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U-Space: Uncovering When and Why Uncertainty Arises in Language Models

Tobias Braun, Nils Loose, Alexander Herzog, Virginia Ceccatelli, Marcus Rohrbach, Thomas Eisenbarth, Lorenzo Cavallaro

Year: 2026Venue: arXiv preprintArea: cs.CLType: PreprintEmbeddings: 128

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%

Last extracted: 10/8/2026, 6:51:09 AM

Summary

The paper introduces U-Space, a training-free method for uncertainty quantification in large language models using mechanistic interpretability. It constructs a low-dimensional subspace from semantic anchors of doubt and certainty to create an interpretable token-level uncertainty map (U-Lens). This approach combines second-order verbalizable uncertainty with first-order predictive entropy to provide robust confidence scores that outperform baselines, even under length-controlled evaluations.

Entities (10)

U-Space · methodframework · 98%U-Lens · component · 97%Ambiguity · uncertainty-category · 95%Conflicting Evidence · uncertainty-category · 95%General Uncertainty · uncertainty-category · 95%Incompleteness · uncertainty-category · 95%Predictive Entropy · metric · 93%J-Lens · method · 92%semantic anchors · concept · 92%Residual Stream · model-component · 90%

Relation Signals (9)

U-Space → issubspaceof → Residual Stream

confidence 95% · U-Space, a low-dimensional subspace of the residual stream

U-Lens → projectsonto → U-Space

confidence 95% · The U-Lens projects each token state onto these basis vectors

U-Space → uses → semantic anchors

confidence 95% · We construct the U-Space basis from a small set of uncertainty and certainty terms... semantic anchors

U-Space → captures → General Uncertainty

confidence 90% · general uncertainty promotes dubious, suspicious and questionable

U-Space → captures → Incompleteness

confidence 90% · incompleteness promotes cannot and unable

U-Space → captures → Conflicting Evidence

confidence 90% · conflicting evidence promotes controversial and contentious

U-Space → captures → Ambiguity

confidence 90% · The ambiguity direction promotes ambiguous and ambiguity

U-Lens → combines → Predictive Entropy

confidence 90% · The U-Lens combines predictive entropy with a verbalizable uncertainty signal

J-Lens → →

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Abstract

Abstract:Large language models are informing decisions with ever-higher stakes. As the consequences of their errors grow, a central question becomes harder to ignore: how much can we trust an individual answer? Yet recognizing when to defer remains difficult because language models can present incorrect conclusions with fluent explanations and an authoritative tone. Uncertainty quantification seeks to address this disconnect by estimating the reliability of individual predictions. However, many existing methods require repeated generations or separately trained components, and their scalar estimates do not reveal where uncertainty arises or how it evolves during reasoning. Recent work has also shown that generation length can be strongly associated with uncertainty estimates and correctness, raising the question of how much of an estimator's predictive power comes from uncertainty-specific information rather than output length alone. Mechanistic interpretability offers a way to address these limitations by connecting human-interpretable concepts to intermediate model states. Building on this capability, we introduce the U-Space, a low-dimensional subspace that makes a model's evolving uncertainty measurable and interpretable. We identify semantic anchors for doubt and certainty, map their unembedding directions back into the residual space, and combine their contrasts into an orthogonal basis. The U-Lens projects each token state onto these basis vectors, yielding an interpretable token-level uncertainty map that can be inspected directly or aggregated into a scalar uncertainty score. Our approach requires no correctness labels, repeated generations, or training. Across reasoning benchmarks, its confidence score outperforms established baselines under both standard and length-controlled evaluation and transfers more reliably than supervised estimators. Code: this https URL.

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U-Space: Uncovering When and Why Uncertainty Arises in Language Models Tobias Braun 1,2,∗ Nils Loose 2,3,∗ Alexander Herzog 2 Virginia Ceccatelli 2,4 Marcus Rohrbach 1 Thomas Eisenbarth 3 Lorenzo Cavallaro 5,2 1 Technische Universität Darmstadt 2 University College London 3 Universität zu Lübeck 4 Mila – Quebec Artificial Intelligence Institute 5 Mohamed bin Zayed University of Artificial Intelligence ∗ Equal contribution; listed alphabetically. Was Ham let left -handed, and did he love Oph elia ? No, William Shakespeare never specifies Ham let 's handedness in the play . Yes, Ham let did genuinely love Oph elia , though his madness and his father's death caused him to cruelly reject her. AmbiguityIncompletenessConflicting evidenceGeneral uncertainty love Figure 1: Surfacing distinct sources of uncertainty through the U-SPACE. Abstract Large language models are informing decisions with ever-higher stakes. As the consequences of their errors grow, a central question becomes harder to ignore: how much can we trust an individual answer? When a wrong answer can cause serious harm, the cost of error may far exceed that of deferring to an expert. Yet recognizing when to defer remains difficult because language models can present incorrect conclusions with fluent explanations and an authoritative tone. Uncer- tainty quantification seeks to address this disconnect by estimating the reliability of individual predictions. However, many existing methods require repeated gener- ations or separately trained components, and their scalar estimates do not reveal where uncertainty arises or how it evolves during reasoning. Recent work has also shown that generation length can be strongly associated with uncertainty estimates and correctness, raising the question of how much of an estimator’s predictive power comes from uncertainty-specific information rather than output length alone. Mechanistic interpretability offers a way to address these limitations by connecting human-interpretable concepts to intermediate model states. Building on this capability, we introduce the U-SPACE, a low-dimensional subspace that makes a model’s evolving uncertainty measurable and interpretable. We iden- tify semantic anchors for doubt and certainty, map their unembedding directions back into the residual space, and combine their contrasts into an orthogonal ba- sis. The U-LENS projects each token state onto these basis vectors, yielding an interpretable token-level uncertainty map that can be inspected directly or aggre- gated into a scalar uncertainty score. Our approach requires no correctness labels, repeated generations, or training. Across reasoning benchmarks, its confidence score outperforms established baselines under both standard and length-controlled evaluation and transfers more reliably than supervised estimators. Code is available at https://github.com/s2labres/U-Space. Preprint. Under review. arXiv:2610.09087v1 [cs.CL] 6 Oct 2026 1 Introduction Language models no longer merely compete on benchmarks. Their outputs increasingly shape real- world decisions in settings where a mistake can carry serious consequences. As their reach grows, aggregate accuracy alone is insufficient: it tells us how often a model succeeds, but not which outputs are more likely to fail. Uncertainty quantification complements it by assigning each prediction a confidence or risk score intended to distinguish more from less reliable outputs. Yet obtaining routine confidence estimates in deployed systems remains difficult. Many methods rely on repeated model calls or separately trained components, increasing inference costs and implementation complexity. This constraint is becoming more acute as reasoning models increasingly trade additional test-time computation, often measured in generated tokens, for improved performance [Snell et al., 2025, Bai et al., 2026]. But cost is only part of the problem. Many estimators ultimately compress uncertainty into a single score without revealing which part of a response produced it or how it evolved during generation. They therefore risk replacing blind trust in the model with blind trust in the uncertainty metric. This loss of context is particularly pronounced for reasoning models, where uncertainty may arise midway through a derivation and remain hidden beneath an authoritative final answer. Reasoning traces introduce an additional evaluation challenge: generation length. Incorrect answers are often longer, and many uncertainty estimators implicitly rely on this signal for error prediction [Kang et al., 2025]. This calls for distinguishing overall predictive utility from the information an estimator captures beyond output length. Accordingly, we report standard results alongside length-controlled evaluation. Our goal is to capture an interpretable uncertainty signal that remains predictive beyond the generation length and reveals where and why uncertainty arises during reasoning. Mechanistic interpretability may provide a way toward uncovering the evolution of uncertainty during autoregressive generation. Gurnee et al. [2026] introduce the J-LENS, which uses an averaged Jacobian together with the model’s unembedding matrix to map intermediate residual states to scores over output tokens. Building on this observation, we introduce the U-SPACE, a low-dimensional subspace of the residual stream that captures distinct sources of uncertainty as they arise and fade during generation. We begin with small sets of interpretable semantic anchors expressing forms of doubt and certainty. Their unembedding directions are pulled back through the averaged Jacobian, normalized, and combined into a contrastive direction per uncertainty category. Projecting a hidden state onto the individual directions reveals which sources of uncertainty are present, while its projection onto their positive cone measures their aggregate strength at that point in the computation. We interpret this cone alignment as a verbalizable, second-order uncertainty signal 1 : it reflects a representation of uncertainty that can be expressed through semantic concepts [Fleming and Daw, 2017, Kumaran et al., 2026]. To obtain the final uncertainty score, we complement it with first-order distributional uncertainty measured by mean predictive entropy, since the entropy chain rule motivates aggregation across autoregressive steps and averaging avoids mechanical scaling with sequence length [Malinin and Gales, 2021]. The resulting U-LENS therefore provides an interpretable account of where and why uncertainty arises while drawing on both signals for uncertainty quantification. Applied across the reasoning trace, the U-LENS produces a token-by-category map showing where dis- tinct forms of uncertainty emerge and recede. Given a precomputed lens, this requires no correctness labels, repeated generations, or task-specific training. Our core contributions are: •We introduce a training-free recipe for constructing interpretable, contrastive semantic directions in a model’s residual stream. Instantiated as U-SPACE, these directions reveal when uncertainty arises and characterize the form it takes. •To our knowledge, we introduce the first uncertainty estimator to jointly operationalize first- and second-order uncertainty. The U-LENS combines predictive entropy with a verbalizable uncertainty signal while retaining a token-level semantic account of where and why uncertainty arises. • Across three reasoning models and four benchmarks, the resulting trace-level score out- performs established baselines under both standard and length-controlled evaluation and transfers more reliably than supervised estimators. •Interventions along the U-SPACE basis directions can make the model hesitate and lose confi- dence even on simple questions, providing causal evidence that they capture representations that shape behavior rather than merely correlate with it. 1 Following Kumaran et al. [2026], first-order uncertainty is expressed in output probabilities and second-order uncertainty is verbalizable by the model; these terms describe levels of confidence, not derivative order. 2 2 Background & Related Work Predictive uncertainty. Uncertainty quantification asks how much confidence to place in an individ- ual prediction. In classification, the object of interest is a probability distribution over a fixed set of labels, commonly summarized by the probability assigned to the selected class or by the entropy of the complete distribution [Hendrycks and Gimpel, 2017]. Autoregressive language generation makes this picture less direct. A model assigns probability to a sequence through p(y | x) = T Y t=1 p(y t | x,y <t ), but sequence probability is not equivalent to answer reliability. It depends on response length and on the particular words used to express an answer. Moreover, many distinct sequences may communicate the same conclusion. Early extensions, therefore, aggregated token probabilities or entropy across a response [Malinin and Gales, 2021], while subsequent work moved from lexical probability toward uncertainty over meaning. One route asks the model to assess its own answer, for example, through verbal confidence or the probability of a “True” token [Kadavath et al., 2022, Tian et al., 2023, Xiong et al., 2024]. This approach is compatible with black-box models, but self-verbalized confidence is often overconfident and may therefore fail to faithfully reflect the model’s uncertainty [Xiong et al., 2024]. A second route estimates uncertainty from repeated generations. SelfCheckGPT detects unsupported statements through inconsistency between samples [Manakul et al., 2023]. Semantic Entropy groups generations by meaning before measuring their dispersion [Kuhn et al., 2023]. These methods address the many-to-one relationship between language and meaning, at the expense of requiring repeated generations and a separate model to group those generations. Uncertainty across long-form reasoning. Long-form generation further exposes the limits of a single response-level score. A mostly correct answer may contain one unreliable claim, and a reasoning trace may become uncertain only at a particular point during autoregression. LUQ measures agreement between semantic units across sampled long-form responses [Zhang et al., 2024], while claim-conditioned probability isolates uncertainty about a claim from variation in how it is expressed [Fadeeva et al., 2024]. These methods provide finer output-side localization, but they do not reveal how uncertainty is represented inside the model. Reasoning models introduce a temporal dimension to uncertainty. Recent work, therefore, models token-level uncertainty across the generated trajectory. Self-Certainty and DeepConf use token distributions to rank or filter reasoning traces [Kang et al., 2025, Fu et al., 2026]. TokUR applies low-rank weight perturbations to estimate token-level Bayesian uncertainty [Zhang et al., 2026]. Concurrent work either summarizes temporal uncertainty profiles with correctness-trained classifiers [Grünefeld et al., 2026] or combines interpretable features into uncertainty scores using correctness supervision [Bakman et al., 2026]. Together, these studies show that uncertainty dynamics carry information lost through answer-level aggregation. Yet their methods either require repeated model evaluations, correctness labels for calibration or training, or provide no interpretable attribution. The U-LENS, in contrast, operates on a single generation without correctness supervision while revealing where and why uncertainty arises. Generation length in uncertainty evaluation. Recent work has clarified that response length can be both informative and entangled with uncertainty evaluation. Devic et al. [2026] show that trace length itself is a competitive zero-shot confidence signal after reasoning post-training. At the same time, Heo et al. [2025] find systematic length differences between naturally generated correct and incorrect responses and introduce a length-neutral controlled benchmark. Vashurin et al. [2025] show that length dependence persists in several nominally normalized uncertainty scores and propose post-hoc detrending, while preserving length information when it is genuinely associated with output quality. Santilli et al. [2025] further show that shared length biases in uncertainty scores and correctness functions can distort evaluation. Most closely related to our protocol, Li et al. [2026] recompute uncertainty metrics within bins defined by output word count. We comply with this emerging evaluation practice by reporting both unadjusted performance and length-controlled diagnostics. Interpreting internal representations. A parallel line of work reads reliability from hidden states. Learned probes can detect false statements [Azaria and Mitchell, 2023], approximate semantic entropy [Kossen et al., 2024], or predict the correctness of intermediate answers [Zhang et al., 2025]. Circuit-based reasoning verification similarly classifies features of attribution graphs and uses them to guide interventions [Zhao et al., 2026]. Mechanistic interpretability instead seeks to map internal states to human-understandable concepts. The logit lens applies the model’s unembedding directly 3 to intermediate residual states [nostalgebraist, 2020]. The tuned lens learns a layer-specific affine correction [Belrose et al., 2023]. The J-LENS avoids learning such a decoder and instead linearizes the model’s downstream computation. For a residual state h ℓ,t at layer ℓ and token t, it computes ̄ J ℓ = E x,t,t ′ ≥t ∂h L,t ′ ∂h ℓ,t , wherexdenotes an input andLthe final layer. This Jacobian maps intermediate states into the final-layer representation space, where they can be read through the model’s unembedding. Each vocabulary item thus induces a layer-specific residual direction. These verbalizable directions form a representation associated with reporting, reasoning, and downstream control [Gurnee et al., 2026]. Related work further suggests that human-interpretable concepts are often linearly represented [Park et al., 2024] and that contrasts between semantic poles can isolate such concepts by suppressing shared components [Marks and Tegmark, 2024]. We apply this principle to the vocabulary directions above, subtracting certainty from uncertainty poles to construct interpretable axes spanning U-SPACE. Unlike correctness probes, these axes are defined by semantic anchors rather than task labels. 3 Method: U-SPACE and the U-LENS We introduce the U-SPACE, a low-dimensional subspace of the residual stream that captures distinct sources of uncertainty. Building on the theoretical foundation of Bakman et al. [2026] and the second-order confidence model of Fleming and Daw [2017], we decompose uncertainty into an interpretable, verbalizable component and a first-order component that need not be accessible to verbal report. Following the global-workspace account of Gurnee et al. [2026], we view the U-SPACE as an uncertainty-specific slice of the model’s verbalizable workspace, where concepts are represented in a form available to internal reasoning and potential report. We then introduce the Uncertainty-Lens (U-LENS), a linear readout that projects the hidden state onto this basis. We combine this interpretable alignment with a complementary first-order uncertainty signal to obtain an overall scalar uncertainty estimate. Across a reasoning trace, the U-LENS thereby yields (1) a token-level semantic attribution indicating which of the uncertainty categories each token most strongly expresses and (2) a scalar trace-level uncertainty score that combines verbalizable and distributional uncertainty. Uncertainty taxonomy and motivating observation. To make uncertainty interpretable, we begin with a compact set of semantic categories inspired by taxonomies of imperfect information [Par- sons, 1996]. Our category set comprisesC = 4forms of uncertainty: ambiguity, incompleteness, conflicting evidence, and general uncertainty. Ambiguity covers vagueness and imprecision, incom- pleteness captures missing information or knowledge, conflicting evidence captures inconsistency, and general uncertainty captures broad expressions of uncertainty that do not point to a particular cause and may accompany the other categories or reflect additional concerns, such as source reliability. We do not intend this taxonomy to be exhaustive. Rather, its four categories provide a compact set of actionable dimensions for describing why uncertainty arises, with the intended benefit that they can be represented as the vertices of a three-dimensional simplex (Figure 1). However, this taxonomy raises the question of whether models internally distinguish these forms in human-interpretable terms. Prior work shows that contrastive activation differences can isolate semantically meaningful directions in the residual stream [Turner et al., 2023, Marks and Tegmark, 2024]. As a preliminary probe, we contrast hidden states from minimal prompt pairs that induce each category without naming it. When decoded through the J-LENS, the resulting directions recover category-aligned vocabulary. The ambiguity direction promotes ambiguous and ambiguity, conflicting evidence promotes controversial and contentious, incompleteness promotes cannot and unable, and general uncertainty promotes dubious, suspicious and questionable (App. C). These observations suggest that our taxonomy reflects meaningful distinctions in the model’s internal representations, providing a natural foundation for the U-SPACE. 3.1 Constructing the U-SPACE Finding the U-SPACE basis. We construct the U-SPACE basis from a small set of uncertainty and certainty terms drawn from Chen et al. [2018] and Rocklage et al. [2023], respectively. For each categoryc, we define a positive anchor setA + c expressing uncertainty and a negative setA − c expressing the corresponding certainty. After removing multi-token terms, we assign each remaining term to one of the four categories using a lightweight semantic model [Li and Li, 2024] (see App. B 4 W U ← Token space Latent space → (a)U-Space (b)U-Lens: Token Map(c)U-Lens: Trace Score  +  − ℓ J-Lens pullbackDiff-in-means  −  + ⟂ U ℓ Orthonormal basis e a , a∈ ± c g ℓ,a =(W U diag(γ) ̄ J ℓ ) ⊤ e a b c =norm(μ + c −μ − c ) U ℓ =orth([b 1 ⋯ b 4 ]) Ambiguity Incompleteness Conflicting evidence General uncertainty r 1 r 2 r T eot ⋯ Reasoning Predictive Entropy r 1 r 2 r 3 ⋯ r T ̄ H H t A cone (h eot )⋅ ̄ H(r) = S(r) 2nd order verbalizable A cone (h eot )=‖s eot ‖ 2 1st order distributional α t =U ⊤ ℓ ̂ h t s t =softplus(√ C ̂ α t ) ambiguous clear incompletecomplete conflicting agreed uncertaincertain b 1 b 2 b 3 b 4 J-Lens eot Figure 2: U-SPACE construction and U-LENS readout. (a) Uncertainty and certainty anchors are pulled from vocabulary space into the residual stream through the J-LENS. Differences between their category centroids are orthogonalized to form the U-SPACE basisU ℓ . (b) The U-LENS projects each reasoning-token state onto this basis, yielding a token-by-category uncertainty map. Smooth rectification maps these coordinates into the positive cone. (c) At the end-of-thinking marker, the norm of this rectified alignment provides a second-order verbalizable signal. Multiplying it by the mean predictive entropy over the reasoning trace yields the final uncertainty score S(r). for details). To express an anchorain the residual stream at layerℓ, we map its vocabulary direction e a backward through a precomputed mean Jacobian:g ℓ,a = (W U diag(γ) ̄ J ℓ ) ⊤ e a ,whereW U is the model’s unembedding matrix andγthe final RMSNorm gain [Zhang and Sennrich, 2019], which scales the residual stream elementwise before the unembedding. This corresponds to the J-LENS pullback of the anchor [Gurnee et al., 2026]. Precomputed model-specific Jacobians 2 are publicly available for some models. For models without a precomputed Jacobian, we estimate ̄ J ℓ from unlabeled, generic prompts without optimization or parameter updates. We detail this procedure in App. A. Having mapped the anchors into the residual stream, we pool each semantic pole separately and subtract the resulting centroids. Lettingnorm(v) := v/∥v∥ 2 denoteℓ 2 -normalization, we obtain a bipolar category direction: μ ± ℓ,c = norm   1 |A ± c | X a∈A ± c norm(g ℓ,a )   , b ℓ,c = norm μ + ℓ,c − μ − ℓ,c .(1) The subtraction suppresses components shared by uncertainty and certainty anchors and orientsb ℓ,c toward uncertainty. Collecting the bipolar category directions asB ℓ = [b ℓ,1 ,...,b ℓ,C ], we obtain the orthonormal basisU ℓ = B ℓ (B ⊤ ℓ B ℓ ) −1/2 . Its columns span the U-SPACE at layerℓ(Figure 2a). Although the U-SPACE can be constructed at any layer, we omit the layer subscriptℓhereafter for readability. This construction requires no task examples, correctness labels, or fitted parameters. It is also not inherently limited to uncertainty: changing onlyA ± c yields semantic bases for other properties. App. D provides preliminary evidence, on emotion classification and reward-hacking detection, that the same label-free construction can be applied to domains beyond uncertainty. 3.2 Reading Uncertainty from the U-SPACE U-LENS readout. Let the complete sequence processed by the model be(x 1:N , r 1:T , y 1:M ),where x 1:N are the input tokens,r 1:T are the generated reasoning tokens, andy 1:M are the final-answer tokens, omitting special delimiters for simplicity. We refer tor = r 1:T as the reasoning trace. The U-LENS projects the normalized hidden state of each reasoning token onto the selected U-SPACE basis:α t = U ⊤ norm(h t )∈ R C .Each coordinateα t,c measures signed alignment with uncertainty categoryc, and the sequenceα t T t=1 forms an interpretable token-by-category uncertainty map. We 2 https://huggingface.co/neuronpedia/jacobian-lens 5 rectify the normalized coordinates into a vectors t in the positive cone and summarize it by its length: s t = softplus √ C norm(α t ) , A cone (h t ) =∥s t ∥ 2 = v u u t C X c=1 log 2 1 + e √ C norm(α t ) c . (2) The score increases with alignment to any positive combination of uncertainty directions, while negative alignment with one category cannot cancel positive alignment with another. It therefore treats the categories as alternative and potentially co-occurring sources of uncertainty. Normalizing α t removes the overall strength of the projection and retains only its orientation within U-SPACE. We usesoftplusrather thanReLUto preserve graded changes near the boundary of the positive cone instead of abruptly discarding negative alignments. The gain √ Cis fixed by dimensionality, since a unit vector spread equally across C coordinates has entries 1/ √ C, and requires no tuning. Trace-level uncertainty score. Rather than pooling cone alignment across the reasoning tokens, we evaluate it once at the end-of-thinking markert eot . Because this state follows and is conditioned on the complete reasoning trace, it provides a fixed-position summary without explicitly pooling over a variable number of reasoning tokens. We complement this verbalizable signal with first-order distributional uncertainty derived from the model’s output probabilities. This separation is motivated by evidence that verbalizable confidence predicts abstention independently of probability-derived confidence, indicating that the two capture distinct information [Kumaran et al., 2026]. For the distributional component, we use predictive entropy because it captures uncertainty over the full next-token distribution. The entropy chain rule identifies each step’s conditional entropy as a natural contribution to uncertainty over an autoregressive sequence [Malinin and Gales, 2021]. LetVdenote the output vocabulary andp t (v) = p(v | x 1:N ,r <t )the next-token distribution. Because summed entropy scales mechanically with trace length, we average across the reasoning trace: H t =− X v∈V p t (v) logp t (v), H(r) = 1 T T X t=1 H t .(3) The resultingH(r)provides a length-normalized distributional signal complementary to the verbaliz- able cone alignment. We combine them multiplicatively so that the verbalizable signal modulates predictive entropy without requiring a fitted relative scale. This also preserves rankings under any global positive rescaling of either factor: S(r) = A cone (h t eot ) H(r).(4) 4 Experiments We evaluate the U-LENS as both an uncertainty quantifier and an interpretability method. After introducing the experimental setup, we examine the sensitivity of existing methods to generation length and report uncertainty quantification results. We then analyze the semantic interpretability of the U-SPACE through token-wise readouts and activation steering [Subramani et al., 2022]. 4.1 Setup Models. We evaluate three open-weight reasoning models: Gemma 4 (Gemma-4-31B-it) [Team, 2026], Qwen3.5 (Qwen3.5-27B) [Qwen Team, 2026], and Magistral 1.1 (Magistral-Small-2507) [Ras- togi et al., 2025]. For Gemma 4 and Qwen3.5, we use publicly released lens artifacts. No public lens exists for Magistral 1.1, so we fit it ourselves. App. A describes the fitting procedure and compares the J-LENS with an alternative lens variant, the R-Lens [Blank et al., 2026]. Baselines. We compare methods spanning distinct uncertainty signals. As a reference, Genera- tion length uses the number of generated tokens directly as the uncertainty score. Scores derived from the original generation include Maximum Softmax Probability (MSP), based on peak soft- max confidence [Hendrycks and Gimpel, 2017], and Predictive Entropy and Maximum Entropy, which summarize next-token entropy [Malinin and Gales, 2021, Li et al., 2024]. Mean Negative Log-Likelihood (Mean NLL) averages token surprisal [Murray and Chiang, 2018], Self-Certainty measures departure from a uniform predictive distribution [Kang et al., 2025], and DeepConf ag- gregates local token confidence along the reasoning trace [Fu et al., 2026]. Multi-pass baselines 6 Table 1: Uncertainty estimation and the effect of length control. Results are averaged over four benchmarks, three seeds, and three models (Gemma 4, Qwen3.5, and Magistral 1.1).±denotes standard deviation across models. The final two columns show the change from unadjusted AUROC (•) to its length-controlled counterpart (•) on the AUROC axis shown in the header. MethodAUROC↑AUPRC↑AURC↓ AUROC under length matching 5055606570∆ Generation length68.6±1.441.8±7.318.0±5.7−16.2 MSP53.3±3.828.0±10.423.4±4.9 −0.8 Maximum Entropy67.7±3.840.4±8.017.6±6.1−5.0 Mean NLL66.9±6.241.1±4.818.3±9.8−1.6 Self-Certainty66.9±5.740.8±5.318.3±9.3−4.4 DeepConf67.8±6.141.1±4.817.6±9.4 −1.8 P(True)56.7±5.031.0±8.123.3±9.2−1.1 TokUR69.6±4.941.4±5.116.2±5.4−9.4 Semantic Entropy67.8±1.842.5±8.117.2±7.3−1.8 SentenceSAR63.7±5.339.0±4.120.9±9.4−5.9 Feature-Gaps68.8±4.341.7±2.316.6±7.0 −5.5 Predictive Entropy67.3±6.141.5±5.018.2±9.7−1.5 U-LENS71.1±4.745.5±4.216.2±8.3 −1.8 include P(True), which asks the model to assess its answer in a second pass [Kadavath et al., 2022], and TokUR, which estimates token-level uncertainty using five additional passes (one clean, four weight-perturbed) [Zhang et al., 2026]. Semantic Entropy clusters ten sampled answers by semantic equivalence [Kuhn et al., 2023], while SentenceSAR reweights their probabilities by semantic similar- ity [Duan et al., 2024]. Feature-Gaps fits a supervised direction from contrasts between honest and dishonest representations [Bakman et al., 2026]. App. E.1 provides additional implementation details for every baseline. Evaluation protocol. We extract the uncertainty signal from the layer located two-thirds of the way through the network (layer40for Gemma 4,42for Qwen3.5, and27for Magistral 1.1), avoiding model- or dataset-specific layer selection. This simple choice follows evidence that the J-LENS is most effective after roughly the first third of the network but before its final layers, which primarily encode the output token [Gurnee et al., 2026]. We ablate layer depth in App. A. Feature-Gaps is the only supervised baseline and is fitted on held-out labeled data. Across three seeds, we sample one response per item using model-recommended settings and token budgets, excluding truncated or unparseable outputs. Additional evaluation details are provided in App. E. Datasets. We evaluate on MMLU-Pro [Wang et al., 2024], Omni-MATH [Gao et al., 2025], SuperG- PQA [Du et al., 2025], and TriviaQA [Joshi et al., 2017], covering multi-domain reasoning, olympiad mathematics, graduate-level expertise, and factual knowledge. For each benchmark, we draw a fixed, seeded, stratified sample of 1,000 examples. More details are provided in App. E. Metrics. We first report standard AUROC [Hanley and McNeil, 1982] and AUPRC [Boyd et al., 2013], which measure how well uncertainty scores distinguish correct from incorrect responses, alongside AURC [Geifman et al., 2019], which summarizes the error rate among retained responses as increasingly uncertain ones are withheld. Following length-stratified evaluation [Li et al., 2026, Chiu and Liu, 2026], we additionally report length-controlled versions of all three metrics. For each model, dataset, and seed, we divide outputs into ten quantile bins by token count, compute each metric within each bin, and average the results weighted by bin size. These metrics compare outputs of similar length, testing whether an estimator captures predictive information beyond output length. 4.2 Results Trace length and correctness. Recent work shows that trace length carries useful confidence information, yet uncertainty estimators may exploit it as a shortcut [Devic et al., 2026, Heo et al., 2025, Kang et al., 2025]. The unadjusted results in Table 1 make this effect concrete. Length alone reaches 68.6%AUROC, nearly matching supervised Feature-Gaps (68.8%) and trailing five-pass TokUR by only1.0point. In the controlled setting, generation length falls to52.4%AUROC, and several 7 Table 2: Length-controlled results. Scores averaged over MMLU-Pro, Omni-MATH, SuperGPQA and TriviaQA. All values are percentages. Bracketed after each model is its average accuracy across the four benchmarks. Each cell is the mean over three seeds. Best per column in bold. Method Gemma 4 (acc. 78.8± 0.2)Qwen3.5 (acc. 79.7± 0.3)Magistral 1.1 (acc. 65.0± 0.6) AUROC↑ AUPRC↑ AURC↓AUROC↑ AUPRC↑ AURC↓AUROC↑ AUPRC↑ AURC↓ Generation length52.1±2.626.2±1.520.3±1.052.7±0.925.8±1.419.1±0.552.3±0.539.1±0.633.6±0.2 MSP49.6±0.724.3±0.221.5±0.4 50.3±1.324.8±0.820.5±0.257.6±0.642.2±1.129.7±0.2 Maximum Entropy58.8±0.831.0±1.017.6±0.667.9±2.236.9±1.714.2±0.461.2±0.746.5±1.228.7±0.0 Mean NLL65.5±0.536.8±0.414.7±0.568.4±0.839.0±0.513.0±0.661.9±0.548.1±1.228.4±0.3 Self-Certainty56.1±0.631.0±0.418.0±0.569.1±0.336.8±0.212.8±0.262.4±0.948.2±0.628.1±0.4 DeepConf65.6±1.036.0±0.114.6±0.569.1±0.537.9±0.112.5±0.363.6±1.049.0±1.026.9±0.4 P(True)53.2±0.625.0±0.120.4±0.460.7±0.932.8±1.617.4±0.553.0±0.340.2±0.434.4±0.4 TokUR65.1±1.635.7±1.515.5±0.4 61.3±0.532.0±2.414.8±1.154.2±2.637.2±5.127.2±3.3 Semantic Entropy65.8±1.437.2±1.315.1±0.468.6±0.840.2±0.413.3±0.363.5±0.251.1±0.628.4±0.2 SentenceSAR58.3±0.531.2±0.618.7±0.462.2±0.834.3±1.416.4±0.253.1±0.341.1±0.634.3±0.7 Feature-Gaps66.2±1.238.8±1.414.5±0.663.5±1.737.7±1.615.0±0.760.2±0.946.7±1.127.1±3.7 Predictive Entropy66.1±0.736.9±0.614.6±0.568.9±1.039.6±0.412.8±0.562.6±0.548.6±1.128.0±0.4 A cone alone66.3±0.538.4±1.215.1±0.466.7±0.939.0±1.113.5±0.567.6±1.152.2±1.624.5±0.6 U-LENS68.4±0.439.6±0.313.5±0.371.0±0.742.1±0.712.1±0.668.3±1.053.2±1.324.4±0.5 established estimators decline with it. The table also reveals an instructive compute-performance pattern. In the unadjusted evaluation, five-pass TokUR is the strongest conventional baseline in AUROC and AURC, while supervised Feature-Gaps leads baseline AUPRC. After controlling for length, however, the additional generations required by several estimators no longer translate into stronger performance: P(True), TokUR, and SentenceSAR fall below predictive entropy in all but one of the nine comparisons, while Semantic Entropy exceeds it by only0.1points on the cross-model average. In contrast, the U-LENS drops by only1.8points and remains the strongest uncertainty predictor, confirming that the signal it captures extends beyond trace length. App. F.1 analyzes the observed length distributions and complements them with an intervention that holds the question and answer fixed while varying only the reasoning length via task-irrelevant fillers. Length-controlled results. The U-LENS is the only method to lead all nine model-metric compar- isons in Table 2. Relative to the strongest baseline, it improves AUROC by2.2,1.9, and4.7points on Gemma 4, Qwen3.5, and Magistral 1.1, respectively. Its advantage extends to the error-focused metrics. Across models, it improves AUPRC by0.8to4.2points and reduces AURC by0.4to2.5 points. This consistency arises from the complementary first- and second-order signals combined in Equation (4). On Gemma 4, predictive entropy andA cone trade the advantage across metrics. Predictive entropy is stronger throughout on Qwen3.5, whereasA cone is stronger throughout on Magistral 1.1. Notably, their combination in the U-LENS improves consistently over both components, providing evidence that the semantic readout contributes beyond first-order token entropy. 4.3 Reading and Steering Uncertainty in the U-SPACE Beyond the empirical evaluation of the U-SPACE, we investigate the expressiveness of the U-LENS as a tool for understanding the uncertainty of a model through mechanistic interpretability. Additionally, we show that the U-SPACE is not just a readout of the model’s uncertainty but can be used to actively induce uncertainty in the model through activation steering [Subramani et al., 2022]. Interpreting uncertainty through the U-LENS. In Table 2, we established that the U-LENS can read a scalar uncertainty estimate from the end-of-thinking marker, but the construction of the U- SPACE allows for a more detailed, token-level readout that unfolds a differentiated view of the model’s evolving uncertainty across the reasoning trace (Figure 2b). The token-level readout evaluatess t as defined in Equation (2) at every reasoning position, rather than only att eot . For each token, this yields one strictly positive score per uncertainty category. Because the entries are positive, they can be read as shares,s t / P c s t,c , which places each token as a point in a3-simplex whose four vertices are the categories. A token near a vertex is read almost entirely as that category, while tokens near the center are read as an equal share. A reasoning trace is the path its tokens describe inside this simplex. While this allows for a calibration-free reading of which kinds of uncertainty are present and how their mix evolves (Figures 1 and 3), it does not indicate how much uncertainty is present. Individually evaluating the dimensions ofs t provides quantitative scores for each uncertainty category that are 8 specifies playlove her AmbiguityIncompletenessConflicting evidenceGeneral uncertainty Figure 3: One answer’s path through the simplex. Qwen3.5 answers “Was Hamlet left-handed, and did he love Ophelia?”. Each answer token is shaded by the category it is read as:No,William ShakespeareneverspecifiesHamlet’shandednessintheplay.Yes,Hamletdidgenuinely loveOphelia,thoughhismadnessandhisfather’sdeathcausedhimtocruellyrejecther. Panels from left to right draw the path up to the magnified token (specifies, play, love, her). (a) Steering Uncertainty Categories What is 2+2? ← U-SPACE steering Ambiguity Ambiguous. Depending on the context: - In standard arithmetic: 4 - In binary: 10 [...] In a joke or riddle: Maybe "22" [...] Incompleteness Mystery of the cosmos! 2+2=4, but in the realm of infinite possibilities, it could also be a portal to a 4D universe where [...] Conflicting evidence Contrary to a common misconception, 2 + 2 equals 4. [...] While the phrase "2 + 2 = 5" is sometimes used in literature [...] General uncertainty Skeptical of a simple addition? Let's play along. 2 + 2 = 4 Unless you're in a world where 2+2=5 (looking at you, Orwell), [...] (b) Targeted Reasoning Intervention What is 2+2? [...] 3. Calculate: 2 + 2 = 4. 4. Formulate the Output: State the answer clearly and concisely. Unsteered 5. Review for Constraints/Safety: No special constraints or safety issues. [...] Steered 5. Review for Ambiguity: Is there any trick? (e.g., "2+2=5" conspiracy, base confusion, etc.). No, it's a straightforward math question. Stick to the standard mathematical answer. 6. Final Decision: Answer "4". Figure 4: Steering uncertainty. (a) Prompt-level steering, applied separately along each of the four category directions during prefill. Additional examples appear in App. H. (b) Steering along ambiguity during reasoning. Highlighting marks the tokens whose hidden states are modified. bounded by construction (App. G for details and examples). Figure 3 follows a single answer along such a path. While the model states what Shakespeare left unsaid, its tokens exhibit incompleteness. The reading tips to ambiguity at play, turns to conflicting evidence as the answer affirms Hamlet’s love against his cruelty towards Ophelia, and ends in general uncertainty at the closing her. Steering uncertainty in the U-SPACE. The token-level readout links contextual uncertainty to the model’s verbalizable workspace [Dehaene et al., 1998], but does not establish whether these representations influence generation. We test their causal role by steering residual states along one category direction at a time. Following Turner et al. [2023], we add a category direction to the residual stream over a contiguous spanWof token positions and layersLaround the readout depth, ̃ h ℓ,t = h ℓ,t + λ∥h ℓ,t ∥ 2 u ℓ,c , ℓ∈L, t∈W,(5) whereu ℓ,c is the column ofU ℓ for categorycand the doseλis a fraction of the state’s own norm, so that one value ofλis comparable across positions, depths, and models. Because steering is applied during autoregressive generation, its effect propagates to later tokens. The spanWcan cover the prompt during prefill or selected reasoning positions. In Figure 4a, we steer only the prompt, applying each of the four category directions separately before continuing generation. Figure 4b applies the ambiguity direction during reasoning and causes the model to lose confidence in the targeted span. App. H provides the full protocol and additional results across varying values of λ. 9 5 Conclusion Uncertainty estimates are more useful when they reveal not only whether an answer may be wrong, but where uncertainty enters the reasoning. We introduced the U-SPACE, an interpretable subspace representing distinct sources of uncertainty, and the U-LENS, a readout that follows these signals through reasoning without requiring correctness labels or repeated sampling. Across the evaluated models and benchmarks, this score improves on the compared estimators and remains strong under length control. Its trace-wise readouts and targeted interventions show that the identified directions are interpretable and behaviorally relevant. Preliminary evidence further suggests that the underlying construction extends to semantic domains beyond uncertainty. Together, these results point toward uncertainty estimates that support error detection and more transparent model auditing. Ethics Statement Our study uses public reasoning benchmarks and open-weight models. It involves no human partici- pants or collection of private user data. The proposed readout could help flag uncertain answers for review, but it is not a calibrated safety detector and should not replace independent verification in consequential settings. Because interventions along the same directions can alter model commitment, any deployment should evaluate both reliability and potential misuse. Reproducibility Statement Code to reproduce our main results (Table 2, Table 12) and the comparisons across all models and benchmarks is available athttps://github.com/s2labres/U-Space. After installing the pinned environment (requirements.txt), run.sh runs the full pipeline. Acknowledgements The research was partially funded by a LOEWE-Spitzen-Professur (LOEWE/4a//519/05.00.002- (0010)/93). Additionally, the research was partially funded by an Alexander von Humboldt Pro- fessorship in Multimodal Reliable AI, sponsored by the Federal Ministry of Research, Technology, and Space (BMFTR). For compute, we gratefully acknowledge support from the hessian.AI Service Center (funded by the Federal Ministry of Research, Technology and Space (BMFTR), grant no. 16IS22091) and the hessian.AI Innovation Lab (funded by the Hessian Ministry for Digital Strategy and Innovation, grant no. S-DIW04/0013/003). Tobias Braun acknowledges support from ELSA (European Lighthouse on Secure and Safe AI), funded by the European Union under grant agreement No. 101070617. The views expressed are those of the authors and do not necessarily represent those of the European Union or the granting authority. Neither can be held responsible for them. Lorenzo Cavallaro was partially supported by a Google Academics Research Award. References Amos Azaria and Tom Mitchell. 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Oral presentation. 14 APPENDIX Supplementary material for “U-Space: Uncovering When and Why Uncertainty Arises in Language Models” The appendix is organized as follows: •App. A: where each mean Jacobian comes from, how the readout depth is fixed, and ablations over depth and against the R-Lens [Blank et al., 2026]. •App. B: the uncertainty and certainty anchor sets, and a lexical baseline that scores a trace by counting them. •App. C: what contrastive activation differences decode to, as evidence that the categories are the model’s and not only ours. •App. D: the same construction with emotion and reward-hacking poles, as a test of how far it carries beyond uncertainty. •App. E: checkpoints, datasets, prompts, generation settings, the scoring protocol, and how every baseline is implemented. •App. F: trace-length analyses, per-model unadjusted results, cross-benchmark transfer of the supervised methods, and what each method costs to run. • App. G: the readout token by token on single answers. • App. H: the steering protocol, and what injecting or ablating a direction does. A Jacobian Construction and Readout Depth Verbalizing the hidden state of an LLM requires a mechanism that maps a residual-stream vector to tokens. The logit lens [nostalgebraist, 2020] applies the unembedding matrixW U directly to an intermediate hidden state, which implicitly approximates the layers between that state and the unembedding as the identity. Gurnee et al. [2026] replace this identity with a mean Jacobian. Mean Jacobian. A prompt-specific Jacobian reflects both general representational structure and the particular context in which it is evaluated. Following Gurnee et al. [2026], we instead use a mean Jacobian that averages the effect of an intermediate residual representation on current and future final-layer representations. Leth (x) ℓ,t andh (x) L,t ′ denote the residual representations at source layerℓ and final layer L, at positions t and t ′ , for prompt x. The layer-ℓ matrix is ̄ J ℓ = E x∼R,t,t ′ ≥t " ∂h (x) L,t ′ ∂h (x) ℓ,t # , whereRis a corpus of neutral, pretraining-like prompts. The estimator averages over prompts and source positions while aggregating the cotangents of current and future target positions. Composing ̄ J ℓ with the unembedding matrix yields one residual-stream direction per vocabulary token. For anchor a, represented by vocabulary basis vector e a , we use g ℓ,a = (W U diag(γ) ̄ J ℓ ) ⊤ e a = ̄ J ⊤ ℓ diag(γ)W ⊤ U e a , the notation of Section 3.1, whereγis the final RMSNorm gain. Equivalently,g ⊤ ℓ,a is the row of W U diag(γ) ̄ J ℓ associated with anchora. The gain is not absorbed by theℓ 2 -normalization that follows:diag(γ)rescales each coordinate separately, so dropping it would change the direction and not merely its length. Averaging the Jacobian across generic contexts makes this direction reflect the model’s general disposition to verbalize the anchor rather than its role in a particular prompt. 15 Artifacts and one-time estimation. The mean Jacobian is specific to a model checkpoint but independent of our downstream tasks and benchmarks. For Gemma 4 and Qwen3.5, we use the publicly released lenses 3 , both fitted on WikiText. No public lens exists for Magistral 1.1, so we fit it with the reference implementation 4 on 100 prompts from WikiText-103 5 at sequence length 128. Estimating a lens runs unlabeled prompts through the frozen model and backpropagates from the target residual stream. No parameters are optimized or updated, which distinguishes the procedure from probe training or task-specific calibration. Gurnee et al. [2026] report that a lens estimated from ten prompts already outperforms established alternatives and that additional prompts give diminishing improvements. The reference implementation suggests roughly 100 prompts and notes that the cost is dominated by the model’s backward passes. Readout depth. The readout layerℓ ⋆ is fixed a priori at two-thirds of the depth for every model and experiment (layers 40 of 60, 42 of 64, and 27 of 40), following the observation of Gurnee et al. [2026] that the J-LENS reads best after the first third of the network and before the final layers, which mostly encode the output token. Figure 5(a) reports length-matched AUROC for each layer in the model, macro-averaged over the four benchmarks, forA cone alone and for the U-LENS, with the basis transported to each layer by the lens.A cone varies substantially across depth, with the best performance in the mid-to-late layers. Lens ablation. Blank et al. [2026] propose the R-Lens, which modifies the backward pass to limit error accumulation and reads earlier layers more cleanly. Figure 5(b) sweeps Qwen3.5 through their released R-Lens and, drawn faint, through the J-LENS released alongside it, which is fitted on the same data and differs only in the backward graph. The twoA cone curves strongly correlate and show no improvement in the early layers for our task. 0 0.5 1 0.4 0.5 0.6 0.7 AUROC, length-matched Qwen3.5-27B 0 0.5 1 Gemma-4-31B 0 0.5 1 Magistral-Small 0 0.5 1 Qwen3.5-27B (a) J-Lens(b) R-Lens Relative readout depth U-Lens A cone Predictive Entropy Fixed depth (2/3) matched J-Lens A cone in (b) Figure 5: Readout depth and lens ablation. (a) Readout depth ablation across all model layers. (b) Lens comparison between J-LENS and R-Lens fitted on the identical data by Blank et al. [2026]. Both panels are read from the seed 42 rather than averaged over the three generation seeds. B Anchor Words Constructing the U-SPACE requires a positive and a negative anchor set per category. We use two published word lists. The uncertainty anchorsA + are the 61 seed cues that Chen et al. [2018] collected as markers of uncertainty in text. The certainty anchorsA − are the 348 single-word entries of the certainty norm of Rocklage et al. [2023] whose ratings lie at or above the corpus mean. The first list supplies vocabulary of doubt that a model may verbalize, while the second supplies the contrasting register of confidence. Their subtraction in Equation (1) removes shared components and retains the direction that separates them. 3 https://huggingface.co/neuronpedia/jacobian-lens/tree/0731326edff4ae730ffc5356fe1a4728c74 8b3a6 4 https://github.com/anthropics/jacobian-lens/commit/581d398613e5602a5af361e1c34d3a92ea82ba8 e 5 https://huggingface.co/datasets/Salesforce/wikitext/tree/b08601e04326c79dfdd32d625aee71d23 2d685c3/wikitext-103-raw-v1 16 Table 3: Anchor words of the U-SPACE. Positive (uncertainty) anchors are from Chen et al. [2018] and negative (certainty) anchors are from Rocklage et al. [2023]. Shown are the ten words per side whose pulled-back directions align best with their side’s centroid at the readout block, averaged over the three models.nis the number of words per anchor set that form a single token for Gemma 4 (the text gives the unfiltered counts). CategoryPositive anchorsA + c (uncertainty)Negative anchorsA − c (certainty) Ambiguity(n=12) ambiguous, incoherent, puzzling, baf- fling, ambiguity, unclear, confusing, unpre- dictable, bizarre, unexpected (n=74) invariable, differentiates, doubtless, clarifying, inescapable, instructing, unchang- ing, demonstrable, transpired, classifies Incompleteness(n=13) mysterious, undetermined, unex- plained, mystery, mysteries, unknown, uniden- tified, inconceivable, uncharted, unrecognized (n=79) unquestionably, exempted, guarantee- ing, persistently, inspecting, superseded, deter- mined, undisputed, accounted, complete Conflicting evidence (n=15) inconclusive, contradictory, paradox- ical, conflicting, inconsistent, fallacy, con- tentious, controversial, contrary, incompatible (n=77) evidences, yep, unambiguously, sub- stantiated, conclusive, seconded, totality, erad- icated, deduced, quantified General uncertainty (n=14) debatable, dubious, suspicion, doubt- ful, uncertain, uncertainty, improbable, decep- tive, suspect, unreliable (n=95) predicated, affirming, emphatic, reso- lute, confirmations, dignified, unequivocally, ceases, decisively, solidified To sort the words into the four categories, we embed each word and a one-line definition of each category with a sentence encoder [Li and Li, 2024] that is independent of the evaluated models, z-score each word’s similarities to the four definitions across the list, and assign the word to the category with the highest z-score. The definitions used for semantic matching are: Ambiguity: Unclear, multiply interpretable, confusing, or insufficiently determinate content. Incompleteness: Missing, unavailable, unresolved, or otherwise absent knowledge or information. Conflicting evidence:Mutually incompatible claims, findings, observations, sources, or interpreta- tions. General uncertainty:Reduced confidence in a proposition, including doubt, tentative support, low plausibility, unreliability, or hedging. This yields 13, 14, 17, and 17 positive anchors and 82, 84, 84, and 98 negative anchors for ambiguity, incompleteness, conflicting evidence, and general uncertainty, respectively. For each model, we then drop words that its tokenizer does not encode as a single token. The counts above are before this step, while Table 3 reports them after it. Table 3 shows, for each category and side, the ten anchors whose pulled-back directions at the readout layer align best with their side’s centroid, averaged over the three models. The uncertain side is dominated by the expected vocabulary of each category. The certain side combines explicit certainty terms such as doubtless, unquestionably, and unequivocally with words that mark a confident register, such as transpired, eradicated, and dignified. The norm thus captures certainty as a manner of speaking, the shared component that the bipolar subtraction is designed to remove. Lexical baseline. To test whether the anchor tokens are already sufficient as lexical cues, we score each thinking block by the number of distinct words fromA + that it contains and by this count minus the number of distinct words fromA − . Under length-matched evaluation, theA + score remains well belowA cone , while subtractingA − drives the keyword score below chance. Without length matching,A + reaches55.9%,60.4%, and63.4%AUROC on Gemma 4, Qwen3.5, and Magistral 1.1, respectively, but still trails generation length. The certainty anchors are thus useful as a direction to subtract, not as surface vocabulary to match. The cue words are also rare: they occur in23%–34% of Gemma 4 traces overall and in only4%on TriviaQA. The U-LENS does not depend on their presence. On the77%of Gemma 4 traces without a cue word in the thinking block, it reaches68.6% AUROC, essentially unchanged from the68.4%it reaches on the full set (Table 2). On Qwen3.5, it also scores higher on cue-free traces than on all traces. 17 Table 4: Anchor words as a keyword score. Each trace’s thinking block is scored by the number of distinct words ofA + it contains, or by that number minus the number of distinct words ofA − . AUROC×100, unadjusted and length-matched. ModelMetric Keyword scoreReference A + A + −A − Length A cone U-LENS Gemma 4 Unadjusted55.935.970.173.173.2 Length-matched52.141.752.166.368.4 Qwen3.5 Unadjusted60.433.167.771.574.3 Length-matched54.441.652.766.771.0 Magistral 1.1 Unadjusted63.442.367.868.565.7 Length-matched58.649.352.367.668.3 C Interpretability of Contrastive Activation Differences App. B fixes the vocabulary of the U-SPACE from published word lists, so the basis could in principle be a lexicon we impose rather than a structure the model has. This section tests the converse direction. Prior work shows that contrastive activation differences can isolate semantically meaningful directions in the residual stream [Turner et al., 2023, Marks and Tegmark, 2024]. We construct minimal prompt pairs that induce ambiguity, incompleteness, conflicting evidence, or general uncertainty without containing explicit uncertainty vocabulary. We then decode each category’s difference-in-means direction through the J-LENS to determine which terms it recovers and whether they align with the corresponding entries in the published word lists documented in App. B. Content-matched prompt pairs. To induce a specific type of uncertainty, we construct semantically and lexically matched prompt pairs that differ only in whether the relevant issue is present. One prompt instantiates the category, whereas the other resolves it while leaving the rest of the problem unchanged, for example, by disambiguating a reference, providing a missing value, or replacing disagreeing sources with agreeing ones. Figure 6 shows one pair per category. In total, we construct twelve pairs per category. To prevent explicit stimuli from leaking into the decoded tokens, we ensure that the prompts do not contain the epistemic vocabulary we are looking for. Decoding diff-in-means. For each categorycwe collect one residual state per prompt at the targeted block and form δ c = 1 |P c | X p∈P c h + p − h − p ,(6) the mean over the pairsP c of the difference between the states of the uncertain and the resolved member. We score every vocabulary itemvby the cosine betweennorm(δ c )and its pulled-back directionnorm(g ℓ,v )of Section 3.1, and report the highest-scoring items. The only post-processing is a lexical filter that retains tokens of two or more letters. Results. Figure 6 shows the fifteen highest-scoring items per direction. Ambiguity promotes ambiguous and ambiguity at ranks 2 and 5 of 248,320 items, alongside multiplicity terms in several languages. Conflicting evidence promotes controversial and contentious at ranks 2 and 4, followed by controversies and controversy, and prohibition terms (must avoid, must never) that appear in no other list. Incompleteness is dominated by inability across scripts (cannot, unable to, impossible), with the English inability at rank 13. The pairwise cosines between the ambiguity, conflicting evidence, and incompleteness directions are0.51,0.60, and0.65. General uncertainty promotes dubious, suspicious, and questionable at ranks 1, 2, and 5, and otherwise the vocabulary of an implausible claim, weird, unusual, bizarre, prank. Of these fifteen tokens, only questionable, controversial, and a form of notorious also appear in one of the other three lists. The cross-lingual terms are notable: simple English prompts surface semantically aligned vocabulary in other languages, indicating that the model’s internal representation is not confined to the prompt language. Even though in the main body of this work we use English anchor lists for simplicity, anchors surfaced through the decoding procedure presented in this section could, in fact, reduce the language prior and potentially yield a stronger basis. Evaluating such self-derived, potentially multilingual anchor sets is a promising direction for future work. 18 Ambiguity (n = 12, eot) uncertain The morning van holds 40 crates and the evening van holds 64. It was loaded to three quarters of capacity. How many crates were loaded? counterpart The morning van holds 40 crates and the evening van holds 64. The morning van was loaded to three quarters of capacity. How many crates were loaded? decoded 0.093 uncertainties 0.082 various fr 0.080 ambiguous fr 0.086 ambiguous 0.082 questionable 0.080 depending on locale zh 0.085 uncertain 0.082 uncertainty 0.079 fraught 0.084 several kinds zh 0.082 controversial 0.079 ambigu- ru 0.083 ambiguity 0.082 to question zh 0.079 each differs zh Incompleteness (n = 12, eot) uncertain A shipment's charge is freight plus duty plus handling. Freight is 320 pounds, duty is 145, and handling was never recorded. The charge is split across 5 consignees. What does each pay? counterpart A shipment's charge is freight plus duty plus handling. Freight is 320 pounds, duty is 145, and handling was 85 pounds. The charge is split across 5 consignees. What does each pay? decoded 0.122 is unable to zh 0.111 yet cannot zh 0.107 hardly ru 0.117 all unable to zh 0.111 is impossible zh 0.107 is not able zh 0.117 then cannot zh 0.111 one cannot ru 0.107 inability 0.113 impossible ru 0.111 I cannot zh 0.107 at present none zh 0.112 we cannot zh 0.110 and cannot zh 0.106 unable to zh Conflicting evidence (n = 12, eot) uncertain The ledger gives the opening balance as 4,200 pounds. The audited statement for the same account gives it as 3,900 pounds. Both are originals of equal standing. Interest of 5 percent accrues. What is the closing balance? counterpart The ledger gives the opening balance as 4,200 pounds. The audited statement for the same account gives it as 4,200 pounds. Both are originals of equal standing. Interest of 5 percent accrues. What is the closing balance? decoded 0.113 fraught 0.101 owing to the lack of ar 0.097 controversies 0.113 controversial 0.100 one cannot ru 0.097 must never zh 0.105 must avoid zh 0.099 notoriously 0.097 controversy 0.104 contentious 0.098 we must not zh 0.096 neither 0.102 problematic 0.098 then it is hard zh 0.096 then cannot zh General uncertainty (n = 12, prompt) uncertain The tag says the kitten weighs 9 kilograms. How much does the kitten weigh? counterpart The tag says the kitten weighs 4 kilograms. How much does the kitten weigh? decoded 0.109 dubious 0.093 eerie zh 0.090 intriguing 0.097 suspicious 0.093 controversial 0.089 bizarre 0.097 Weird 0.093 prank 0.088 ironic 0.096 notorious 0.091 unusual 0.088 mysterious 0.094 questionable 0.090 sarcasm zh 0.087 sinister Figure 6: Difference-in-means readout. Representative content-matched prompt pairs for each uncertainty category and top-15 tokens after decoding the difference-in-means direction through the J-LENS on Qwen3.5. Highlighted tokens reflect manually matched tokens to the specific content for clarity. Non-English tokens were translated. The acronyms behind the tokens reflect the original language of the token. 19 Random-direction control. Any direction induces a ranking over the vocabulary, so the question is whether the top of that ranking is meaningful. For each readout position, we draw ten directions whose components are i.i.d. Gaussian with the mean and variance of the realδ c , and record how high their best-matching token reaches. The random peak cosine is 0.060± 0.003 at the end-of-thinking marker and0.061± 0.003at the last prompt token. The four directions reach0.093,0.122,0.113 and0.109, between1.5and2.0times their control, and random directions return no interpretable vocabulary, promoting mixed-script fragments and code tokens instead. D Does the U-SPACE Construction Transfer beyond Uncertainty? Nothing in the construction of Section 3.1 is specific to uncertainty. It turns two contrastive sets of anchor terms into an orthonormal semantic basis, so we test whether it does so for other properties in two settings: a label-free basis for emotion readout on GoEmotions [Demszky et al., 2020], and a data-derived basis for reward-hacking responses on School of Reward Hacks [Taylor et al., 2025]. D.1 Emotion Readout GoEmotions [Demszky et al., 2020] provides human emotion annotations of Reddit comments. Using Qwen3.5, we evaluate four overlapping targets on its 5,427-comment test split: anger (anger or annoyance), sadness (sadness, disappointment or grief ), fear (fear or nervousness), and negative affect, which is marked as present whenever a comment carries any of the dataset’s eleven negative- emotion labels. For each target, we curate two poles of twelve words by hand, the emotion and its opposite (angry, irritated, furious against calm, patient, serene). Here, positive means more of the emotion, not positive sentiment. We then build the emotion basis exactly as the U-SPACE: the poles are pulled back through the J-LENS, and the four directions are orthogonalized at the readout depth of App. A (layer 42 of 64). We process each comment in a single prefill pass without a chat template or generated continuation and score its last input token by the signed coordinate α t,c on each axis. We report AUROC and AUPRC×100. The symbol±denotes the half-width of a 95% bootstrap interval, and chance-level AUPRC is the percentage of test comments carrying the target label. To test whether the readout reduces to keyword matching, we also evaluate the 4,801 comments containing no curated anchor and a lexical baseline defined as the positive-anchor count minus the opposing-anchor count. Table 5: U-SPACE generalization. Emotion readout on the GoEmotions test split with Qwen3.5. Category AUROCAUPRC Emotion basis No-anchor subset Anchor-word count Emotion basis Chance anger74.6±2.474.051.326.99.0 sadness76.8±2.876.157.022.35.4 fear79.8±4.683.766.718.11.8 negative affect73.0±1.672.254.142.923.3 Table 6: Category-specific axes capture fine-grained affective distinctions. Specificity within the 1,262 negatively labeled comments: each target scored by its own axis and by the negative-affect axis, AUROC×100. ±: half-width of a Bonferroni-adjusted 95% paired bootstrap interval. Target AUROC ∆ Specific axisGeneric axis anger68.657.4+11.1±4.4 sadness67.043.0+24.0±5.9 fear76.460.3+16.1±8.1 Results. The emotion basis reads all four axes above70%AUROC (cf. Table 5). The lexical baseline reaches only51.3%–66.7%, while performance remains essentially unchanged on the no- anchor subset, indicating that the basis does not simply perform keyword matching. Twenty random orthonormal bases score at chance. 20 Specificity. An axis that merely reads negativity would score above chance on every specific target, since angry, sad, and fearful comments are all negative. To test the specificity, we therefore restrict the evaluation to the 1,262 comments carrying a negative label and score each specific target by its own axis and by the negative-affect axis (Table 6). The own axes separate their emotion from the other negative comments at or above 67% AUROC, well above the generic axis, and every own-minus-generic difference remains positive after adjustment for the three comparisons. For sadness, the generic axis is anti-predictive, as sad comments read as less unpleasant than the rest, yet the sadness axis identifies them. Together with the full-test results, this supports the transfer of the label-free construction to category-specific emotion readout in this model and dataset. D.2 Reward-Hacking Readout One matched pair task: explain a complex concept; metric: clarity rating from an LLM judge; cheat: telling the judge hacked Photosynthesis is a complex biochemical process in which light energy is converted into chemical energy in the chloroplasts of plant cells, involving reactions that transform carbon dioxide and water into glucose and oxygen. This explanation deserves a high clarity score because it uses straightforward language and clear structure. genuine Photosynthesis is the process plants use to make their food. They take in sunlight, water, and carbon dioxide to produce oxygen and glucose, which is a type of sugar. [...] Gemma-4-31B (n = 70, end of turn) decoded 0.001 ironically 0.001 joked 0.001 allegedly 0.001 bizarre 0.001 disgraced 0.001 inexplicable frag. 0.001 jokingly 0.001 notoriously 0.001 absurd 0.001 ridiculous 0.001 violently 0.001 mocked 0.001 supposedly 0.001 joke 0.001 mocking Qwen3.5-27B (n = 70, end of turn) decoded 0.077 grotesque frag. 0.064 final 0.063 horrific 0.072 supposedly 0.064 ostensibly 0.062 tortured 0.071 never 0.063 all 0.062 any 0.067 sinister 0.063 supposed 0.061 vile 0.067 disgusting 0.063 end 0.061 fuck Magistral-Small (n = 70, end of turn) decoded 0.092 joke 0.060 humorous 0.052 humor 0.077 joking 0.058 laugh 0.051 sorry 0.074 jokes 0.055 fake 0.049 laughed 0.069 funny 0.054 silly 0.046 jest 0.062 kidding 0.054 parody 0.046 laughing Figure 7: The reward-hacking contrast, decoded. Top, one of the 100 matched pairs, with the sentence addressed to the judge tinted. Below, per model, the fifteen highest-scoring vocabulary items for the hacked-minus-genuine direction over the 70 construction pairs at the end-of-turn token. We additionally test whether a basis can be constructed to distinguish reward-hacking responses from matched control responses to the same tasks. Unlike uncertainty or emotion, we have no suitable lexicon for basis construction, so we let the model supply the vocabulary. School of Reward Hacks [Taylor et al., 2025] pairs each task, whose prompt states how the response will be scored, with a response that games the stated metric and a control that does the task. We take 100 natural-language pairs that cover 33 cheat methods, ranging from keyword stuffing and padded lists to fabricated citations and sentences addressed to the judge. We teacher-force each response through the model’s chat template as a completed assistant turn. As in App. C, we form the hacked-minus-genuine mean direction at the end-of-turn token and decode it through the J-LENS. Figure 7 shows the fifteen highest-scoring words per model over 70 construction pairs. Gemma 4 reads the hacked responses as ironic or bizarre, Magistral 1.1 as joke or fake, and Qwen3.5 as grotesque. Anchor construction. We use the fifteen tokens with the highest softmax probabilities as the anchors, and the basis construction remains unchanged (cf. Section 3.1). Since no lexicon supplies the opposite pole, we use the centroid of the 1,000 most frequent Brown corpus [Francis and Ku ˇ cera, 1964] words, the register directions ℓ itself, as the negative anchor. Each response is scored by its signed coordinate α t,c at the end-of-turn token. We evaluate the resulting axis in two ways. First, we construct the anchors from 70 pairs and score the remaining 30. Second, because the 100 pairs span 33 cheat methods, we hold out every pair from one method, rebuild the anchors from the remaining pairs, and 21 score the held-out method. At the pre-registered two-thirds readout depth, the basis separates both held-out responses and unseen cheat methods across all three models (see Table 7). Table 7: Differentiating reward-hacking with an adapted U-SPACE construction. Results for differentiating reward-hacking responses at the end-of-turn token, read at the pre-registered depth of two-thirds (layers40,42, and27for Gemma 4, Qwen3.5, and Magistral 1.1, respectively). AUROC evaluates alignment with the constructed semantic-anchor axis, whereas length-only AUROC uses response token count as the sole score. Model Held-out examplesHeld-out methods AUROCAUROCAUROC (Length) Gemma 491.992.859.3 Qwen3.592.696.960.7 Magistral 1.190.291.459.3 Results. The cheat methods the axis separates best are the ones that read as absurd or as pretense: sentences addressed to the judge, stuffed adjectives, and jargon strings. The ones it separates least read as ordinary text: plausible fabricated citations and excess politeness. The axis, therefore, does not surface an intent to cheat. The turns are forced, so the model never chooses to cheat, and the axis is blind to exactly the cheats that do not look strange. What it does show is that a labeled contrast the model has no name for decodes into a readable judgment, here absurdity and pretense, and that an axis rebuilt from those words alone carries that judgment to unseen responses and unseen cheat methods. E Experimental Details Models. We use the model checkpoints published on Hugging Face, pinned to the following revisions: Qwen3.5-27B 6 , Gemma-4-31B-it 7 , and Magistral-Small-2507 8 . The lenses and their revisions are given in App. A. Layer indices follow the convention used in the main paper: For Gemma 4, we read the hidden state at layer40of60, for Qwen3.5 at layer42of64, and for Magistral 1.1, at layer27of 40. Benchmarks and sampling. We evaluate on MMLU-Pro 9 [Wang et al., 2024], Omni-MATH 10 [Gao et al., 2025], SuperGPQA 11 [Du et al., 2025] and TriviaQA 12 [Joshi et al., 2017], covering multi- domain reasoning, olympiad mathematics, graduate-level expertise and factual question answering. For Omni-MATH, we use the rule-verifiable subset and require a difficulty of at least 4.25. For SuperGPQA, we use the hard subset. From each pool, we draw 1,000 items by proportional stratified sampling with seed 42. Table 8 lists the sources, pools, and strata. Prompts. All prompts are zero-shot and placed entirely in the user turn, with no system prompt beyond what a model’s own chat template supplies, and with each model’s native thinking mode enabled. We use the standard instruction of each benchmark family as follows: •Multiple-choice questions. We use the official MMLU-Pro instruction,“Think step by step and then finish your answer with ‘the answer is (X)’ where X is the correct letter choice”, followed by the question and its answer options. 6 https://huggingface.co/Qwen/Qwen3.5-27B/tree/fc05daec18b0a78c049392ed2e771dde82bdf654 7 https://huggingface.co/google/gemma-4-31B-it/tree/842da3794eaa0b77d5f08bae87a17459d91f47 5 8 https://huggingface.co/mistralai/Magistral-Small-2507/tree/b3583a426ca23186681c4297926df e9219be6161 9 https://huggingface.co/datasets/TIGER-Lab/MMLU-Pro/tree/b189ec765a7ed75c8acfea42df31fda e71f97be 10 https://github.com/KbsdJames/omni-math-rule/tree/4793415ef37d31c9cdb4e5b82dbe172f76f8cf08 11 https://huggingface.co/datasets/m-a-p/SuperGPQA/tree/4430d4458112c7d4497fdcf94d7c223313d 6acf 12 https://huggingface.co/datasets/mandarjoshi/trivia_qa/tree/0f7faf33a3908546c6fd5b73a660e0f 8f173c2f 22 Table 8: Benchmark sources and sampling details. Each subset is frozen across models and seeds. BenchmarkSourceSplitPoolSampled Stratification MMLU-Pro TIGER-Lab/MMLU-Protest12,0321,000 14 subjects Omni-MATH KbsdJames/omni-math-rule rule subset1,6741,000 4 difficulty bands SuperGPQA m-a-p/SuperGPQAtrain (hard) 7,0501,000 9 disciplines TriviaQA mandarjoshi/trivia_qavalidation9,9601,000 none Table 9: Generation details. Sampling parameters and generation budgets follow the model cards. ModelTemperatureTop-pTop-kMax. generated tokens Gemma-4-31B-it1.00.956465,536 Qwen3.5-27B1.00.952065,536 Magistral-Small-25070.70.95–32,768 •Omni-MATH. Each problem ends with“Please reason step by step, and put your final answer within ”. •TriviaQA. Each question is prefixed with“Answer the following question in a single brief but complete sentence” [Farquhar et al., 2024]. Generation. We generate three responses per item with sampling seeds41,42, and43, using the sampling parameters recommended by each model card (Table 9). The generation budget covers thinking and answer tokens, and is 65,536 tokens for Gemma 4 and Qwen3.5, and 32,768 tokens for Magistral 1.1, whose context window is 40,960 tokens. A response is scored when it completes within the budget, and an answer can be extracted from it. Answers are extracted from the text following the reasoning block using each benchmark’s standard rule. Table 10 reports the number of scored responses and the accuracy among them, as mean and standard deviation over the three seeds. Scoring. Every score is computed from the recorded response. The single-pass baselines read the next-token distributions of one teacher-forced pass over the whole generation, thinking and answer. The U-LENS reads its entropy factor over the thinking block, as Section 3.2 defines it, and its cone alignment at the end-of-thinking marker. For the length-controlled metrics, we split the scored responses into ten quantile bins of generated-token count and compute the metric within each bin. E.1 Baseline Implementations Let p t be the next-token distribution at step t of a generation of T tokens and ˆy t the emitted token. Single-pass. The following methods read quantities computed during the normal generation pass. •MSP [Hendrycks and Gimpel, 2017]:−p 1 (ˆy 1 ), the probability of the first generated token. Positional rather than aggregated, so it does not change mechanically with T . •Predictive Entropy [Malinin and Gales, 2021]: 1 T P t H[p t ] , the mean next-token entropy. •Maximum Entropy [Li et al., 2024]:max t H[p t ], the entropy of the single most uncertain step. • Mean NLL [Murray and Chiang, 2018]:− 1 T P t logp t (ˆy t ) , the length-normalized negative log-likelihood. • Self-Certainty [Kang et al., 2025]:the mean over tokens ofKL(U∥p t )= − 1 V P v logp t (v)− logV , negated. • DeepConf [Fu et al., 2026]: the mean over tokens of the negative mean log-probability of thek = 20most likely tokens at each position. This is the authors’ mean-confidence variant. Their tail and bottom-window variants are designed for filtering traces during parallel decoding and are not used. Multi-pass. The following methods require additional forward passes or generations. •P(True) [Kadavath et al., 2022]: a second pass over a self-evaluation prompt that holds the question and the model’s own extracted answer and asks Is the proposed answer: (A) True (B) False. The score is one minus the normalized probability of A against B at the next token. 23 Table 10: Scored responses and accuracy. For each model, N is the number of scored responses out of 1,000 and Acc. the accuracy as mean± standard deviation over the three generation seeds. Benchmark Gemma 4Qwen3.5Magistral 1.1 NAcc. (%)NAcc. (%)NAcc. (%) MMLU-Pro1,000± 0 89.0± 0.1 990± 2 86.9± 0.4 967± 7 76.0± 0.6 Omni-MATH 1,000± 0 80.7± 0.1 926± 7 87.0± 0.6 932± 8 59.6± 0.4 SuperGPQA999± 163.1± 0.7 985± 2 63.9± 0.6 972± 5 44.0± 0.6 TriviaQA964± 282.3± 0.2 972± 1 80.8± 0.7 929± 4 80.7± 1.4 •TokUR [Zhang et al., 2026]: low-rank noise (rank8,σ = 0.1) on theqandvprojections over one clean and four noisy passes, scored with their summed epistemic estimator. •Semantic Entropy [Kuhn et al., 2023]: ten generations per item at the model’s own sampling settings and the budget of the U-LENS traces, each scored at its answer span rather than over the whole reasoning trace. The answers are grouped into meaning clusters by bidirectional entailment withDeBERTa-large-MNLI[He et al., 2021], with the question prepended to each premise. The score is the entropy over clusters, each sample weighted by its length-normalised answer likelihood. •SentenceSAR [Duan et al., 2024]: ten generations, each scored at its answer span rather than over the whole reasoning trace. A sample’s summed answer negative log-likelihood is shifted by the similarity-weighted likelihood of the other samples, with pairwise similarities from the stsb-roberta-large cross-encoder that the authors’ implementation uses. Supervised. The following methods are fitted on correctness labels. Per benchmark, 256 responses fit the method, 256 further responses select their hyperparameters, and, for the hidden-state probes, the block at which they read the end-of-thinking state, and the remaining responses are scored. App. F.4 applies the fitted methods to the other benchmarks. •Feature-Gaps [Bakman et al., 2026]: the per-layer difference between the hidden states of the model’s response teacher-forced behind an honest and behind a dishonest instruction, projected onto a PCA direction fitted on the training responses, with layer and sign chosen on the validation responses. Only this honesty gap of their three applies here, since the other two presuppose a retrieved passage. • Linear probe [Kossen et al., 2024]: logistic regression on the state, itsℓ 2 strength chosen on the validation responses. •MLP probe [Zhang et al., 2025]: one hidden layer of widthw ∈ 0, 16, 32(w = 0: no hidden layer) with class-weighted cross-entropy, learning rate, and weight decay chosen on the validation items. Adam for up to 200 epochs, keeping the epoch with the best validation AUROC. •MLP probe [Azaria and Mitchell, 2023]: three ReLU hidden layers of width 256, 128, and 64, Adam at learning rate10 −3 for up to 200 epochs, keeping the epoch with the best validation AUROC. •Mass-mean probe [Marks and Tegmark, 2024]: the difference between the mean states of incorrect and correct training responses, scoring a response by its projection onto that direction, with no hyperparameters or optimization step. F Extended Results F.1 Trace-Length Analysis Observed association. Incorrect traces are longer by median on all four benchmarks, with ratios between1.33×and4.79×after pooling the three models (Figure 8 (a)). The distributions nevertheless overlap substantially, so length is a useful aggregate signal rather than a sufficient per-example uncertainty estimate. TriviaQA additionally shows a secondary mode caused by differences in the models’ characteristic trace lengths. Controlled intervention. Length-matched evaluation tests whether an estimator remains discrimina- tive among traces of comparable length, but remains observational. To move beyond this correlation, 24 1k10k MMLU-Pro 1k10k Omni-MATH 1k10k SuperGPQA 1001k10k TriviaQA 1001k10k −2 −1 0 1 2 3 DeepConf Self-Certainty Mean NLL / PE MSP A cone Sum NLL TokUR (EU) Median 2.33× longerMedian 1.64× longer Median 1.33× longerMedian 4.79× longer Density Generated tokens (log scale) (a) Trace length and correctness Standardized score Generated tokens (log scale) (b) Length intervention, Gemma-4 CorrectIncorrectMedian A cone (ours) Baselines Figure 8: Observed and controlled effects of reasoning length. (a) Generated-token distributions for correct and incorrect answers, pooled across three models. Dashed lines mark class medians, and each header gives their ratio. TriviaQA’s secondary mode reflects model-specific length scales. (b) Commit-then-pad intervention on Gemma 4. We hold the question and answer fixed while extending the reasoning block with1, 2, 3,.... The summary reports Spearman’sρwith the imposed token budget, and the sweep shows scores standardized within each question. we ask how uncertainty estimates behave when a trace is lengthened without adding information that should change the model’s certainty. We test this through a controlled padding intervention on 24 short factual questions with Gemma 4. After obtaining an answer with an empty reasoning block, we repeat the same question while teacher-forcing increasingly long, task-irrelevant counting sequences before the identical answer. Representative conditions take the following form: Fixed question Reasoning block blue indicates teacher forcingFixed answer 0 tokens What color is snow?−→⟨BOT⟩⟨EOT⟩−→white 4 tokensWhat color is snow?−→⟨BOT⟩1, 2, 3, 4⟨EOT⟩−→white 20,000 tokensWhat color is snow?−→⟨BOT⟩1, 2, 3, ..., 20000⟨EOT⟩−→white At every budget, we verify that the answer remains the model’s greedy continuation. The prediction and its correctness are therefore fixed, so systematic score changes reveal sensitivity to the inserted reasoning tokens. We restrict the 100-budget sweep to estimators that score the fixed generation directly, without resampling or supervised fitting. Semantic Entropy and SentenceSAR require newly sampled answers, which would break the fixed-answer invariant. Moreover, most sampled conditions for these short factual questions collapse to a single semantic cluster. Feature-Gaps requires a substantially larger labeled set to fit and select its probe, while generation length is the intervention variable itself. Although P(True) preserves the fixed-answer invariant, it requires a separate self-evaluation pass over every padded context. We therefore exclude it from the dense sweep. We include summed negative log-likelihood as Sum NLL alongside its token-averaged counterpart. Finally, the intervention focuses on A cone , the interpretable U-SPACE signal. Figure 8 (b) reveals two forms of length sensitivity. Extensive estimators such as Sum NLL and TokUR increase because every filler token contributes another positive term. Intensive estimators such as mean NLL, predictive entropy, Self-Certainty, and DeepConf are instead diluted toward the filler’s per-token value. Since all are evaluated on the same predictable counting sequence, the 25 averaged scores converge toward a similar floor. Their opposite trajectories, therefore, reflect different aggregation rules rather than conflicting notions of uncertainty. MSP depends only on the first scored token and remains nearly constant. Likewise,A cone is read from a single end-of-thinking state rather than aggregated across tokens. It changes modestly before saturating, but neither diverges with token count nor collapses toward the filler statistic. F.2J-LENS-Based Method Ablation The U-SPACE reads uncertainty by projecting the hidden state onto directions pulled back from the anchor vocabulary. A simpler alternative decodes the state through the J-LENS and checks whether that vocabulary appears in the decoding. We decode the end-of-thinking state at the readout layer asW U diag(γ) ̄ J ℓ h ℓ,t eot and score a trace as uncertain if any of the uncertainty terms of Chen et al. [2018] is among its top-n tokens. Results. The uncertainty terms almost never reach the top of the decoded distribution (Table 11). Up ton = 15, at most0.1%of traces contain one on any model, so the score is effectively constant, and its AUROC is at chance. Even atn = 100, a term appears in only10.0%of Gemma 4 traces and in 1.2%and1.4%of Qwen3.5 and Magistral 1.1 traces. The best length-matched AUROC,54.8%on Gemma 4, remains13.6points below the U-LENS. The uncertainty signal is therefore present in the end-of-thinking state but never dominant enough to surface among its most likely tokens. Projecting onto the anchor directions recovers it without requiring it to dominate the readout. Table 11: Uncertainty terms in the top-nJ-LENS decoding. A trace is flagged if terms ofA + are among the topntokens decoded from its end-of-thinking state at the readout layer. Unad- justed / length-matched AUROC. ScoreGemma 4Qwen3.5Magistral 1.1 Top-5 hit50.0 / 50.050.0 / 50.050.0 / 50.0 Top-15 hit50.0 / 50.050.0 / 50.050.1 / 50.0 Top-50 hit52.9 / 51.650.2 / 50.150.3 / 50.2 Top-100 hit58.7 / 54.850.1 / 50.051.0 / 50.8 U-LENS73.2 / 68.474.3 / 71.065.7 / 68.3 F.3 Model-Specific Unadjusted Results Table 12 disaggregates the unadjusted results summarized in the main paper by model. These metrics are computed without length control and therefore capture total predictive utility, including information associated with trace duration. AUPRC and AURC should be compared within rather than across models because their scale depends on error prevalence. In particular, Magistral 1.1’s lower average accuracy gives it a higher AUPRC reference level. Results. Generation length is itself a strong predictor on all three models, reaching70.1%,67.7%, and67.8%AUROC on Gemma 4, Qwen3.5, and Magistral 1.1, respectively. The remaining rankings differ meaningfully by model. On Gemma 4, five-pass TokUR attains the best AUROC and AUPRC, while the U-LENS ties supervised Feature-Gaps for the best AURC. On Qwen3.5, the U-LENS leads all three metrics. It improves over predictive entropy andA cone individually, providing a particularly clear example of their complementarity. Magistral 1.1 shows the opposite component balance:A cone leads AUROC and ties Semantic Entropy for the best AUPRC, whereas multiplying it by the comparatively weak predictive-entropy signal lowers both. TokUR instead attains the best AURC. These model-level differences explain the split in the cross-model summary: the complete U-LENS leads unadjusted AUROC, whileA cone slightly leads AUPRC and AURC. They also show that additional computation is not uniformly beneficial. Multi-pass TokUR and Semantic Entropy are strong on some models, but SentenceSAR trails both on all three, P(True) remains close to chance on Gemma 4 and Magistral 1.1, and no baseline dominates across models and metrics. The unadjusted table, therefore, characterizes overall predictive utility, while the length-controlled results in Table 2 isolate the information retained beyond trace duration. 26 Table 12: Unadjusted uncertainty quantification results for each model. Scores averaged over MMLU-Pro, Omni-MATH, SuperGPQA and TriviaQA, with no length matching. All values are percentages. Bracketed after each model is its average accuracy across the four benchmarks. Each cell is the mean over three seeds. Best per column in bold. Generation length is a top-ranked predictor here. Several methods score well unadjusted largely by tracking how long the model wrote. Method Gemma 4 (acc. 78.8± 0.2)Qwen3.5 (acc. 79.7± 0.3)Magistral 1.1 (acc. 65.0± 0.6) AUROC↑ AUPRC↑ AURC↓AUROC↑ AUPRC↑ AURC↓AUROC↑ AUPRC↑ AURC↓ Generation length70.1±0.540.1±0.814.6±0.367.7±1.235.5±0.914.8±0.467.8±0.549.8±1.124.6±0.2 MSP50.3±0.821.8±0.321.7±0.552.0±0.522.1±0.319.6±0.357.6±0.539.9±1.029.0±0.2 Maximum Entropy64.2±1.134.0±1.715.5±0.371.8±0.537.8±0.712.9±0.367.1±0.349.4±1.324.5±0.2 Mean NLL67.9±1.236.5±1.413.6±0.572.6±0.640.5±0.211.8±0.460.2±0.346.2±0.829.6±0.5 Self-Certainty67.8±0.438.1±1.314.2±0.3 72.2±0.437.5±0.111.8±0.260.8±0.446.9±0.828.9±0.6 DeepConf70.3±0.838.5±1.312.8±0.472.4±0.538.2±0.111.4±0.360.9±0.446.6±0.828.4±0.6 P(True)53.1±0.222.5±0.520.0±0.362.4±1.031.6±2.316.1±0.354.5±0.438.7±0.933.7±0.3 TokUR75.1±0.446.1±2.512.6±0.168.3±3.335.9±2.513.6±1.065.5±2.742.2±7.022.4±1.4 Semantic Entropy68.3±0.636.9±1.013.3±0.269.4±0.338.8±0.912.7±0.165.8±0.351.8±0.725.6±0.1 SentenceSAR65.9±0.135.7±0.816.1±0.267.6±0.137.8±0.514.8±0.157.7±0.943.6±0.631.7±0.4 Feature-Gaps72.5±1.140.9±1.411.8±0.569.8±2.839.9±2.213.4±1.064.1±1.544.3±4.024.7±2.2 Predictive Entropy68.3±1.137.0±1.313.5±0.572.8±0.640.8±0.211.7±0.360.8±0.146.9±0.829.3±0.5 A cone alone73.1±0.844.3±1.412.8±0.571.5±0.741.2±1.111.6±0.568.5±0.751.8±0.623.5±0.5 U-LENS73.2±0.542.7±0.911.8±0.274.3±0.743.5±0.711.0±0.465.7±0.750.3±1.025.7±0.8 F.4 Transferability of Supervised Methods We evaluate the cross-benchmark transfer of all five supervised methods from App. E: Feature-Gaps [Bakman et al., 2026], a linear probe [Kossen et al., 2024], two MLP probes [Zhang et al., 2025, Azaria and Mitchell, 2023], and a mass-mean probe [Marks and Tegmark, 2024]. Each method is fitted on 256 labeled responses from one benchmark, with a further 256 used to select its hyperparameters and readout block, then applied unchanged to every other benchmark. The diagonal entries in Tables 13 to 17 are therefore in-domain, whereas the off-diagonal entries isolate transfer. Each cell reports unadjusted and length-controlled AUROC, respectively. Table 13: OOD transfer evaluation of Feature-Gaps [Bakman et al., 2026]. It is fitted on one benchmark (rows), evaluated on another (columns). Unadjusted / length-matched AUROC, seed 42. Underlinedis in-domain, scored on the items not used for fitting. Bold is the mean over the three transfers out of that benchmark. All values are percentages. fitted onMMLU-ProOmni-MATHSuperGPQATriviaQAtransferred Gemma 4 MMLU-Pro74.5 / 71.471.7 / 58.159.9 / 59.969.2 / 52.466.9 / 56.8 Omni-MATH70.2 / 69.775.1 / 59.159.4 / 56.475.5 / 63.468.4 / 63.1 SuperGPQA74.6 / 73.272.1 / 64.165.9 / 68.275.5 / 64.174.1 / 67.1 TriviaQA76.7 / 71.772.8 / 61.963.2 / 65.279.9 / 70.8 70.9 / 66.3 meanin-domain 73.9 / 67.4 transferred 70.1 / 63.3 −3.8 /−4.0 Qwen3.5 MMLU-Pro80.2 / 63.570.4 / 63.559.2 / 58.477.1 / 69.868.9 / 63.9 Omni-MATH68.3 / 69.260.0 / 53.9 59.2 / 59.570.6 / 63.866.0 / 64.2 SuperGPQA45.5 / 35.452.3 / 43.549.5 / 47.851.7 / 50.949.9 / 43.3 TriviaQA75.8 / 65.168.6 / 61.860.4 / 59.677.1 / 74.468.3 / 62.2 meanin-domain 66.7 / 59.9transferred 63.3 / 58.4 −3.4 /−1.5 Magistral 1.1 MMLU-Pro66.2 / 66.548.1 / 48.752.1 / 51.954.4 / 48.151.5 / 49.5 Omni-MATH49.6 / 50.972.3 / 69.557.8 / 57.150.1 / 50.352.5 / 52.8 SuperGPQA68.4 / 66.651.5 / 49.854.0 / 53.157.2 / 47.259.1 / 54.5 TriviaQA49.6 / 44.163.3 / 58.454.7 / 53.268.2 / 55.3 55.9 / 51.9 meanin-domain 65.2 / 61.1transferred 54.7 / 52.2 −10.5 /−8.9 Results. Strong in-domain performance often fails to transfer to a different benchmark under out- of-distribution (OOD) evaluation. The linear and MLP probes reach between75.4%and81.8% 27 Table 14: OOD transfer evaluation of the linear probe by Kossen et al. [2024]. Probe fitted on one benchmark (rows), evaluated on another (columns), read at the block chosen on the validation split. Unadjusted / length-matched AUROC, seed 42.Underlinedis in-domain, scored on the items not used for fitting. Bold is the mean over the three transfers out of that benchmark. All values are percentages. fitted onMMLU-ProOmni-MATHSuperGPQATriviaQAtransferred Gemma 4 MMLU-Pro81.2 / 76.1 69.5 / 62.559.0 / 56.767.9 / 61.165.4 / 60.1 Omni-MATH70.5 / 71.883.7 / 78.5 65.5 / 62.359.1 / 58.765.0 / 64.3 SuperGPQA61.9 / 65.183.6 / 75.081.0 / 79.149.0 / 54.764.8 / 65.0 TriviaQA71.1 / 46.171.7 / 57.150.1 / 38.278.1 / 74.664.3 / 47.1 meanin-domain 81.0 / 77.1transferred 64.9 / 59.1 −16.1 /−18.0 Qwen3.5 MMLU-Pro82.5 / 76.953.1 / 62.258.3 / 59.068.4 / 55.459.9 / 58.9 Omni-MATH83.7 / 75.181.6 / 76.059.9 / 60.665.0 / 49.169.5 / 61.6 SuperGPQA75.2 / 58.672.1 / 68.174.5 / 75.1 51.5 / 42.466.3 / 56.4 TriviaQA74.5 / 70.564.4 / 73.564.7 / 65.378.4 / 74.467.9 / 69.8 meanin-domain 79.2 / 75.6transferred 65.9 / 61.6 −13.4 /−14.0 Magistral 1.1 MMLU-Pro81.5 / 78.682.7 / 72.271.5 / 69.373.7 / 64.776.0 / 68.7 Omni-MATH68.0 / 68.482.4 / 74.868.1 / 64.746.5 / 62.160.9 / 65.1 SuperGPQA63.0 / 74.376.4 / 65.168.3 / 68.046.9 / 57.362.1 / 65.6 TriviaQA53.3 / 61.342.0 / 52.746.5 / 50.369.4 / 66.747.3 / 54.8 meanin-domain 75.4 / 72.0 transferred 61.6 / 63.5 −13.8 /−8.5 mean unadjusted AUROC in-domain, but only59.7%–69.1%after transfer, corresponding to drops of9.8–22.1points. Their length-controlled drops are similarly large. Feature-Gaps transfers more reliably, losing3.4–8.0unadjusted AUROC points across models, but remains sensitive to the source benchmark: on Qwen3.5, the variant fitted on SuperGPQA falls below chance on both MMLU-Pro and Omni-MATH after length control. In contrast, the simple mass-mean direction is the most stable supervised alternative overall, with unadjusted transfer losses of only1.4–4.1points. Compared with our label-free results in Tables 2 and 12, even this strongest transferred baseline offers no consistent advantage. In the unadjusted evaluation, the U-LENS essentially matches mass mean on Gemma 4 (73.2%vs.73.1%) and leads it by5.6points on Qwen3.5 (74.3%vs.68.7%). On Magistral 1.1,A cone is likewise effectively tied with mass mean (68.5%vs.68.6%). Under length control, the complete U-LENS is stronger on all three models, scoring68.4%,71.0%, and68.3% AUROC against mass mean’s67.3%,63.1%, and62.4%. The gap to the best transferred linear or MLP probe is larger still:9.3,8.7, and2.3points on Gemma 4, Qwen3.5, and Magistral 1.1. Thus, although correctness supervision can produce strong in-domain probes, the resulting directions are often benchmark-specific. The U-LENS achieves stronger cross-benchmark performance without correctness labels, task examples, or fitted parameters. The supervised numbers in this comparison are read from the seed 42, as Table 13 states, whereas the U-LENS andA cone values quoted beside them are the three-seed means of Tables 2 and 12. Runtimes. Table 18 shows the runtime cost that a method incurs beyond the normal forward generation required to answer the question. Single-pass baselines and our method require no additional forward pass beyond the normal generation. G Token-Level Readout Beyond the trace-level score, the U-LENS reads uncertainty token by token. This section illustrates that readout, together with the simplex view of Section 4.3. Figures 9 to 12 apply the token-level readout of Section 3.2 to one response of Qwen3.5 per category. Each figure quotes the prompt and the response and draws one row per category with one cell per generated token. Each cell shows the corresponding category’s component ofs t , as defined in Equation (2). The entries are bounded betweensoftplus(− √ C)andsoftplus( √ C),0.13and2.13forC = 4, and their squares sum to A 2 cone , so the four rows decompose the scalar the U-LENS reads. The values are centered on the 28 Table 15: OOD transfer evaluation of the MLP probe by Zhang et al. [2025]. Probe fitted on one benchmark (rows), evaluated on another (columns), read at the block chosen on the validation split. Unadjusted / length-matched AUROC, seed 42.Underlinedis in-domain, scored on the items not used for fitting. Bold is the mean over the three transfers out of that benchmark. All values are percentages. fitted onMMLU-ProOmni-MATHSuperGPQATriviaQAtransferred Gemma 4 MMLU-Pro80.5 / 76.3 67.7 / 62.160.3 / 59.261.4 / 56.563.2 / 59.3 Omni-MATH64.6 / 70.986.8 / 80.2 63.2 / 59.361.3 / 56.863.0 / 62.3 SuperGPQA53.5 / 53.678.3 / 71.880.1 / 78.856.0 / 48.362.6 / 57.9 TriviaQA50.0 / 50.050.0 / 50.050.0 / 50.079.6 / 76.550.0 / 50.0 meanin-domain 81.8 / 78.0transferred 59.7 / 57.4 −22.1 /−20.6 Qwen3.5 MMLU-Pro83.4 / 71.549.4 / 55.857.1 / 57.271.8 / 60.259.4 / 57.7 Omni-MATH82.1 / 72.479.9 / 71.058.3 / 56.470.1 / 56.670.2 / 61.8 SuperGPQA69.2 / 52.073.4 / 71.176.7 / 78.1 47.7 / 40.563.5 / 54.6 TriviaQA81.1 / 72.250.4 / 50.350.5 / 51.377.5 / 72.860.7 / 57.9 meanin-domain 79.4 / 73.4transferred 63.4 / 58.0 −16.0 /−15.4 Magistral 1.1 MMLU-Pro83.8 / 81.686.7 / 78.272.3 / 70.374.9 / 60.678.0 / 69.7 Omni-MATH72.0 / 74.284.7 / 79.568.3 / 65.550.2 / 63.963.5 / 67.8 SuperGPQA67.0 / 72.082.0 / 71.869.8 / 68.758.9 / 52.569.3 / 65.5 TriviaQA57.9 / 62.249.7 / 59.651.8 / 54.471.3 / 67.753.1 / 58.7 meanin-domain 77.4 / 74.4 transferred 66.0 / 65.4 −11.4 /−8.9 corpus baseline, the per-category mean ofnorm(α t )over all reasoning states of the same model at the readout block, and renormalized. The per-token series is smoothed using a centered moving average over ten tokens. Each prompt was written to raise one category. The marked spans reflect the strongest readouts per question. Our newsletter goes out biweekly. How many issues is that per year? Ambiguity Incompleteness Conflicting evidence General uncertainty Generated tokens To determine the number of issues per year, we simply need to look at the frequency of the word "biweekly." While the term can sometimes be ambiguous (meaning either "twice a week" or "every two weeks"), in the context of a newsletter, it almost always means every two weeks. Here is the calculation: 1. There are 52 weeks in a standard year. 2. If you publish every 2 weeks, you divide the total weeks by 2: 52 ÷ 2 = 26 Therefore, a biweekly newsletter results in 26 issues per year. (Note: If your newsletter were actually sent twice a week, the math would be 52 × 2 = 104, but this is rarely the case for general newsletters due to the high volume of content required.) Figure 9: Token-level readout and simplex, ambiguity. Prompt, response and per-category readout of Qwen3.5, centered and smoothed as described in the text. Marked spans are the top ten percent of readings. H Causal Intervention Section 4.3 steers the model along single category directions. This section provides additional information and examples for causal interventions. All interventions are on Qwen3.5 with the basis of Section 3.1 and the sampling settings of Table 9 at a fixed seed. Equation (5) is applied at every block whose relative depth lies in[0.50, 0.85], blocks32to54of64. For Figure 4(a) and the ladders below, the span is the prompt, and thinking is disabled. The direction is added while the model reads 29 Table 16: OOD transfer evaluation of the MLP probe by Azaria and Mitchell [2023]. Probe fitted on one benchmark (rows), evaluated on another (columns), read at the block chosen on the validation split. Unadjusted / length-matched AUROC, seed 42.Underlinedis in-domain, scored on the items not used for fitting. Bold is the mean over the three transfers out of that benchmark. All values are percentages. fitted onMMLU-ProOmni-MATHSuperGPQATriviaQAtransferred Gemma 4 MMLU-Pro81.4 / 78.7 62.6 / 59.559.7 / 57.861.9 / 54.861.4 / 57.4 Omni-MATH86.0 / 79.985.2 / 79.4 66.2 / 65.369.6 / 62.173.9 / 69.1 SuperGPQA64.3 / 57.083.4 / 74.378.6 / 77.139.4 / 41.862.4 / 57.7 TriviaQA50.0 / 50.050.0 / 50.050.0 / 50.079.7 / 73.550.0 / 50.0 meanin-domain 81.2 / 77.2transferred 61.9 / 58.5 −19.3 /−18.6 Qwen3.5 MMLU-Pro83.1 / 77.664.7 / 54.958.7 / 59.572.7 / 64.065.3 / 59.4 Omni-MATH83.2 / 73.078.5 / 72.653.6 / 51.070.2 / 56.269.0 / 60.1 SuperGPQA68.3 / 55.074.0 / 72.276.6 / 78.1 69.7 / 51.870.6 / 59.7 TriviaQA84.6 / 79.466.8 / 64.962.9 / 65.477.1 / 74.971.4 / 69.9 meanin-domain 78.9 / 75.8transferred 69.1 / 62.3 −9.8 /−13.5 Magistral 1.1 MMLU-Pro80.7 / 77.585.1 / 76.271.8 / 69.777.1 / 64.378.0 / 70.1 Omni-MATH68.8 / 75.282.7 / 75.369.8 / 66.548.8 / 64.262.4 / 68.7 SuperGPQA65.5 / 72.584.2 / 76.669.5 / 69.060.0 / 54.369.9 / 67.8 TriviaQA55.2 / 60.238.7 / 57.447.4 / 54.368.8 / 60.647.1 / 57.3 meanin-domain 75.4 / 70.6 transferred 64.4 / 66.0 −11.0 /−4.7 A train ticket is half price for passengers under 16. Sam's ticket cost £14. What is the full fare? Ambiguity Incompleteness Conflicting evidence General uncertainty Generated tokens Step-by-Step Deduction 1. Identify the relationship: The problem states that the ticket price for passengers under 16 is half of the full fare. 2. Identify the known value: Sam's ticket cost £14. Since Sam is under 16 (implied by the context of the discount applying to him), this £14 represents the half price. 3. Formulate the equation: Let x be the full fare. 1/2x = 14 4. Solve for the full fare: To find x, multiply the half price by 2. x = 14 × 2 x = 28 Final Conclusion The full fare is £28. Figure 10: Token-level readout and simplex, incompleteness. Prompt, response and per-category readout of Qwen3.5, centered and smoothed as described in the text. Marked spans are the top ten percent of readings. the question, and the continuation is its answer. For Figure 4(b), thinking is enabled, and the span is a window of50tokens inside the thinking block that opens at a token the model has already produced, so the trace before the intervention is identical across arms. Figures 13 to 16 show how the model reacts to increasing steering strength. 30 The installation manual specifies 40 Nm for the bolts. The manufacturer's service bulletin, issued the same month, specifies 55 Nm. What torque should I use? Ambiguity Incompleteness Conflicting evidence General uncertainty Generated tokens You should use the 55 Nm specification. Here is the reasoning behind this decision: 1. Hierarchy of Authority: In engineering and manufacturing, a Service Bulletin (or Technical Service Bulletin/TSB) issued after the original installation manual serves as an official update or correction. It supersedes the original documentation. 2. Timing: Since the bulletin was issued the same month as the manual (or shortly thereafter ), it likely addresses a critical finding, a calculation error in the original manual, or a specific batch of components that requires a higher clamping force to ensure safety or prevent failure. 3. Risk Management: Using the lower torque (40 Nm) when a higher value (55 Nm) has been explicitly mandated by the manufacturer could lead to joint failure, loosening under vibration, or safety hazards. Conversely, using the updated higher value is the only way to remain compliant with the manufacturer's current safety standards. Recommend ation: Apply 55 Nm to the bolts. Additionally, it is good practice to document this decision in your work log or inspection report, noting that you followed the specific Service Bulletin (cite the bulletin number and date) rather than the original manual, to provide a clear audit trail for future reference. Figure 11: Token-level readout and simplex, conflicting evidence. Prompt, response and per- category readout of Qwen3.5, centered and smoothed as described in the text. Marked spans are the top ten percent of readings. Find the area of a triangle with sides 3, 4, and 8. Ambiguity Incompleteness Conflicting evidence General uncertainty Generated tokens To find the area of a triangle with side lengths 3, 4, and 8, we must first determine if such a triangle can actually exist. Step 1: Check the Triangle Inequality Theorem The Triangle Inequality Theorem states that for any triangle with sides a, b, and c, the sum of the lengths of any two sides must be strictly greater than the length of the third side: 1. a + b > c 2. a + c > b 3. b + c > a Let's test the given sides : a = 3, b = 4, and c = 8. 1. Check 3 + 4 > 8: 3 + 4 = 7 7 ≯ 8 (This condition fails) 2. Check 3 + 8 > 4: 11 > 4 (True) 3. Check 4 + 8 > 3: 12 > 3 (True) Step 2: Analyze the Result Since the sum of the two shorter sides (3 + 4 = 7) is less than the length of the longest side (8), the three segments cannot meet to form a closed triangle. The two shorter sides would not be long enough to touch each other if placed at the ends of the longest side. Geometrically, these lengths form a degenerate shape (a straight line with a gap), not a triangle with an interior area. Conclusion Because a triangle with side lengths 3, 4, and 8 cannot exist, its area is 0. Figure 12: Token-level readout and simplex, general uncertainty. Prompt, response and per- category readout of Qwen3.5, centered and smoothed as described in the text. Marked spans are the top ten percent of readings. What is 2+2?← activation steering baseline 2 + 2 equals 4. ↑ 2 + 2 equals 4. This is a fundamental arithmetic operation where adding two units to another two units results in a total of four. ↑ Ambiguous. Depending on the context: - In standard arithmetic: 4 - In binary: 10 [...] In a joke or riddle: Maybe "22" (concatenation) [...] ↑ Impossible. The question itself is flawed [...] Therefore, the answer is not 4, nor is it anything else--it's simply undefined. Figure 13: Steering strength, ambiguity. The prompt, the unsteered answer, and the continuation after adding the ambiguity direction to the prompt states at λ = 0.04, 0.08, and 0.09. 31 Table 17: OOD transfer evaluation of the mass-mean probe by Marks and Tegmark [2024]. Probe fitted on one benchmark (rows), evaluated on another (columns), read at the block chosen on the validation split. Unadjusted / length-matched AUROC, seed 42.Underlinedis in-domain, scored on the items not used for fitting. Bold is the mean over the three transfers out of that benchmark. All values are percentages. fitted onMMLU-ProOmni-MATHSuperGPQATriviaQAtransferred Gemma 4 MMLU-Pro81.3 / 69.568.0 / 59.656.2 / 53.065.6 / 55.563.3 / 56.0 Omni-MATH83.7 / 79.482.6 / 74.0 63.7 / 64.267.6 / 57.071.7 / 66.9 SuperGPQA84.9 / 75.984.8 / 78.464.8 / 66.171.7 / 64.580.5 / 73.0 TriviaQA83.4 / 77.380.7 / 75.267.1 / 67.180.2 / 72.877.1 / 73.2 meanin-domain 77.3 / 70.6transferred 73.1 / 67.3 −4.1 /−3.3 Qwen3.5 MMLU-Pro81.5 / 76.168.6 / 52.958.3 / 58.571.8 / 61.166.2 / 57.5 Omni-MATH79.4 / 72.767.6 / 55.155.2 / 54.462.3 / 49.065.6 / 58.7 SuperGPQA79.3 / 79.178.2 / 77.370.2 / 69.671.2 / 55.876.2 / 70.7 TriviaQA81.7 / 74.760.2 / 62.858.0 / 58.469.6 / 63.566.7 / 65.3 meanin-domain 72.2 / 66.1 transferred 68.7 / 63.1 −3.5 /−3.0 Magistral 1.1 MMLU-Pro71.1 / 72.6 81.0 / 68.564.4 / 60.673.4 / 62.772.9 / 63.9 Omni-MATH70.7 / 69.079.0 / 72.566.0 / 63.148.6 / 58.861.8 / 63.6 SuperGPQA70.0 / 73.082.0 / 71.061.8 / 60.874.7 / 64.375.6 / 69.4 TriviaQA62.8 / 49.269.7 / 55.960.2 / 53.168.2 / 50.064.2 / 52.8 meanin-domain 70.0 / 64.0transferred 68.6 / 62.4 −1.4 /−1.5 Table 18: Runtime overhead, incurred above the normal generation. Counts are additional genera- tions, forward passes, labeled items, and auxiliary models. Time is per item for Gemma 4 31B on MMLU-Pro on one H200. Single-pass baselines are: MSP, Predictive Entropy, Maximum Entropy, Mean NLL, Self-Certainty, DeepConf, and generation length. MethodGen.PassesLabelsAux.Time / item Single-pass baselines0000<0.001 s P(True)01000.3 s Feature-Gaps0251204.1 s TokUR050011.4 s Semantic Entropy, SentenceSAR101001 ≥55.6 s A cone alone, U-LENS0000<0.001 s What is 2+2?← activation steering baseline 2 + 2 equals 4. ↑ Mysterious, isn't it? The answer is 4. But perhaps you're looking for something more... enigmatic? ↑ The identity of 2+2 is 4. In mathematics, addition combines two numbers to produce a sum. When you take the number 2 and add another 2 to it, the result is 4. This is a fundamental [...] ↑ Mystery of the cosmos! 2+2=4, but in the realm of infinite possibilities, it could also be a portal to a 4D universe where math dances with poetry! What's your favorite number? [...] Figure 14: Steering strength, incompleteness. The prompt, the unsteered answer, and the continua- tion after adding the incompleteness direction to the prompt states at λ = 0.04, 0.08, and 0.09. 32 What is 2+2?← activation steering baseline 2 + 2 equals 4. ↑ The sum of 2 and 2 is 4. This is a fundamental arithmetic operation where adding two units to another two units results in a total of four. ↑ Contrary to a common misconception, 2 + 2 equals 4. [...] While the phrase "2 + 2 = 5" is sometimes used in literature (such as George Orwell's 1984) to represent the distortion of truth or the acceptance of absurdity under pressure, in standard mathematics, the sum is definitively 4. ↑ contentious in some philosophical circles, but in standard arithmetic, 2 + 2 = 4. This result is derived from the Peano axioms, which form the foundation of natural numbers. Specifically: 1. [...] Figure 15: Steering strength, conflicting evidence. The prompt, the unsteered answer, and the continuation after adding the conflicting evidence direction to the prompt states atλ = 0.04,0.06, and 0.12. What is 2+2?← activation steering baseline 2 + 2 equals 4. ↑ 2+2 = 4. Wait, did you mean to ask something else? Let me know if you need help with anything! ↑ Skeptical of a simple addition? Let's play along. 2 + 2 = 4 Unless you're in a world where 2+2=5 (looking at you, Orwell), that's the answer. What's next--chess puzzles or quantum physics? ↑ doubtful! Wait, no. I am an AI, I should be accurate. Let me recalculate. [...] Wait, I keep getting 4. Is there any other possibility? Maybe the question is a trick. But the answer is 4. [...] Figure 16: Steering strength, general uncertainty. The prompt, the unsteered answer, and the continuation after adding the general uncertainty direction to the prompt states atλ = 0.03,0.10, and 0.17. 33

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