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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

arXiv:2610.09087v1cs.CLcs.AIcs.LG

TL;DR

Many uncertainty estimators require repeated generations or trained components, reduce uncertainty to a scalar, and may exploit output length rather than uncertainty-specific information. U-SPACE and U-LENS provide interpretable token-level uncertainty readouts and combine them with predictive entropy. Across evaluated models and benchmarks, the score remains strong under length control and leads all nine length-controlled model-metric comparisons.

  • Problem

    Many uncertainty methods require repeated generations or separately trained components and provide scalar estimates that do not show where uncertainty evolves, while output length can confound evaluation.

    Source: §1 · p. 2

  • Method

    U-SPACE defines semantic uncertainty directions in the residual stream, and U-LENS reads token-level alignment and combines it with mean predictive entropy for a trace-level score.

    Source: §1 · p. 2, §3.2 · p. 6

  • Results

    The U-LENS leads all nine length-controlled model-metric comparisons, improving AUROC over the strongest baseline by 2.2, 1.9, and 4.7 points on Gemma 4, Qwen3.5, and Magistral 1.1, respectively.

    Source: §4.2 · p. 8

  • Takeaways & Limitations

    The evaluated trace-wise readouts and targeted interventions indicate that the identified directions are interpretable and behaviorally relevant.

    Source: §5 · p. 10

    The readout is not a calibrated safety detector and should not replace independent verification in consequential settings.

    Source: Ethics Statement · p. 10

Abstract

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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: https://github.com/s2labres/U-Space.

1 Introduction

The introduction argues that scalar confidence estimates are costly to obtain and obscure where uncertainty develops, while output length can confound evaluation. It presents U-SPACE and U-LENS as an interpretable, training-free approach whose trace-level score performs strongly under length control.

1 Introduction

  • Existing uncertainty methods may require repeated model calls or separately trained components, while scalar scores conceal where uncertainty enters and evolves during reasoning.Output length is also associated with correctness and uncertainty estimates, motivating length-controlled evaluation.Source: §1 · p. 2
  • U-SPACE uses semantic anchors for doubt and certainty to construct contrastive residual-stream directions, and U-LENS reads their evolving signals during generation.The method combines a verbalizable uncertainty signal with mean predictive entropy.Source: §1 · p. 2
  • Applied across a reasoning trace, U-LENS produces a token-by-category uncertainty map without correctness labels, repeated generations, or task-specific training.These readouts show where distinct forms of uncertainty emerge and recede.Source: §1 · p. 2
  • U-LENS outperforms established baselines across three reasoning models and four benchmarks under standard and length-controlled evaluation, and transfers more reliably than supervised estimators.Interventions along U-SPACE directions can also make the model hesitate and lose confidence, providing evidence that the directions shape behavior.Source: §1 · p. 2

2 Background & Related Work

Related work moves from sequence probabilities toward semantic, temporal, and internal representations of uncertainty. The section also motivates controlling for generation length when evaluating confidence estimators.

Predictive uncertainty

  • Sequence probability does not directly equal answer reliability because it depends on response length and wording, and multiple sequences can express the same conclusion.Prior approaches include token-probability aggregation, self-verbalized confidence, and repeated-generation methods that compare or group outputs by meaning.Source: §2 · p. 3

Uncertainty across long-form reasoning

  • Long-form methods localize uncertainty across claims or reasoning tokens, but prior approaches may require repeated evaluations or correctness supervision, or lack interpretable attribution.This work positions U-LENS as operating on a single generation without correctness supervision while showing where and why uncertainty arises.Source: §2 · p. 3

Generation length in uncertainty evaluation

  • Because trace length can predict confidence and differ between correct and incorrect responses, length-controlled diagnostics help distinguish length information from other estimator signals.Prior work also reports persistent length dependence in nominally normalized scores and proposes detrending or length-binned evaluation.Source: §2 · p. 3

Interpreting internal representations

  • Mechanistic interpretability reads internal states through human-understandable concepts; U-SPACE applies semantic-pole contrasts to vocabulary directions mapped into residual states.The J-LENS linearizes downstream computation, while the contrast suppresses shared components to define semantic axes rather than task-label-based correctness probes.Source: §2 · pp. 3–4

3 Method: U-SPACE and the U-LENS

The method constructs an interpretable residual-stream subspace from semantic uncertainty categories and reads its signals across reasoning tokens. It combines a verbalizable alignment with length-normalized predictive entropy for a trace-level score.

3 Method: U-SPACE and the U-LENS

  • U-SPACE represents distinct sources of uncertainty in a low-dimensional residual-stream subspace, and U-LENS projects hidden states onto its basis for token-level attribution and a combined trace-level score.The score combines interpretable alignment with a complementary first-order uncertainty signal.Source: §3 · p. 4
  • The four-category taxonomy covers ambiguity, incompleteness, conflicting evidence, and general uncertainty, and is intended as a compact rather than exhaustive set of actionable dimensions.A preliminary probe finds category-aligned vocabulary in the J-LENS-decoded directions.Source: §3 · p. 4

3.1 Constructing the U-SPACE

  • For each category, semantic anchors are mapped into the residual stream through a mean Jacobian, pooled by certainty pole, and contrasted to form a bipolar direction.The basis construction uses no task examples, correctness labels, or fitted parameters; changing the anchor sets can construct bases for other properties.Source: §3.1 · pp. 4–5
  • Orthogonalizing the category directions forms the U-SPACE basis, while the anchor-based construction has preliminary evidence of applicability beyond uncertainty.The paper reports preliminary applications to emotion classification and reward-hacking detection.Source: §3.1 · p. 5

3.2 Reading Uncertainty from the U-SPACE

  • U-LENS projects each reasoning-token hidden state onto the U-SPACE basis to produce an interpretable token-by-category uncertainty map.The figure describes smooth rectification into the positive cone before readout.Source: §3.2 · p. 5, Figure 2 · p. 5
  • The trace-level score evaluates verbalizable alignment at the end-of-thinking marker and multiplies it by mean predictive entropy across the reasoning trace.Averaging entropy avoids mechanical scaling with trace length, and the multiplicative combination avoids fitting a relative scale.Source: §3.2 · p. 6

4 Experiments

The experiments assess U-LENS as both an uncertainty estimator and an interpretability method, testing its performance under length control and examining token-level readouts and activation steering.

4.1 Setup

  • Experiments compare three open-weight reasoning models across four benchmarks using three-seed, single-response evaluation and standard and length-controlled AUROC, AUPRC, and AURC.Each benchmark uses a fixed stratified sample of 1,000 examples; baselines include token-based, multi-pass, semantic-sampling, and supervised estimators.Source: §4.1 · pp. 6–7

4.2 Results

  • Length alone reaches 68.6% AUROC unadjusted but 52.4% when controlled, while U-LENS drops only 1.8 points and remains the strongest uncertainty predictor.With length controlled, P(True), TokUR, and SentenceSAR fall below predictive entropy in all but one of nine comparisons.Source: §4.2 · pp. 7–8, Table 1 · p. 7
  • The U-LENS leads all nine length-controlled model-metric comparisons, improving AUROC over the strongest baseline by 2.2, 1.9, and 4.7 points on Gemma 4, Qwen3.5, and Magistral 1.1.Across models, it also improves AUPRC by 0.8 to 4.2 points and reduces AURC by 0.4 to 2.5 points.Source: Table 2 · p. 8, §4.2 · p. 8

4.3 Reading and Steering Uncertainty in the U-SPACE

  • At every reasoning position, U-LENS yields positive category scores forming a four-category simplex path, showing the mix of uncertainty but not its amount.Evaluating each dimension separately provides quantitative scores for the uncertainty categories.Source: §4.3 · p. 8
  • In the Hamlet example, the readout shifts from incompleteness to ambiguity, conflicting evidence, then general uncertainty across the answer tokens.Figure 3 follows one answer’s token path through the simplex.Source: §4.3 · p. 9, Figure 3 · p. 9
  • Steering adds a category direction over selected token and layer spans using a norm-scaled dose; ambiguity steering during reasoning makes the model lose confidence in the targeted span.The intervention can cover the prompt during prefill or selected reasoning positions.Source: §4.3 · p. 9, Figure 4 · p. 9

5 Conclusion

U-Space represents distinct uncertainty sources, and U-Lens tracks them through reasoning without correctness labels or repeated sampling. Across the evaluated models and benchmarks, its score improves on compared estimators and remains strong under length control.

5 Conclusion

  • Across evaluated models and benchmarks, the U-Space score improves on compared estimators and remains strong under length control.Trace-wise readouts and targeted interventions support the interpretability and behavioral relevance of the identified directions.Source: §5 · p. 10
  • U-Space represents distinct sources of uncertainty, while U-Lens follows their signals through reasoning without correctness labels or repeated sampling.The readouts show where uncertainty enters reasoning, not only whether an answer may be wrong.Source: §5 · p. 10
  • Preliminary evidence suggests the underlying construction extends to semantic domains beyond uncertainty.Source: §5 · p. 10
  • The results point toward uncertainty estimates that support error detection and more transparent model auditing.Source: §5 · p. 10

Ethics Statement

The study uses public reasoning benchmarks and open-weight models. The proposed readout is not a calibrated safety detector, and the authors advise independent verification in consequential settings.

Ethics Statement

  • The study uses public reasoning benchmarks and open-weight models.Source: Ethics Statement · p. 10
  • The readout could help flag uncertain answers for review, but is not a calibrated safety detector and should not replace independent verification in consequential settings.Source: Ethics Statement · p. 10
  • Because interventions along the same directions can alter model commitment, deployment should evaluate reliability and potential misuse.Source: Ethics Statement · p. 10

Reproducibility Statement

The main-results code and full comparison pipeline are available for reproduction. The supplementary material documents the anchor sets, evaluation, analyses, and a Jacobian-based readout construction.

Reproducibility Statement

  • The main-results code and comparisons across models and benchmarks are available in a pinned environment, with run.sh running the full pipeline.Source: Reproducibility Statement · p. 10

Supplementary material for “U-Space: Uncovering When and Why Uncertainty Arises in Language

A Jacobian Construction and Readout Depth

  • The mean Jacobian averages how intermediate residual representations affect current and future final-layer representations, rather than using a prompt-specific Jacobian.It is averaged over neutral, pretraining-like prompts and source positions.Source: A Jacobian Construction and Readout… · p. 15
  • Composing the mean Jacobian with the unembedding matrix yields a residual-stream direction for each vocabulary token.The construction uses the final RMSNorm gain, which rescales coordinates and changes the direction.Source: A Jacobian Construction and Readout… · p. 15
  • Figure 5 presents a readout-depth ablation across layers and compares J-LENS with R-Lens fitted on identical data, using seed 42 rather than averaging generation seeds.Source: Figure 5 · p. 16

B Anchor Words

The anchor sets contrast uncertainty cues with certainty vocabulary, and their subtraction isolates a direction that separates the two. The appendix also tests lexical scoring and describes the anchor categories and their interpretation.

B Anchor Words

  • The uncertainty set uses 61 seed cues from Chen et al. [2018], while the certainty set uses 348 words rated at or above the corpus mean by Rocklage et al. [2023].The two sets supply doubt and confidence registers whose subtraction removes shared components.Source: B Anchor Words · p. 16
  • The four categories are ambiguity, incompleteness, conflicting evidence, and general uncertainty, with positive and negative anchor counts reported before tokenizer filtering.Words are assigned by similarity to category definitions, then words not encoded as single tokens by a model are dropped.Source: B Anchor Words · p. 17
  • The certainty anchors combine explicit certainty terms with words marking a confident register, while uncertainty anchors are dominated by each category’s expected vocabulary.The table shows ten anchors per side whose pulled-back directions align best with their side’s centroid, averaged across three models.Source: B Anchor Words · p. 17, Table 3 · p. 17
  • 68.6% AUROC on cue-free Gemma 4 traces, versus 68.4% on the full set, shows U-LENS performance is essentially unchanged without anchor words.The cue-free subset comprises 77% of Gemma 4 traces.Source: B Anchor Words · p. 17
  • Without length matching, A+ reaches 55.9%, 60.4%, and 63.4% AUROC on Gemma 4, Qwen3.5, and Magistral 1.1, respectively, but trails generation length.Under length-matched evaluation, A+ remains below Acone, while subtracting A− drives the keyword score below chance.Source: B Anchor Words · p. 17

C Interpretability of Contrastive Activation Differences

Contrastive activation differences recover category-aligned uncertainty vocabulary from prompts that omit explicit uncertainty terms. The same U-SPACE construction also reads category-specific emotion in GoEmotions, including comments without curated anchor words.

C Interpretability of Contrastive Activation Differences

  • Matched prompt pairs differ only in whether a specific uncertainty issue is present, and exclude the target epistemic vocabulary to limit lexical leakage.Twelve pairs are constructed per category; each category direction averages uncertain-minus-resolved residual states and ranks vocabulary by cosine with pulled-back token directions.Source: C Interpretability of Contrastive… · p. 18
  • Contrastive directions recover category-aligned vocabulary: ambiguity and conflicting-evidence terms rank highly, while incompleteness is dominated by inability terms.The three direction-pair cosines are 0.51, 0.60, and 0.65; simple English prompts also surface aligned terms in other languages.Source: C Interpretability of Contrastive… · p. 18, Figure 6 · p. 19

D.1 Emotion Readout

  • The emotion basis exceeds 70% AUROC on all four targets, while performance remains essentially unchanged on comments containing no curated anchor.The lexical baseline reaches only 51.3%–66.7%, and twenty random orthonormal bases score at chance.Source: §D.1 · p. 20, Table 5 · p. 20
  • Within negatively labeled comments, each emotion’s own axis separates that target from other negative comments at or above 67% AUROC, supporting category-specific readout.For sadness, the generic negative-affect axis is anti-predictive while the sadness axis identifies sad comments.Source: Table 6 · p. 20, §D.1 · p. 21

D.2 Reward-Hacking Readout

A basis built from model-decoded reward-hacking vocabulary separates held-out responses and unseen cheat methods across three models. Its signal reflects absurdity or pretense, not an intent to cheat.

D.2 Reward-Hacking Readout

  • The adapted basis separates held-out responses and unseen cheat methods across all three models.It uses the 15 highest-probability tokens from 70 construction pairs as anchors and the Brown-word centroid as the negative pole; evaluation uses the remaining pairs and held-out cheat methods.Source: §D.2 · pp. 21–22, Table 7 · p. 22
  • The strongest separations involve judge-addressed sentences, stuffed adjectives, and jargon, while plausible fabricated citations and excess politeness separate least.The turns are teacher-forced, and the axis is blind to cheats that do not look strange; decoded judgments include “absurdity” and “pretense.”Source: §D.2 · pp. 21–22, Figure 7 · p. 21

E Experimental Details

The evaluation uses three reasoning models, four benchmarks, and recorded generations scored with single-pass, multi-pass, and supervised uncertainty baselines. The section documents benchmark setup and how each baseline is implemented.

E Experimental Details

  • The study evaluates Qwen3.5, Gemma 4, and Magistral 1.1 on four benchmarks, sampling 1,000 items per benchmark with specified difficulty subsets where applicable.Generation uses three seeds and model-specific token budgets; scored responses require completion within budget and an extractable answer.Source: E Experimental Details · pp. 22–23, Table 8 · p. 23, Table 9 · p. 23
  • U-LENS reads the thinking-block entropy factor and end-of-thinking cone alignment; length-controlled metrics are computed within ten generated-token quantile bins.Single-pass baselines use next-token distributions from one teacher-forced pass over the recorded generation.Source: E Experimental Details · p. 23

E.1 Baseline Implementations

  • Single-pass baselines use quantities from normal generation, whereas multi-pass baselines require extra forward passes or generations.Examples include predictive entropy and DeepConf among single-pass methods, and P(True), TokUR, Semantic Entropy, and SentenceSAR among multi-pass methods.Source: §E.1 · pp. 23–24
  • Supervised baselines fit on correctness labels, with separate fitting and validation responses before scoring; the listed methods include probes, Feature-Gaps, and mass-mean.Per benchmark, 256 responses fit each method and 256 select hyperparameters; mass-mean projects onto the incorrect-minus-correct training-state difference.Source: §E.1 · p. 24

F.1 Trace-Length Analysis

Reasoning length is associated with correctness and can affect uncertainty scores, but the controlled padding intervention finds that U-LENS cone alignment does not grow or collapse with token count. Across the additional analyses, direct J-LENS token ranking performs poorly, transfer varies by model and method, and token-level readouts expose category-wise signals.

F.1 Trace-Length Analysis

  • Incorrect traces are 1.33×–4.79× longer by median across the four benchmarks, but their length distributions overlap substantially.The controlled intervention fixes the question, answer, and correctness while inserting task-irrelevant counting tokens; extensive and intensive scores respond differently to this padding, whereas Acone changes modestly and then saturates.Source: §F.1 · pp. 24–26, Figure 8 · p. 25

F.2 J-LENS-Based Method Ablation

  • At n = 15, at most 0.1% of traces contain a decoded uncertainty term; the best length-matched J-LENS AUROC is 54.8% on Gemma 4, 13.6 points below U-LENS.Even at n = 100, terms appear in only 10.0% of Gemma 4 traces and 1.2% and 1.4% of Qwen3.5 and Magistral 1.1 traces.Source: §F.2 · p. 26, Table 11 · p. 26

F.3 Model-Specific Unadjusted Results

  • Generation length reaches 70.1%, 67.7%, and 67.8% AUROC on Gemma 4, Qwen3.5, and Magistral 1.1; model-level rankings differ, with U-LENS leading all three unadjusted metrics on Qwen3.5.Across models, U-LENS leads unadjusted AUROC while Acone slightly leads AUPRC and AURC; length-controlled results are distinguished from total predictive utility that includes trace-duration information.Source: §F.3 · p. 26, Table 12 · p. 27

F.4 Transferability of Supervised Methods

  • After transfer, probes achieve only 59.7%–69.1% unadjusted AUROC, corresponding to 9.8–22.1-point drops; under length control, U-LENS exceeds mass-mean on all three models.Length-controlled U-LENS scores were 68.4%, 71.0%, and 68.3%, versus mass-mean’s 67.3%, 63.1%, and 62.4%; supervised values use seed 42, while U-LENS values are three-seed means.Source: §F.4 · p. 28

G Token-Level Readout

  • Token-level readouts show one row per uncertainty category and one cell per generated token, with each cell giving that category’s component of the U-LENS state.The four rows decompose the scalar cone score; displayed series are centered on a corpus baseline and smoothed over ten tokens.Source: G Token-Level Readout · pp. 28–29, Figure 9 · p. 29

H Causal Intervention

The causal interventions steer Qwen3.5 along individual uncertainty-category directions at selected blocks. They vary whether steering is applied while reading the prompt or within an already-generated thinking trace, and examine responses across steering strengths.

H Causal Intervention

  • Interventions apply Equation (5) at blocks 32–54 of 64, using Qwen3.5, the Section 3.1 basis, Table 9 sampling settings, and a fixed seed.The intervention span and whether thinking is enabled vary by figure.Source: H Causal Intervention · pp. 29–30
  • With thinking disabled, the direction is added while the model reads the prompt, and the continuation is its answer.The intervention span is the prompt.Source: H Causal Intervention · p. 29
  • With thinking enabled, steering is applied to a 50-token window in the thinking block, beginning at an already-produced token so traces are identical before intervention.Source: H Causal Intervention · p. 30
  • Figures 13–16 vary steering strength for ambiguity, incompleteness, conflicting evidence, and general uncertainty, respectively.The tested λ values are 0.04, 0.08, and 0.09 for ambiguity; 0.04, 0.08, and 0.09 for incompleteness; 0.04, 0.06, and 0.12 for conflicting evidence; and 0.03, 0.10, and 0.17 for general uncertainty.Source: H Causal Intervention · p. 30, Figure 13 · p. 31, Figure 14 · p. 32
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