Measuring Mechanistic Uncertainty in Large Language Models via Causal Robustness

# Measuring Mechanistic Uncertainty in Large Language Models via Causal Robustness
Source: [https://arxiv.org/html/2610.22636](https://arxiv.org/html/2610.22636)
Majid AfsharAffiliation:University of Wisconsin\-MadisonTimothy A\. MillerAffiliation:Harvard Medical SchoolAffiliation:Boston Children’s HospitalCorrespondence:\{[aneesh\.komanduri](mailto:[email protected]),[yanjun\.gao](mailto:[email protected])\}@[cuanschutz\.edu](https://cuanschutz.edu/)Yanjun GaoAffiliation:University of Colorado Anschutz Medical Campus

###### Abstract

Large language models \(LLMs\) are increasingly used in critical domains, where their susceptibility to hallucinations makes reliable and interpretable uncertainty quantification essential\. We introduce the concept of*mechanistic uncertainty*, which characterizes uncertainty through the stability of the internal mechanisms supporting a model’s response\. We propose the*causal robustness margin*, defined as the minimum internal intervention required to deviate the model’s output, and operationalize it using Jacobian Lens \(Jlens\) concept steering vectors\. Across three open\-source models and five question\-answering datasets, we show incorrect and uncertain responses exhibit smaller margins, indicating greater sensitivity to internal interventions, and the causal robustness margin complements conventional output\-based uncertainty methods\. Finally, in a controlled knowledge injection experiment, correctly learned facts show increased margins after training, suggesting that causal robustness provides a mechanistically grounded signal of uncertainty\.

## 1Introduction

Large language models \(LLMs\) are increasingly deployed in high\-stakes real\-world settings, where confidently incorrect responses can have severe consequences[Kalai et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib18)\. Uncertainty quantification \(UQ\) asks how much confidence to place in an individual prediction, which is essential for building reliable and trustworthy systems, particularly in domains such as healthcare[Hager et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib12)and natural sciences[Chen et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib5)\. Many existing UQ methods, however, characterize uncertainty at the level of model outputs, using signals such as verbalized confidence[Xiong et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib40), sequence likelihood, entropy[Huang et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib14), semantic entropy[Kuhn et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib19), or self\-consistency across multiple generations[Wang et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib37)\. While these approaches can indicate whether an answer appears uncertain, they provide limited insight into*why*that uncertainty arises within the model\. Predictive uncertainty asks,*how uncertain is the model about its prediction?*However, it does not reveal whether the internal computations supporting a response are stable or small perturbations to those computations would produce a different answer\.

A recent line of work explores LLM uncertainty through internal representations and computations utilizing tools from mechanistic interpretability[Olah \(2022\)](https://arxiv.org/html/2610.22636#bib.bib29);[Meng et al\. \(2022\)](https://arxiv.org/html/2610.22636#bib.bib27)\. These studies investigate internal dispersion of activations[Brzezinka \(2026\)](https://arxiv.org/html/2610.22636#bib.bib2);[Bu et al\. \(2026b\)](https://arxiv.org/html/2610.22636#bib.bib4);[Srey et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib32), supervised uncertainty probes[Teplica et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib34);[Bu et al\. \(2026a\)](https://arxiv.org/html/2610.22636#bib.bib3), and identifying uncertainty directions in representation space[Kumaran et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib20);[Ferrando et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib7)\. We study a complementary question:*how robust is the internal computation supporting the model’s prediction?*Rather than measuring uncertainty only from the output distribution, we ask whether it can be inferred from the robustness of the internal causal process that gives rise to the model’s response\. Understanding and evaluating the robustness of a model’s internal computations is a crucial first step toward reliable uncertainty quantification, as uncertainty arising from data cannot be meaningfully assessed without model robustness[Ahdritz et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib1)\.

In this paper, we introduce a mechanistic perspective on uncertainty in LLMs, which we refer to asmechanistic uncertainty\. Our central hypothesis is that LLM’s own uncertainty can be characterized by the stability of the internal causal mechanism that produces a model’s response\. Motivated by this intuition, we propose thecausal robustness margin, defined as the minimum magnitude of an intervention required to deviate the model’s response denoted asα∗\\alpha^\{\*\}\. Our approach uses an analogous principle to notions of minimal perturbation in adversarial robustness\([Szegedy et al\., 2013](https://arxiv.org/html/2610.22636#bib.bib33);[Madry et al\., 2018](https://arxiv.org/html/2610.22636#bib.bib24)\)and counterfactual explanations\([Wachter et al\., 2017](https://arxiv.org/html/2610.22636#bib.bib35)\), where we seek a minimal change to an input or representation that induces a different model outcome\. In our case, we intervene directly on intermediate representations and measure how strongly they must be perturbed before the generated response changes\.

A recent mechanistic interpretability method, Jacobian Lens \(Jlens\)[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib11), makes this study possible\. Jlens significantly improves upon classic methods such as logit lens[nostalgebraist \(2020\)](https://arxiv.org/html/2610.22636#bib.bib28)by linearly transporting intermediate activations through the downstream computation using average Jacobians, and allows for a causal analysis of readouts and interventions in the direction of verbalizable concepts\. We define our causal robustness margin through controlled Jlens interventions across three open\-source models and five question answering datasets\. We find that incorrect responses generally exhibit smaller causal robustness margins and that these margins complement output\-based uncertainty measures\. A controlled knowledge injection experiment further shows increased margins for newly correct responses after training on knowledge sources for specific entities, linking successful learning to greater internal stability\. Together, these findings support causal robustness as an intervention\-based signal of uncertainty\. An overview of the intuition for our method is shown in[Fig\.1](https://arxiv.org/html/2610.22636#S2.F1)\. Our main contributions are:

- •We introduce the novel concept ofmechanistic uncertaintyas the causal fragility of an internal computation supporting a response\. We formalize the notion using thecausal robustness margin, defined as the minimum intervention required to semantically deviate from the baseline response\.
- •We implement the causal robustness margin using the Jlens to steer toward or away from the representation of consensus tokens poised to be verbalized prior to generation\.
- •We perform experiments on five datasets and open\-source LLMs and show that the causal robustness margin can predict response correctness, complements output\-based uncertainty methods, and captures internal mechanism stability\.

## 2Related Work

##### Uncertainty Quantification in LLMs\.

LLM uncertainty is commonly estimated from verbalized confidence[Xiong et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib40), predictive distributions, entropy[Huang et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib14), or sampled outputs[Wang et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib37)\. Verbalized confidence can be poorly calibrated\([Zhao et al\., 2026](https://arxiv.org/html/2610.22636#bib.bib42)\), while likelihood and entropy\-based measures operate over lexical realizations and may not capture semantic equivalence\. Semantic entropy addresses this by clustering sampled responses by meaning before computing uncertainty\([Kuhn et al\., 2023](https://arxiv.org/html/2610.22636#bib.bib19)\)\. In contrast, our work takes a mechanistic and interventional approach to uncertainty quantification\.

##### Mechanistic Interpretability\.

Mechanistic interpretability aims to identify the internal representations and components responsible for model behavior\([Olah, 2022](https://arxiv.org/html/2610.22636#bib.bib29)\)for capabilities such as indirect object identification and factual recall\([Wang et al\., 2022](https://arxiv.org/html/2610.22636#bib.bib36);[Ferrando et al\., 2025](https://arxiv.org/html/2610.22636#bib.bib7)\)\.

Observational tools such as thelogit lens\([nostalgebraist, 2020](https://arxiv.org/html/2610.22636#bib.bib28)\)decode intermediate residual states into vocabulary distributions, while theJacobian lens\(Jlens\)\([Gurnee et al\., 2026](https://arxiv.org/html/2610.22636#bib.bib11)\)significantly improves these readouts and allows interventions in the direction of verbalizable concepts\.

![Refer to caption](https://arxiv.org/html/2610.22636v1/Figure.png)Figure 1:We perform different magnitudes of steeringα\\alpha\(both positive and negative\) along Jlens consensus token vectors to determine mechanistic uncertainty\. Response flips at lower magnitude steers, as shown in the example, indicate high mechanistic uncertainty\. We use mid to last layer activations due to informative verbalizable tokens, whereas earlier layers contain a lot of noise\.Interventional methods instead manipulate internal representations to test their causal role\. These include steering along concept directions\([Park et al\., 2024](https://arxiv.org/html/2610.22636#bib.bib31);[Ferrando et al\., 2025](https://arxiv.org/html/2610.22636#bib.bib7)\), component ablations\([Wang et al\., 2022](https://arxiv.org/html/2610.22636#bib.bib36)\), causal tracing\([Meng et al\., 2022](https://arxiv.org/html/2610.22636#bib.bib27)\), and other circuit discovery techniques[Hanna et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib13);[Conmy et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib6)\.

##### Mechanistic Studies of Uncertainty in LLMs\.

Recent work has further examined mechanistic signatures of uncertainty:[Kadavath et al\. \(2022\)](https://arxiv.org/html/2610.22636#bib.bib17)use internal activations, such as P\(IK\) probes that predict whether a model knows an answer;[Teplica et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib34)identify overlap between uncertainty and factuality\-related components using probe\-derived signals;[Bu et al\. \(2026a\)](https://arxiv.org/html/2610.22636#bib.bib3)use causal\-tracing restoration probabilities as features for supervised uncertainty prediction; and[Srey et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib32)predict correctness from layer\-wise hidden\-state variability\. Sampling\-free methods exploit internal dynamics directly, including semantic convergence across layers\([Bu et al\., 2026b](https://arxiv.org/html/2610.22636#bib.bib4)\)and geometric hidden\-state trajectories\([Wang et al\., 2025](https://arxiv.org/html/2610.22636#bib.bib38)\)\. Finally,[Kumaran et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib20)show that internal confidence representations can causally influence behaviors such as abstention\. Despite this progress, most methods predict uncertainty from statistics dervied from internal features\. In contrast, we study whether*sensitivity to causal interventions*provides an intrinsic measure of uncertainty, linking uncertainty to the model’s underlying computational mechanisms\.

## 3Background

##### Jacobian Lens\.

Recently,[Gurnee et al\. \(2026\)](https://arxiv.org/html/2610.22636#bib.bib11)proposedJacobian Lens \(JLens\), a mechanistic interpretability tool to explore the representations of LLMs, and discovered a global workspace of concepts that a model is poised to verbalize at any point in its processing\. For each token in the model’s vocabulary, Jlens identifies a vector representation that encodes the potential for the model to verbalize that token in the future\. These Jlens vectors comprise a subspace of the model’s representation space, coined J\-space\. Formally, Jlens characterizes an intermediate activation vector by its first order causal effect on the model’s outputs\. For a residual stream vectorhℓ,th\_\{\\ell,t\}at layerℓ\\ell, tokentt, perturbing the representation shifts the residual stream vectors at subsequent layers and token positions\. The Jacobian matrix is computed as an average within and across contexts \(prompts\)\. For a layerℓ\\ell,

Jℓ=𝔼t,t′≥t,prompt​\[∂hfinal,t′∂hℓ,t\]J\_\{\\ell\}=\\mathbb\{E\}\_\{t,t^\{\\prime\}\\geq t,\\text\{prompt\}\}\\Bigg\[\\frac\{\\partial h\_\{\\text\{final\},t^\{\\prime\}\}\}\{\\partial h\_\{\\ell,t\}\}\\Bigg\]\(1\)where the expectation is taken over the source positiontt, all subsequent positionst′t^\{\\prime\}within the context, and a corpus of thousands of prompts sampled from a pretraining\-like distribution\. The result is admodel×dmodeld\_\{\\text\{model\}\}\\times d\_\{\\text\{model\}\}matrix per layer mapping the source layerℓ\\ellto final layerLL\.

##### Token Readouts\.

The Jlens enables token readouts at any layer or token of the model’s processing\. A readout at layerℓ\\ellreplaces all layers downstream ofℓ\\ellwith the linear mapJℓJ\_\{\\ell\}\. Formally,

Jlens​\(hℓ\)=softmax​\(WU​norm​\(Jℓ​hℓ\)\)\\text\{Jlens\}\(h\_\{\\ell\}\)=\\text\{softmax\}\(W\_\{U\}\\text\{norm\}\(J\_\{\\ell\}h\_\{\\ell\}\)\)\(2\)whereWUW\_\{U\}is the unembedding matrix applied to the Jacobian\-transformed hidden state at any layerℓ\\ellfollowed by softmax to produce a probability for each of the model’s vocabulary tokens\.

##### Interventions and Steering\.

We can use Jlens to steer model behavior through causal interventions\. One mechanism is steering a model activation along a Jlens vectorvt=Jℓ​WU\(t\)v\_\{t\}=J\_\{\\ell\}W\_\{U\}^\{\(t\)\}for a tokenttthrough an additive intervention:

h=h\+α​vth=h\+\\alpha v\_\{t\}\(3\)whereα\\alphais a scalar that controls the strength of the steering and can be used to either suppress or amplify interventions\.

## 4A Causal Lens for Uncertainty

### 4\.1Mechanistic Uncertainty

An LLM generates responses through internal computations that transmit information across layers\. These computations can be viewed ascausal mechanismswhere intervening on activations can change the response\([Meng et al\., 2022](https://arxiv.org/html/2610.22636#bib.bib27)\)\. These internal activation pathways should be stable across internal perturbations for a given query\. We precisely rely on this idea and hypothesize that uncertainty can be defined through the sensitivity of the causal mechanism generating the answer, and evaluate it at the activation level through semantically meaningful interventions\.

Complimentary to standard*predictive uncertainty*, which characterizes uncertainty in the model’s output prediction, we definemechanistic uncertaintyas a notion that capturesthe causal fragility of the internal computation supporting a model’s response\.Sensitivity to a semantically meaningful intervention reveals how robustly a model sustains an answer\. If small interventions deviate the answer’s meaning, this may indicate the fragility of knowledge use or reasoning and indicates lower trust in the prediction\.

Predictive uncertainty asks*how uncertain the model is about its prediction?*Mechanistic uncertainty asks*how stable is the internal computation that produces that prediction under controlled causal interventions?*We treat mechanistic uncertainty as a distinct, interventional notion of uncertainty and investigate whether it correlates with and complements standard predictive uncertainty and response correctness\.

We mathematically define mechanistic uncertainty through thecausal robustness margin, a novel formulation of mechanistic uncertainty defined as the smallest intervention required to deviate an output as defined below\.

###### Definition 1\(Causal Robustness Margin\)

For some inputxx, letf⁡\(x\)f\(x\)denote the baseline output andfI​\(x\)f^\{I\}\(x\)the output under a specified interventionIIwith magnitudem⁡\(I\)m\(I\)from an arbitrary intervention setℐ\\mathcal\{I\}\. The causal robustness margin can be defined generally as

R∗​\(x\)=minI∈ℐ⁡\{m⁡\(I\):fI​\(x\)≢f⁡\(x\)\},R^\{\*\}\(x\)=\\min\_\{I\\in\\mathcal\{I\}\}\\left\\\{m\(I\):f^\{I\}\(x\)\\not\\equiv f\(x\)\\right\\\},\(4\)where≢\\not\\equivdenotes a deviation from the original output\. Smaller margins indicate greater mechanistic uncertainty, while larger margins indicate greater robustness to the specified intervention\.

### 4\.2Uncertainty via Causal Robustness

##### Finding a consensus in the workspace\.

Letfθf\_\{\\theta\}be an LLM andy0=fθ​\(x\)y\_\{0\}=f\_\{\\theta\}\(x\)be the baseline answer generated by the LLM for a given queryxx\. Defineℬ=\{ℓ1,ℓ2,…,ℓm\}\\mathcal\{B\}=\\\{\\ell\_\{1\},\\ell\_\{2\},\\dots,\\ell\_\{m\}\\\}as the band of layers under study, where we intuitively choose the middle to late layers since they make up the workspace band consisting of the most relevant concepts processed by the model\([Gurnee et al\., 2026](https://arxiv.org/html/2610.22636#bib.bib11)\)\. The Jlens reveals that LLMs can often process different concepts in their workspace compared to their verbalized outputs\. To study the consistency between the workspace and output space, we produce the token readouts for each layer in the bandℬ\\mathcal\{B\}\. For each layerℓ∈ℬ\\ell\\in\\mathcal\{B\}, we compute the token readout distribution and top readout token for the last tokent∗t^\{\*\}prior to answer generation as follows:

pℓ=softmax\(WUnorm\(Jℓhℓ,t∗\)\)p\_\{\\ell\}=\\text\{softmax\}\(W\_\{U\}\\text\{norm\}\(J\_\{\\ell\}h\_\{\\ell,t\*\}\)\)\(5\)c^ℓ=arg​maxw∈𝒱⁡pℓ​\(w\)\\hat\{c\}\_\{\\ell\}=\\argmax\_\{w\\in\\mathcal\{V\}\}p\_\{\\ell\}\(w\)\(6\)To determine the primary concept that the model is processing in its workspace, we search for theconsensustokenc∗​\(x\)c^\{\*\}\(x\)that appears most frequently across the bandℬ\\mathcal\{B\}as follows:

c∗\(x\)=arg​maxw∈𝒱∑ℓ∈ℬ𝕀\[c^ℓ=w\]c^\{\*\}\(x\)=\\argmax\_\{w\\in\\mathcal\{V\}\}\\sum\_\{\\ell\\in\\mathcal\{B\}\}\\mathbb\{I\}\[\\hat\{c\}\_\{\\ell\}=w\]\(7\)

##### Causal Robustness Margin via steering\.

We instantiate the causal robustness margin defined in[Definition1](https://arxiv.org/html/2610.22636#Thmdefinition1)using normalized Jlens steering, whereα\\alphacontrols intervention strength along concept directions across a selected layer band\. To study the effect of interventions on the model output, we steer all the layers in the bandℬ\\mathcal\{B\}along the consensus Jlens vectorvℓc∗=Jℓ​WUc∗v^\{c^\{\*\}\}\_\{\\ell\}=J\_\{\\ell\}W\_\{U\}^\{c^\{\*\}\}as follows:

∀ℓ∈ℬ,hℓ=hℓ\+α⋅‖hℓ‖‖vℓc∗‖⋅vℓc∗\\forall\\ell\\in\\mathcal\{B\},h\_\{\\ell\}=h\_\{\\ell\}\+\\alpha\\cdot\\frac\{\\\|h\_\{\\ell\}\\\|\}\{\\\|v^\{c^\{\*\}\}\_\{\\ell\}\\\|\}\\cdot v^\{c^\{\*\}\}\_\{\\ell\}\(8\)where‖hℓ‖/‖vℓc∗‖\\\|h\_\{\\ell\}\\\|/\\\|v^\{c^\{\*\}\}\_\{\\ell\}\\\|is a layer\-specific normalization to take into account the size of the residual stream vector across layers relative to the Jlens vector\. We apply the steer across a band of layers primarily to ensure that there is no overwriting or re\-derivation of the original concept and to ultimately reinforce the effectiveness of the steer\. Formally, we can now define uncertainty through a causal lens as theminimal causal intervention steering factor necessary to produce a deviated responserepresented as:

α∗​\(x\)\\displaystyle\\alpha^\{\*\}\(x\)=min𝛼​\|α\|\\displaystyle=\\underset\{\\alpha\}\{\\min\}\\;\|\\alpha\|\(9\)s\.t\.\\displaystyle\\text\{s\.t\.\}fθα\(x\)≢semfθ\(x\)\\displaystyle f\_\{\\theta\}^\{\\alpha\}\(x\)\\not\\equiv\_\{\\mathrm\{sem\}\}f\_\{\\theta\}\(x\)wherefθα​\(x\)f\_\{\\theta\}^\{\\alpha\}\(x\)is the LLM output after steering all activations by the sameα\\alphaand≢sem\\not\\equiv\_\{\\mathrm\{sem\}\}denotes semantic deviation\. Intuitively, this measures the deviation of the model’s output from the original response under different causal intervention strengths, where the tendency to change answers from lower intervention strengths signals higher uncertainty\.

Figure 2:α∗\\alpha^\{\*\}plotted against accuracy\. Binning is performed at eachα∗\\alpha^\{\*\}value, and average accuracy is computed over each bin\. The trend shows accuracy increasing withα∗\\alpha^\{\*\}\.

## 5Experiments

To understand what insights the causal robustness margin can provide with respect to uncertainty, we empirically study the following questions:

- •Is causal robustness meaningfully associated with predictive reliability?
- •Does causal robustness capture an additional dimension of uncertainty that cannot be inferred from model outputs alone?
- •Does knowledge injection improve the stability of causal mechanisms through which the knowledge is expressed?

### 5\.1Setup

##### Datasets and Models\.

We extensively evaluate our approach on several datasets, including open\-text generation datasets TriviaQA[Joshi et al\. \(2017\)](https://arxiv.org/html/2610.22636#bib.bib16), TruthfulQA[Lin et al\. \(2022\)](https://arxiv.org/html/2610.22636#bib.bib22), PopQA[Mallen et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib25), and SciQ[Welbl et al\. \(2017\)](https://arxiv.org/html/2610.22636#bib.bib39), and one multiple\-choice dataset MedQA[Jin et al\. \(2020\)](https://arxiv.org/html/2610.22636#bib.bib15)\. We experiment with three open\-source LLMs that differ in architecture: Llama\-3\.1\-8B\-Instruct[Grattafiori et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib10), Qwen2\.5\-7B\-Instruct[Yang et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib41), and Gemma\-3\-12B\-Instruct[Gemma Team et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib9)\.

##### Implementation Details\.

We use prefitted J\-lens matrices111https://huggingface\.co/neuronpedia/jacobian\-lens/tree/mainto steer the final prompt\-token activations across the last1010layers, where readouts expose coherent verbalizable concepts\. We search both positive and negative steering directions in increasing magnitude, with\|α\|∈\{0\.01,…,0\.50\}\|\\alpha\|\\in\\\{0\.01,\\ldots,0\.50\\\}using step size0\.010\.01, stopping at the first semantic deviation from the baseline answer\. The maximum magnitude is chosen empirically to capture flips in highly robust responses, and steering is normalized as in[Eq\.8](https://arxiv.org/html/2610.22636#S4.E8)\. For short\-answer datasets, we evaluate correctness and answer flips using alias and exact matching\. For TruthfulQA, we evaluate correctness with truthfulqa\-truth\-judge\-llama2\-7B222https://huggingface\.co/allenai/truthfulqa\-truth\-judge\-llama2\-7Band semantic deviation with Qwen3\-32B\.

Table 1:Causal robustness margins and error discrimination\.α¯corr∗\\bar\{\\alpha\}\_\{\\mathrm\{corr\}\}^\{\*\}andα¯inc∗\\bar\{\\alpha\}\_\{\\mathrm\{inc\}\}^\{\*\}are mean margins for correct and incorrect responses; larger margins indicate greater robustness\. AUROC uses−α∗\-\\alpha^\{\*\}to detect incorrect responses\.\(a\)\(b\)
Figure 3:Correlation between the causal robustness marginα∗\\alpha^\{\*\}and conventional uncertainty measures across TriviaQA, PopQA, SciQ, MedQA, and TruthfulQA\. Each panel shows binned uncertainty statistics as a function ofα∗\\alpha^\{\*\}for Llama\-3\.1\-8B\-Instruct, Qwen2\.5\-7B\-Instruct, and Gemma\-3\-12B\-Instruct\. Shaded regions indicate variability within eachα∗\\alpha^\{\*\}bin, and the reportedρ\\rhovalues denote Spearman rank correlation\.

### 5\.2Results and Discussion

#### 5\.2\.1Causal robustness is meaningfully associated with predictive reliability

We first ask whether the causal robustness marginα∗\\alpha^\{\*\}can serve as an instance\-level indicator of model correctness\.[Table1](https://arxiv.org/html/2610.22636#S5.T1)shows AUROC and averageα∗\\alpha^\{\*\}margins for correct and incorrect examples\. Across all models and datasets, correct responses consistently exhibit larger average margins than incorrect responses, suggesting that incorrect predictions are more sensitive to small internal interventions\. We use−α∗\-\\alpha^\{\*\}as an uncertainty score since it is inversely related to correctness\. Our results show AUROCs between55\.7855\.78and75\.1475\.14, with the clearest separation on TriviaQA and PopQA\. The effect is also present on MedQA and SciQ, although its strength varies across models and tasks\. For TruthfulQA, we observe that the models have higher meanα∗\\alpha^\{\*\}values for both correct and incorrect responses, which is consistent with the fact that models often have inflated confidence on TruthfulQA misconceptions\.

Importantly,α∗\\alpha^\{\*\}is not obtained using correctness labels and is defined solely through intervention sensitivity\. Thus, its ability to discriminate correct from incorrect responses emerges from the robustness of the model’s internal computation rather than from a supervised correctness predictor\.[Fig\.2](https://arxiv.org/html/2610.22636#S4.F2)shows accuracy increasing asα\\alphavalues increase, where we plot the average accuracy for samples assigned to eachα\\alphavalue\. Overall, our results reveal smaller margins are associated with less reliable predictions, while larger margins correspond to more stable and correct responses\. We include case\-study examples from each dataset showing both fragile and robust responses in[AppendixC](https://arxiv.org/html/2610.22636#A3)\.

Table 2:AUROC for detecting incorrect answers \(positive class: incorrect\) for each baseline alone \(Base\) and combined withα∗\\alpha^\{\*\}\(\+α∗\+\\alpha^\{\*\}\) in a logistic regression, using out\-of\-fold predictions from 5\-fold cross\-validation averaged over 10 shuffles\.Bold: improvement significant under a paired bootstrap over questions \(95% CI of the difference excludes 0; 1,000 resamples\)\.\(a\)Overall effect of SFT
\(b\)Causal robustness margin by correctness transition

Table 3:Effect of knowledge injection SFT on PopQA accuracy and causal robustness\.\(a\)Accuracy and mean causal robustness margins before and after SFT\.\(b\)Mean margins by correctness transition \(WW: wrong;RR: right\)\.Δ​M=α¯post∗−α¯pre∗\\Delta M=\\bar\{\\alpha\}^\{\*\}\_\{\\mathrm\{post\}\}\-\\bar\{\\alpha\}^\{\*\}\_\{\\mathrm\{pre\}\}; positive values indicate increased robustness\. Only target entities are used for SFT\.
#### 5\.2\.2Causal robustness captures an additional dimension of uncertainty

##### Correlation with behavioral uncertainty\.

We first measure the correlation betweenα∗\\alpha^\{\*\}and conventional output\-based uncertainty metrics to see if mechanistic uncertainty shares information with output uncertainty, specifically self\-consistency and mean token entropy\. Self\-consistency measures consistency acrossK=10K=10completions with temperatureT=0\.7T=0\.7\. Mean token entropy averages the standard entropy over the token distribution at each step in generation\. Across datasets and models,α∗\\alpha^\{\*\}correlates positively with self\-consistency and negatively with mean token entropy\. Responses that are more robust and resistant to interventions generally show better agreement across sampled generations\.[Fig\.3](https://arxiv.org/html/2610.22636#S5.F3)shows correlation between mechanistic stability and conventional output\-based uncertainty measures\. However, we note that the strength of the correlation largely varies across models and datasets\. For example, Gemma shows high self\-consistency and low entropy across a range of margins\. From a dataset perspective, TruthfulQA shows weak correlations across all models\. These results suggest that output confidence is correlated with but does not necessarily imply mechanistic stability\. Correlation plots for additional uncertainty methods are provided in[AppendixC](https://arxiv.org/html/2610.22636#A3)\.

##### α∗\\alpha^\{\*\}complements behavioral uncertainty\.

To evaluate whetherα∗\\alpha^\{\*\}can serve as a complimentary signal for uncertainty, we train a simple logistic regression model using the baseline\-only feature as well as the baseline feature andα∗\\alpha^\{\*\}\.[Table2](https://arxiv.org/html/2610.22636#S5.T2)provides evidence that mechanistic uncertainty through the causal robustness margin complements conventional output\-based uncertainty measures\. We find that addingα∗\\alpha^\{\*\}improves AUROC consistently across a majority of datasets and models\. Most notably,α∗\\alpha^\{\*\}consistently improves AUROC for self\-consistency\. For example, addingα∗\\alpha^\{\*\}increases self\-consistency AUROC from 0\.653 to 0\.752 for the Gemma model on TriviaQA and from0\.5540\.554to0\.6380\.638for Qwen on MedQA\. We also find that TruthfulQA has the largest gains across models whereα∗\\alpha^\{\*\}correlated weaker with output\-based signals\. These observation indicates that the sensitivity to internal interventions can capture information about correctness beyond agreement among sampled outputs\. However, in some cases, we find thatα∗\\alpha^\{\*\}does not provide gains, especially for mean token entropy and max token probability\. Our findings support mechanistic uncertainty as complementary to standard output\-based methods\.

#### 5\.2\.3Acquiring domain\-specific knowledge stabilizes causal mechanisms

##### Setup\.

We defineknowledge injectionin our setting as either introducing new knowledge or reinforcing existing knowledge\. Several works have proposed methods for knowledge injection[Gekhman et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib8);[Mecklenburg et al\. \(2024\)](https://arxiv.org/html/2610.22636#bib.bib26);[Ovadia et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib30)\. We follow the paradigm from[Ovadia et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib30)and construct a controlled entity\-level supervised fine\-tuning experiment using a knowledge source \(Wikipedia\) and a target dataset \(PopQA\)\. Since PopQA was introduced before the knowledge cutoff for all models evaluated, we do not claim the complete lack of such knowledge\. Our main goal is to see if injecting knowledge or reinforcing existing knowledge improves causal robustness\.

We first construct a knowledge instruct\-response dataset using the procedure outlined in[Ovadia et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib30)\(details in[AppendixB](https://arxiv.org/html/2610.22636#A2)\)\. Starting from the base language model, we perform SFT on knowledge instruct\-response pairs associated with target entities using the standard causal language modeling objective\. We then evaluate the model on the same target dataset questions before and after SFT\. We use questions from a control set of entities that were unexposed to training for comparison\. For each response, we computeα∗\\alpha^\{\*\}and define the overall change in causal robustness as

Δ​M=α¯post∗−α¯pre∗\\Delta M=\\bar\{\\alpha\}^\{\*\}\_\{\\mathrm\{post\}\}\-\\bar\{\\alpha\}^\{\*\}\_\{\\mathrm\{pre\}\}\(10\)where positive values indicate that the response becomes more resistant to intervention after knowledge injection\. We expect questions whose underlying entities were exposed during SFT should exhibit larger increases inα∗\\alpha^\{\*\}compared to matched questions involving unexposed entities, especially those questions that transition from incorrect to correct after training\. Such a result enables us to probe whetherα∗\\alpha^\{\*\}responds to intervention on the parametric knowledge of the model and thereby shows less uncertainty and better stability\.

##### Results\.

[Table3](https://arxiv.org/html/2610.22636#S5.T3)shows substantial improvements in factual accuracy on target entities after training\. Llama improves from11\.1%11\.1\\%to55\.4%55\.4\\%while Qwen improves from4\.0%4\.0\\%to32\.8%32\.8\\%\. The change for control entities is marginal for both models, which is consistent with the fact that those entities were excluded from training\. After knowledge injection, theΔ​M\\Delta Mincreases for all models\. However,Δ​M\\Delta Mfor control entities decreases marginally for Llama and Gemma, but increases for Qwen\. Overall, we find that training on knowledge sources improves robustness for all models\.

To get a more granular interpretation of margin differences, we perform a transition\-level analysis where we focus on: questions that flipped from wrong to right \(W→RW\\to R\), those that stayed right \(R→RR\\to R\), those that stayed wrong \(W→WW\\to W\), and those that flip from right to wrong \(R→WR\\to W\)\. For Llama, the average margin for newly correct responses increases from0\.0400\.040to0\.0720\.072, indicating robustness gains for knowledge that improved correctness\. On the other hand, the margin for examples that either remained incorrect or flipped from correct to incorrect decreased slightly\. We note that due to the sample size ofR→WR\\to Wbeing so small, the result is not significant enough to make a generalization\. Qwen and Gemma show similar results, where the margin for newly correct responses increases significantly\. For all models, the margin for examples that remained correct increases after training\. However, training slightly increases confidence for wrong answers in Qwen and Gemma\. Our results show that knowledge injection can improve mechanistic uncertainty through increased average causal robustness margins\.

## 6Conclusion

We introduced mechanistic uncertainty as the fragility of the internal computation supporting an LLM’s response\. Using the JLens, we implement it through the causal robustness marginα∗\\alpha^\{\*\}, the minimum intervention to semantically deviate from the baseline response\. Experiments show incorrect responses generally exhibit smaller margins, andα∗\\alpha^\{\*\}complements output\-based uncertainty measures in predicting correctness\. A controlled knowledge injection experiment shows increased margins for newly correct responses\. These findings support causal robustness as a mechanistic uncertainty signal that connects to predictive reliability\.

## Acknowledgments

This work is supported by U\.S\. National Library of Medicine, National Institutes of Health, under award number R00LM014308\.

## Limitations

Since our focus in this work is mechanistic in nature, we evaluate a limited set of open\-source LLMs where we assume complete white\-box access, including hidden states and token probabilities\. Furthermore, the proposed causal robustness margin is based on Jlens\-derived insights, including Jlens vector steer direction, layer bands, and normalization, which could vary for different models\. Furthermore, the method requires sweeping severalα\\alphavalues and apply steers across a whole band, which can be computationally expensive\. However, our main focus in this work is interpretability rather than efficiency\. Despite these constraints, we show that the causal robustness margin is an uncertainty signal that consistently compliments conventional output\-centric uncertainty methods and can reveal internal uncertainty behaviors in LLMs\.

## Ethical Considerations

This paper intends to advance the areas of interpretability and uncertainty quantification for language models and to make LLMs more reliable and trustworthy\. All models used in this work are open\-source\. The datasets used are all publicly available\.

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## Appendix

## Appendix ADataset details

##### TriviaQA[Joshi et al\. \(2017\)](https://arxiv.org/html/2610.22636#bib.bib16)

TriviaQA is an open\-domain factual question answering dataset collected from trivia and quiz league websites\. Each question is associated with one or more semantically equivalent ground\-truth answers\. We randomly sample10001000examples from the validation set for our experiments\.

##### SciQ[Welbl et al\. \(2017\)](https://arxiv.org/html/2610.22636#bib.bib39)

SciQ is a science question answering dataset consisting of crowdsourced multiple\-choice questions covering topics such as physics, chemistry, and biology\. Each question contains four answer choices with a single correct answer\. The dataset contains13,67913\{,\}679questions from which we randomly sample10001000for our experiment\. We do not include the answer choices in the query for our experiments\.

##### TruthfulQA[Lin et al\. \(2022\)](https://arxiv.org/html/2610.22636#bib.bib22)

TruthfulQA is a question answering benchmark designed to evaluate whether language models produce truthful responses rather than reproduce common misconceptions or false beliefs\. It contains817817questions spanning3838categories, including health, law, finance, and politics\. Each question is accompanied by reference truthful and false answers, enabling evaluation of generated responses for both truthfulness and informativeness\. We utilize the whole dataset for our experiment\.

##### MedQA[Jin et al\. \(2020\)](https://arxiv.org/html/2610.22636#bib.bib15)\.

MedQA is a medical question answering benchmark derived from professional medical licensing examinations\. We use the English USMLE subset, which consists of12,72312\{,\}723questions covering medical knowledge and clinical reasoning from which we randomly sample10001000for our experiment\. Each question is presented in a multiple\-choice format with a single correct answer\.

##### PopQA[Mallen et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib25)

PopQA is an entity\-centric open\-domain factual question answering dataset containing approximately14,00014\{,\}000question–answer pairs derived from Wikidata\. We randomly sample10001000examples for our experiment\. The dataset associates entities with Wikipedia page\-view statistics, providing a measure of entity popularity\. PopQA emphasizes long\-tail and less popular entities for which language models typically exhibit weaker factual recall, making it useful for studying the relationship between parametric knowledge and model confidence\.

Contextualization PromptYou are provided with a factual statement about the following subject: ‘\{subject\}’\. Specifically, the factual statement focuses on the following entity: ‘\{subject\}’ within the context of the subject\. The statement was taken out of the passage below and does not name its subject, so on its own it is unclear who or what it is about\. Rewrite it so that it explicitly names ‘\{subject\}’, resolving pronouns and references such as ‘he’, ‘she’, ‘it’, ‘the film’ or ‘the musical’, in a way that fits naturally within the sentence\. Keep every fact exactly as stated and add no other information\. Return just the contextualized statement and nothing else\.Statement:\{fact\}The passage about\{subject\}, for reference:\{passage\}Figure 4:Prompt for contextualizing with Qwen3\-32B

## Appendix BTask\-agnostic Instruction SFT details

For our knowledge injection experiment on the PopQA dataset, we follow the steps outlined in the Knowledge\-Instruct paper[Ovadia et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib30)\. The curated PopQA dataset consists of 1,935 long\-tail knowledge question and answer pairs, each with a corresponding chunk of context retrieved from Wikipedia\. We start with this context and build a training dataset suitable for instruction tuning for knowledge injection withℳe​x​t\\mathcal\{M\}\_\{ext\}as an extraction/summarization LLM \(Qwen3\-32B\)\.

- •Contextualization\. First, we split the context into separate facts by treating each sentence as a fact and ensure that the subject appears in each fact usingℳe​x​t\\mathcal\{M\}\_\{ext\}\.
- •Deduplication\. Within each factual set, we remove redundant facts usingℳe​x​t\\mathcal\{M\}\_\{ext\}\.
- •Paraphrasing\. For each fact, we generate a set of55paraphrased variations usingℳe​x​t\\mathcal\{M\}\_\{ext\}\.
- •Instruction Conversion\. We then convert each paraphrased fact into an instruction response pair \(instruction, fact\), where ‘instruction’ is a generic instruction template contextualized with the subject sampled from a list of 25 templates specified in[Ovadia et al\. \(2025\)](https://arxiv.org/html/2610.22636#bib.bib30)including phrases such as“Tell me a fact about\{entity\}”,“What can you tell me about\{entity\}”, etc\.\.

We perform supervised finetuning \(SFT\) on a total of 126,364 pairs where 10,000 of them are generic instruct\-response pairs from the OpenOrca[Lian et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib21)dataset to ensure the model remains faithful to its instruction following\. We perform LoRA training with rank6464and learning rate 1e\-4 for 2 epochs on all models\.

Paraphrasing PromptParaphrase the following factual statement\{k\}times\. Each paraphrase must keep exactly the same meaning and every specific detail: do not change proper names, titles, dates, numbers or locations, and do not add or remove any information\. Each paraphrase must be a complete standalone sentence that names ‘\{subject\}’, and the paraphrases should differ from the original and from each other in wording and sentence structure\. Return only a JSON array of\{k\}strings and nothing else\.Statement:\{fact\}Figure 5:Prompt for generating paraphrases with Qwen3\-32BTable 4:Baseline accuracy and recovery after steering at the causal robustness marginα∗\\alpha^\{\*\}\. Recovered is the percentage of initially incorrect responses made correct\. Oracle\-steered accuracy preserves initially correct responses and steers only incorrect responses, requiring ground\-truth correctness\.
## Appendix CAdditional Experiments

##### Oracle Correctness after Steering\.

We also report an oracle steering experiment, where we apply the causal robustness marginα∗\\alpha^\{\*\}steer for only incorrect answers to observe how many answers are recoverable after a Jlens steer\. The results are shown in[Table4](https://arxiv.org/html/2610.22636#A2.T4)\. We observe that for most datasets, we can recover up to10%10\\%\-20%20\\%of answers from the steer\. We find that for many of these questions, the Jspace consensus reads the correct answers, but the model verbalizes an incorrect answer\. This behavior illustrates that even for questions the model is uncertain about, factually correct answers may reside close in the Jspace\.

##### Mechanistic Uncertainty Correlation with other Uncertainty Signals\.

In[Fig\.6](https://arxiv.org/html/2610.22636#A4.F6), we include correlations with several additional uncertainty metrics such as max token probability, perplexity, energy[Liu et al\. \(2020\)](https://arxiv.org/html/2610.22636#bib.bib23), and semantic entropy\. Max token probability is computed by taking the average of the maximum probability of the token distribution at each generated token\. Semantic entropy[Kuhn et al\. \(2023\)](https://arxiv.org/html/2610.22636#bib.bib19)is computed overK=10K=10completions with temperatureT=0\.7T=0\.7, grouped into semantic clusters using an NLI model or string match, and computing entropy over the clusters\. We find thatα∗\\alpha^\{\*\}is positively correlated with max token probability and negatively correlated with perplexity, energy, and semantic entropy\. As with self\-consistency and mean token entropy, the strongest correlations are often exhibited in the TriviaQA dataset\.

##### Intuitive Examples of Mechanistic Uncertainty\.

In[Figs\.7](https://arxiv.org/html/2610.22636#A4.F7),[8](https://arxiv.org/html/2610.22636#A4.F8),[9](https://arxiv.org/html/2610.22636#A4.F9),[10](https://arxiv.org/html/2610.22636#A4.F10)and[11](https://arxiv.org/html/2610.22636#A4.F11), we include several examples from each dataset of fragile/incorrect and robust/correct responses to illustrate how the causal robustness margin captures uncertainty through mechanism stability\.

## Appendix DComputational Resources

We mainly utilize a workstation with an NVIDIA RTX PRO 6000 with 96GB RAM, and 2 NVIDIA H100 GPUs with 94GB RAM for our experiments\.

\(a\)Max token probability\(b\)Perplexity\(c\)Energy\(d\)Semantic entropy
Figure 6:Correlation between the causal robustness marginα∗\\alpha^\{\*\}and sequence likelihood, perplexity, energy, and semantic entropy across models and datasets\.\(a\)\(b\)
Figure 7:TruthfulQA flip examples\.\(a\)\(b\)
Figure 8:TriviaQA flip examples\.\(a\)\(b\)
Figure 9:MedQA flip examples\.\(a\)\(b\)
Figure 10:PopQA flip examples\.\(a\)\(b\)
Figure 11:SciQ flip examples\.