-3.2 1B		Qwen2.5-1.5B		Gemma-3 1B
Method	LMB	IWS	LMB	IWS	LMB	IWS
Random	$-0.41\pm 0.01$	$-0.36\pm 0.01$	$-0.51\pm 0.02$	$-0.43\pm 0.02$	$-0.63\pm 0.01$	$-0.85\pm 0.01$
Integrated Grads	$-1.36\pm 0.02$	$-1.27\pm 0.02$	$-2.06\pm 0.02$	$-2.08\pm 0.03$	$\mathbf{-0.81\pm 0.01}$	$-1.02\pm 0.01$
Input $\times$ Grad	$-1.61\pm 0.02$	$-1.46\pm 0.02$	$-2.06\pm 0.02$	$-2.09\pm 0.03$	$-0.80\pm 0.01$	$-1.03\pm 0.01$
Semantic Scope	$-1.63\pm 0.02$	$-1.49\pm 0.02$	$-2.00\pm 0.02$	$-1.92\pm 0.03$	$-0.80\pm 0.01$	$-1.03\pm 0.02$

Appendix

Table 3: AOPC under PAD-token ablation (higher magnitude is better) on LAMBADA (LMB) and IWSLT2017 DE

\to

EN (IWS), averaged over

k\in\{5\%,10\%,20\%\}

via the trapezoidal rule. Best or tied-best entries in each column are bolded.

Model	Dataset	Scope	Baseline	Mean $\Delta$ [95% CI]	$p$
LLaMA-3.2 1B	LAMBADA	Fisher	Input $\times$ Grad	$\mathbf{-0.036\ [-0.048,-0.024]}$	$\mathbf{1.2{\times}10^{-6}}$
LLaMA-3.2 1B	IWSLT	Temperature	IG	$\mathbf{-0.049\ [-0.073,-0.025]}$	$\mathbf{4.0{\times}10^{-4}}$
LLaMA-3.2 3B	LAMBADA	Fisher	Input $\times$ Grad	$\mathbf{-0.055\ [-0.071,-0.039]}$	$\mathbf{1.1{\times}10^{-11}}$
LLaMA-3.2 3B	IWSLT	Fisher	Input $\times$ Grad	$\mathbf{-0.029\ [-0.042,-0.015]}$	$\mathbf{3.9{\times}10^{-4}}$
Qwen2.5 1.5B	LAMBADA	Temperature	Input $\times$ Grad	$\mathbf{-0.141\ [-0.168,-0.114]}$	$\mathbf{3.1{\times}10^{-21}}$
Qwen2.5 1.5B	IWSLT	Temperature	IG	$-0.025\ [-0.052,\phantom{-}0.002]$	$0.12$

Appendix

Table 4: Paired AOPC tests (best Scope vs. best baseline from table 1 ). Negative

\Delta

favors the Scope; bold entries are significant at

p<0.05

(two-sided Wilcoxon).

Figure 7: Semantic Scope visualizes how LLaMA-3.2 understands system-prompt-style instructions. The logit assigned to the token “truthful” is most prominently attributed to the input token “deceive”, and “nice” to “argue”.

Figure 8: Fisher Scope reveals that translated words, highlighted in red, are primarily influenced by their lexical counterparts in the source language.

Figure 9: Semantic Scope attribution for detective micro-fiction. The attribution graph illustrates how LLaMA-3.2 3B connects suspects to their associated clues when reasoning about potential perpetrators and weapons in a novel-like setting. Attributions clarify the model’s process for linking evidence to conclusions across a long narrative context.

Figure 10: Temperature Scope shows LLaMA-3.2 attending to similar motifs in history when extrapolating a logistic map trajectory.

Figure 11: Temperature Scope attribution graph of LLaMA-3.2 extrapolating a partially observed Lorenz system, with a different initial condition from that in Figure 4 (a).

Figure 12: Temperature Scope attribution graph of LLaMA-3.2 extrapolating (a) a partially observed Lorenz system, and (b) the same Lorenz system with uniform drift.

Figure 13: Additional Temperature Scope attributions on LLaMA-3.2 extrapolating Brownian motion in context.

Figure 14: Left: (a) Temperature Scope, (b) Semantic Scope, and (c) Fisher Scope attributions for the same Lorenz system. Right: The predicted probability mass over the top 20 candidate tokens at the target position, ordered by their numerical values. The ground-truth token (shown in red) is used as the target for Semantic Scope.

Figure 15: Causal attribution using (a) Semantic Scope and (b) its path-integrated variant based on Integrated Gradients.

Figure 16: Token-wise $\ell_{2}$ norm of the Integrated Gradient integrand $\nabla z(\bm{\tilde{X}}(\alpha))$ along the interpolation path $\alpha\in[0,1]$ .

Figure 17: Justification for the rank-1 approximation used in Fisher Scope, evaluated on LLaMA-3.2 1B and LAMBADA. Top: Eigenvalue spectrum of $\bm{F}$ across 10 prompts on a log-log scale, exhibiting a consistent power-law decay spanning 6 orders of magnitude. Middle: Mean cumulative explained variance $\sum_{i=1}^{r}\lambda_{i}/\mathrm{tr}(\bm{F})$ averaged over 1000 passages (shaded band: $\pm 1\sigma$ ); $\bm{u}_{1}$ alone accounts for $\approx$ 37% of total sensitivity (1 $\sigma$ : 25–50%). Bottom: AOPC as a function of rank $r\in\{1,4,16,64,256\}$ , with baselines shown as dashed lines (reproduced in Figure 5 ). Rank-1 Fisher Scope already outperforms Input $\times$ Gradient and Semantic Scope; increasing $r$ yields only marginal improvement at $r$ -fold computational cost.

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The generative nature of Large Language Models (LLMs) is reflected in the conditional probabilities they compute to sample each response token given the previous tokens. These probabilities encode the distributional structure that the model learns in training and exploits in inference. In this work, we use these probabilities to situate LLMs within the mathematical theory of stochastic processes. We use this framework to design a model-agnostic probabilistic token attribution measure, using Bayes rule to invert the next-token log-probabilities so as to capture the models internal representation of the distribution over token sequences. The representation is independent of the models computational structure. This representation yields the conditional probability of the response given the prompt, and of the response given the prompt with a token marginalized away. Our attribution score is the log of the ratio of these probabilities. We further compute the entropies of a single prompts token distributions, conditioned on the remaining context. The interplay between entropy and attribution score sheds light on LLM behavior. We evaluate 8 models across 7 prompts and investigate anomalies, token sensitivity, response stability, model stability, and training convergence, thereby improving interpretability and guiding users to focus on uncertain or unstable parts of the generation.

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