From Tokens to Semantics: Leveraging Complementary Signals for Hallucination Detection in Black-Box LLMs

# From Tokens to Semantics: Leveraging Complementary Signals for Hallucination Detection in Black-Box LLMs
Source: [https://arxiv.org/html/2609.02679](https://arxiv.org/html/2609.02679)
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AI4Good Workshop, NeurIPS 2026

###### Abstract

When LLMs support public\-facing or high\-stakes workflows, missed fabrications can harm users and institutions, while false alarms consume limited human\-review capacity\. When no trusted context or reference document is available, we study two signals accessible through black\-box model APIs: semantic entropy, which measures disagreement among sampled response meanings, and uncertainty derived from token log\-probabilities\. Their failure modes can be complementary: semantic entropy becomes uninformative when responses form one semantic cluster, while token uncertainty can miss consistently confident errors\. We extend token\-based uncertainty detection by aggregating token\-level signals across sampled responses through ourTopKmethod, evaluate the hybrid CoCoA method, which combines target\-response uncertainty with semantic dissimilarity, and propose and study two supervised methods:Gated, which routes single\-cluster cases to an aggregated\-token\-feature classifier, andStacked, which learns jointly from semantic uncertainty and broader token features\. We evaluate seven benchmarks—including five public benchmarks \(four text datasets and multimodal handwritten\-cheque extraction\) and two constructed benchmarks, Financial Summaries and Long\-Text QA with four language models\. In our evaluation across models and datasets, Stacked gave the best performance in nearly half of the cases, while TopK and CoCoA remain competitive without supervised training labels, although their thresholds require careful calibration\. No method is universally strongest\. We therefore evaluate performance at false\-positive\-rate budgets from 1% to 15%, assess their sensitivity to generation and calibration choices, and examine variation across dataset characteristics\.

## 1Introduction

Large language models increasingly support information workflows in regulated organizations and public\-facing services\([Sajadieh et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib25)\)\. In these settings, fabricated claims or incorrect figures can mislead downstream reviewers and users, while false alarms consume limited human\-review capacity\. Reference\-based methods can check outputs against authoritative context, but open\-ended tasks may provide no trusted context or complete reference document at inference time\. Proprietary models are also commonly accessed through black\-box APIs that expose generated text and sometimes token log\-probabilities, but not hidden states\([Kossen et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib23);[Wang et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib18)\)\. We therefore study whether these observable signals can identify hallucinated responses without trusted reference evidence or model\-internal access\.

Two API\-visible signal families are especially relevant\. Semantic entropy \(SE\) samplesNNresponses, groups them by meaning, and measures uncertainty from their distribution across semantic clusters\([Kuhn et al\., 2023](https://arxiv.org/html/2609.02679#bib.bib11);[Farquhar et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib10)\)\. Token\-log\-probability methods instead estimate confidence within a response using sequence likelihood, perplexity, top\-kkentropy, and candidate margins \(the differences between the highest and second\-highest token log\-probabilities\)\([Kadavath et al\., 2022](https://arxiv.org/html/2609.02679#bib.bib15);[Duan et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib19);[Shapiro et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib16)\)\. Because SE already requires multiple responses, we compute token signals for each response and aggregate them across the same samples\. Our scalarTopKscore combines mean top\-kkentropy with cross\-response confidence variation\. We also evaluate CoCoA\([Vashurin et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib8)\), a hybrid score that combines target\-response confidence with semantic dissimilarity to sampled alternatives\.

The two signal families fail differently\. SE becomes zero when all samples form one semantic cluster, even if they are incorrect\. Token features can be informative in some such cases but can miss consistently confident hallucinations; conversely, semantic disagreement can expose errors that token probabilities do not\. This complementarity motivates two supervised combinations\. The*Gated cascade*uses semantic uncertainty when multiple clusters are present and routes single\-cluster cases to an aggregated\-token classifier\. The*Stacked classifier*instead learns jointly from both signal families for every query \(see Figure[1](https://arxiv.org/html/2609.02679#S1.F1)\)\.

We evaluate seven benchmarks—including five public benchmarks \(four text datasets and multimodal handwritten\-cheque extraction\) and two author\-constructed benchmarks, Financial Summaries and Long\-Text QA—using four language models where supported\. In our evaluation across models and datasets, Stacked gave the best performance in nearly half of the cases, while TopK and CoCoA remain competitive without supervised training labels, although their thresholds require careful calibration\. No method is universally strongest\. We therefore evaluate performance at false\-positive\-rate budgets from 1% to 15%, assess sensitivity to generation and calibration choices, and examine variation across dataset characteristics \(see Appendix[E](https://arxiv.org/html/2609.02679#A5)\)\. We document the construction of our two benchmarks for reproducibility but do not present them as openly released datasets\.

##### Research questions\.

We ask where semantic and token\-log\-probability signals fail across datasets; whether their complementary information can improve black\-box hallucination detection; and how consistently TopK, Gated, and Stacked perform across datasets, models, false\-positive\-rate budgets, and data characteristics\. Our findings provide conditional guidance for the evaluated setting types\.

\(a\) Semantic clusteringC1C\_\{1\}C2C\_\{2\}C3C\_\{3\}high SE→\\rightarrowhallucination✓\(b\) Token log\-probabilitiesone generationrir\_\{i\}high token uncertainty→\\rightarrowhallucination✓\(c\) Semantic\-clusteringfailureC1C\_\{1\}\(all wrong\)r1r\_\{1\}r2r\_\{2\}rNr\_\{N\}SE=0\\mathrm\{SE\}\{=\}0✗token uncertaintyhigh✓\(d\) Token\-log\-probabilityfailureHtopkH\_\{\\mathrm\{topk\}\}low✗\|S\|=2\|S\|\{=\}2✓\(e\) Multi\-response TopKNNresp\.tokenlog\-probsaggregateacrossNNUtopk=mean⁡\(Htopk\)U\_\{\\mathrm\{topk\}\}=\\operatorname\{mean\}\(H\_\{\\mathrm\{topk\}\}\)\+Var⁡\(log⁡p¯\)\+\\operatorname\{Var\}\(\\overline\{\\log p\}\)\(f\) Gated cascadeNNresp\.\|S\|\|S\|?SE scoretoken\-featureclassifier\>1\>1=1=1P⁡\(hallu\)P\(\\mathrm\{hallu\}\)\(g\) Stacked classifiersemantic \+spectralaggregatedtokencombinefeaturesPCA \+logisticP⁡\(hallu\)P\(\\mathrm\{hallu\}\)Figure 1:Complementary black\-box signals and methods\.\(a–d\)Semantic and token uncertainty expose complementary failures\.\(e–g\)TopK aggregates token uncertainty across responses, Gated routes by cluster count, and Stacked learns jointly from semantic and token features\.

## 2Complementary Black\-Box Hallucination Signals

### 2\.1Related work

##### Semantic Entropy\.

Semantic entropy \(SE\)\([Kuhn et al\., 2023](https://arxiv.org/html/2609.02679#bib.bib11);[Farquhar et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib10)\)drawsNNresponses to a promptxx, groups them into semantic equivalence classes𝒞=\{C1,…,CK\}\\mathcal\{C\}=\\\{C\_\{1\},\\ldots,C\_\{K\}\\\}, and measures uncertainty over the empirical class distribution:

SE\(x\)=−∑k=1Kp^\(Ck\)logp^\(Ck\),p^\(Ck\)=\|Ck\|/N\.\\mathrm\{SE\}\(x\)=\-\\sum\_\{k=1\}^\{K\}\\hat\{p\}\(C\_\{k\}\)\\log\\hat\{p\}\(C\_\{k\}\),\\qquad\\hat\{p\}\(C\_\{k\}\)=\|C\_\{k\}\|/N\.\(1\)Standard SE has two limitations: finite sampling may omit plausible semantic classes, while hard clustering discards graded similarities\. Alphabet\-size corrections estimate classes unobserved in the sample\([McCabe et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib7)\)\. Pairwise\-similarity methods retain graded information: Kernel Language Entropy uses a semantic\-similarity kernel, while Semantic Nearest Neighbor Entropy \(SNNE\) uses nearest\-neighbour similarities without discrete classes\([Nikitin et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib12);[Nguyen et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib13)\)\. Spectral Uncertainty, which we evaluate, uses the response\-similarity graph spectrum to retain graded information and separate aleatoric and epistemic uncertainty\([Walha et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib9)\)\.

Semantic Energy augments semantic classes with penultimate\-layer logits but requires white\-box access\([Ma et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib22)\)\. Semantically Diverse Language Generation \(SDLG\) elicits diverse alternatives from one model, while cross\-model disagreement compares similarities within and across models\([Aichberger et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib21);[Hamidieh et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib14)\)\. These methods enrich semantic variation but may provide limited evidence when the same wrong meaning repeats, motivating the token\-level evidence considered next\.

##### Token\-level uncertainty\.

Token\-probability methods assess confidence within a generation rather than disagreement among response meanings\. Basic approaches use sequence likelihood or predictive entropy, while self\-evaluation uses the probability assigned toTrueas confidence\([Malinin and Gales, 2021](https://arxiv.org/html/2609.02679#bib.bib24);[Kadavath et al\., 2022](https://arxiv.org/html/2609.02679#bib.bib15)\)\. Shifting Attention to Relevance \(SAR\) weights uncertainty associated with semantically important tokens and responses\([Duan et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib19)\)\. EAS accumulates entropy along a reasoning sequence, whereas Hallucination Assessment via Log\-probs as Time Series \(HALT\) learns temporal patterns from top\-kktoken log\-probabilities\([Zhu et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib20);[Shapiro et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib16)\)\. HaluNet also uses hidden\-state features, placing it outside our black\-box setting\([Tong et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib17)\)\. Unlike HALT, which models uncertainty across token positions within one response, our methods aggregate token evidence across the multiple responses already sampled for semantic entropy; consistently confident hallucinations nevertheless remain difficult to identify\.

##### Hybrid and higher\-access methods\.

CoCoA is a closely related hybrid method\. It multiplies a target response’s scalar model uncertainty by its average semantic dissimilarity to sampled alternatives, deriving the combination through a Minimum Bayes Risk formulation\([Vashurin et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib8)\)\. We instead aggregate broader token\-probability features across allNNresponses and combine them with query\-level semantic signals\. Other approaches require either additional context or hidden\-state access\([Wang et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib18);[Chen et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib27);[Sriramanan et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib28);[Min et al\., 2023](https://arxiv.org/html/2609.02679#bib.bib26);[Es et al\., 2024](https://arxiv.org/html/2609.02679#bib.bib29)\)\.

### 2\.2Proposed methods

The different failure modes of semantic and token signals motivate three methods using the sameNNsampled responses and API\-exposed log\-probabilities\. TopK aggregates token uncertainty across responses\. Gated and Stacked combine token and semantic evidence: Gated routes queries based on whether multiple semantic clusters are present, whereas Stacked learns jointly from both signal families\. None requires reference evidence or model\-internal access\.

##### Multi\-response token representation\.

For each response, we extract token confidence, sequence confidence, positional confidence changes, and response\-length features, then summarise their distributions across theNNresponses using means, variances, quantiles, ranges, and extreme values\. TopK uses only mean top\-kkentropy and across\-response confidence spread, whereas Gated and Stacked use the broader aggregated feature vector\. For analysis, we group log\-probabilities, candidate margins, and entropy astoken confidence; positional confidence changes and response length asresponse dynamics; cross\-sample spreads and extremes asacross\-response aggregation; and response diversity and Von Neumann entropy assemantic diversity\.

##### Gated cascade\.

The Gated cascade uses the semantic\-cluster countKKfor routing\. WhenK≥2K\\geq 2, it returns the selected semantic entropy score; whenK=1K=1, empirical SE is zero by construction, so it uses a logistic\-regression classifier over the aggregated token features:

score​\(x\)=\{SEm​\(x\),K≥2,σ⁡\(𝐰⊤​𝐟tok​\(x\)\+b\),K=1\.\\text\{score\}\(x\)=\\begin\{cases\}\\mathrm\{SE\}\_\{m\}\(x\),&K\\geq 2,\\\\ \\sigma\\\!\\left\(\\mathbf\{w\}^\{\\top\}\\mathbf\{f\}\_\{\\mathrm\{tok\}\}\(x\)\+b\\right\),&K=1\.\\end\{cases\}\(2\)Here,𝐟tok​\(x\)\\mathbf\{f\}\_\{\\mathrm\{tok\}\}\(x\)is the token\-feature vector aggregated across theNNresponses, andσ\\sigmaconverts the classifier output into a hallucination probability\.Gated \(Hybrid\)uses Hybrid SE on the semantic branch, whereasGated \(Spectral\)uses Von Neumann entropy\. They share the same token classifier and differ only in their semantic entropy score\.

##### Stacked classifier\.

The Stacked classifier combines the aggregated token profile and cluster count with one semantic feature block:Hybrid, Spectral,orVon Neumann\. The features are standardised, reduced using principal component analysis \(PCA\), and passed toL2L\_\{2\}\-regularised logistic regression, allowing both signals to influence every prediction \(for details on feature inputs see Appendix[C\.4](https://arxiv.org/html/2609.02679#A3.SS4)\)\.

## 3Methodology and Experimental Setup

For each query, we compute several semantic uncertainty scores and the multi\-response token representation from Section[2\.2](https://arxiv.org/html/2609.02679#S2.SS2.SSS0.Px1), evaluating the measures independently as well as jointly via the Gated and Stacked approaches\. This section defines the variants, datasets, models, training procedure, and metrics\.

### 3\.1Methods and Variants

Table[1](https://arxiv.org/html/2609.02679#S3.T1)provides an overview of the selected methods and how their measures are used through the Gated and Stacked approaches\.

Table 1:Overview of the 13 evaluated methods\.Descriptions summarise the uncertainty score or classifier inputs used by each method\.We embed theNNresponses usingtext\-embedding\-3\-largeand use pairwise cosine similarity to measure their semantic similarity\. For Standard SE, UEigV, and Hybrid SE, a dataset\-specific thresholdτ\\taudetermines whether a response is assigned to an existing semantic class\. We selectτ\\tauusing separate validation data and fix it before five\-fold cross\-validation; the validation examples are excluded from the evaluation folds\. Von Neumann and spectral scores use the same similarity matrix directly\. Appendix[A\.1](https://arxiv.org/html/2609.02679#A1.SS1)contains full implementation details\.

### 3\.2Datasets

We evaluate seven benchmarks spanning factual, ambiguous, multi\-hop, extractive, long\-context, financial, and multimodal settings\. Five are public, while Financial Summaries and Long\-Text QA are author\-constructed\. We retain only queries with a valid binary hallucination label; Table[2](https://arxiv.org/html/2609.02679#S3.T2)summarises the datasets, citations, task descriptions, and resulting sample sizes\. It also characterises each dataset by context complexity, answer complexity, reasoning depth, and domain specificity\. These human\-assigned ratings use a 0–5 ordinal scale and provide qualitative context \(further details on dataset construction and licenses are in Appendix[C](https://arxiv.org/html/2609.02679#A3)\)\.

Table 2:Dataset profiles and hyperparameters\.nnis the number of valid labelled examples used in five\-fold cross\-validation\. Complexity values are qualitative 0–5 ratings defined below\.τ\\tauis the cosine\-similarity clustering threshold, andCCis theL2L\_\{2\}regularisation strength for Gated \(G\) and Stacked \(S\)\.Ctx\(context\): 0 = none, 1 = inline, 2 = short doc, 3 = long doc, 4 = multi\-doc, 5 = multi\-modal\.Ans\(answer\): 0 = binary, 1 = short factual, 2 = structured, 3 = multi\-answer, 4 = paragraph, 5 = multi\-part\.Reas\(reasoning\): 0 = lookup, 1 = single\-hop, 2 = multi\-hop, 3 = synthesis, 4 = judgement, 5 = creative\.Dom\(domain\): 0 = general, 1 = semi\-general, 2 = domain\-specific, 3 = expert, 4 = niche expert, 5 = regulated\.

### 3\.3Models, Training, and Evaluation

##### Models\.

We evaluate GPT\-4\.1\-mini, GPT\-5\.1, GPT\-5\.4, and Llama 3\.3 70B where supported\. Of the 28 possible dataset–model pairs, 26 contain both label classes and yield AUROC estimates\. The labelled Llama 3\.3 70B outputs for Long\-Text QA contain only the hallucinated class, so AUROC is undefined, while Cheque Generation requires vision capability and is not evaluated with the vision\-incompatible Llama model\.

##### Generation and training\.

The default setting usesN=10N=10responses per prompt, temperature1\.01\.0, andk=5k=5requested log\-probability candidates per token\. GPT\-5\.4 returned only one candidate, giving effectivek=1k=1token results\. All transformations and classifiers are fitted on the training portion of each stratified five\-fold split and evaluated on held\-out predictions\.

##### Baselines\.

We compare thirteen methods: four SE baselines \(Standard SE, SE UEigV, SE Hybrid, and SE Von Neumann\); TopK, Spectral Epistemic, CoCoA SP, and CoCoA PPL, which do not use supervised training labels; and five supervised methods \(Gated Hybrid, Gated Spectral, Stacked Hybrid, Stacked Spectral, and Stacked Von Neumann\)\. CoCoA SP and CoCoA PPL combine sampled\-response consistency with sequence\-probability \(SP\) and perplexity confidence \(PPL\), respectively\.

##### Evaluation metrics\.

For each dataset, we compute AUROC separately within each of the five held\-out folds and report the mean across folds\. We separately pool the out\-of\-fold predictions from all five folds into a single set, which we use for query\-level bootstrap confidence intervals and other supporting analyses\. We also report true\-positive rate at false\-positive\-rate budgets from 1% to 15%\. These operating points trade hallucination coverage against false positives, review capacity, and the relative costs of the two error types\. The leading method’s 95% confidence interval overlaps at least one competitor on every dataset, so close rankings are not conclusive \(Appendix[F\.2](https://arxiv.org/html/2609.02679#A6.SS2)reports confidence intervals, and Appendix[F\.1](https://arxiv.org/html/2609.02679#A6.SS1)reports significance tests\)\.

##### Sensitivity analyses\.

We vary the response countN∈\{3,5,7,10,15,20\}N\\in\\\{3,5,7,10,15,20\\\}, temperature in\{0\.3,0\.5,0\.7,1\.0,1\.2\}\\\{0\.3,0\.5,0\.7,1\.0,1\.2\\\}, returned candidatesk∈\{1,…,5\}k\\in\\\{1,\\ldots,5\\\}, clustering thresholdτ∈\{0\.80,0\.85,0\.90,0\.95,0\.98\}\\tau\\in\\\{0\.80,0\.85,0\.90,0\.95,0\.98\\\}, and regularisation strengthC∈\{0\.1,1,10\}C\\in\\\{0\.1,1,10\\\}\. Cross\-model comparisons reuse the selectedτ\\tauandCCvalues\. Section[4](https://arxiv.org/html/2609.02679#S4)summarises the findings, and Appendix[E](https://arxiv.org/html/2609.02679#A5)reports the complete ablation results\.

## 4Results and Discussion

Across seven benchmarks and four models, we find complementary failure modes for semantic disagreement and token uncertainty\. Among hallucinated queries, the percentage of queries where sampled responses form one semantic cluster ranges from 39% on AmbigQA to 99% on Financial Summaries, forcing standard SE to zero\. Within these single\-cluster cases, median TopK uncertainty is higher for hallucinated than non\-hallucinated queries on six of seven datasets\. The distributions still overlap: depending on the dataset, 21–56% of hallucinated single\-cluster queries have lower TopK uncertainty than the median non\-hallucinated query\. Thus, token evidence remains informative in many cases where semantic disagreement disappears, although both signals can fail when a hallucination is semantically consistent and generated with high token confidence \(See Appendix[D\.1](https://arxiv.org/html/2609.02679#A4.SS1)for the failure mode analysis\)\.

### 4\.1Performance patterns across methods, datasets, and models

We group the methods into five families: semantic or spectral scores, TopK, CoCoA scores, Gated classifiers, and Stacked classifiers\. Supervised classifiers lead four of seven GPT\-4\.1\-mini comparisons, four of five valid Llama comparisons, and three of seven GPT\-5\.1 comparisons\. Methods without supervised training labels—TopK, CoCoA, and semantic or spectral scores—lead or share the lead on six of seven GPT\-5\.4 comparisons\. The GPT\-5\.4 results suggest that graded semantic measures can become more informative in particular datasets, although this pattern is not uniform and does not establish that model capability causes the improvement\. SE Von Neumann changes sharply with the generating model, rising from 0\.468 to 0\.750 on Financial Summaries and from 0\.581 to 0\.774 on Cheque Generation between GPT\-4\.1\-mini and GPT\-5\.4\. AmbigQA is the clearest exception to the supervised pattern: CoCoA leads all four valid model comparisons\. Among models with broad text\-dataset coverage, Llama shows the weakest overall separation: the best method for each of its five valid comparisons averages about 0\.64 AUROC\. Together, these shifts show that the most informative uncertainty signal depends on the generating model as well as the dataset\.

Table 3:GPT\-5\.4 AUROC by method and dataset\.Values are mean held\-out AUROC over five folds\. Text datasets useN=10N=10and Cheque Generation usesN=20N=20\. The API returned one candidate per token, giving an effectivek=1k=1for token\-based methods\. Bold marks the highest value per dataset\.Figure[2](https://arxiv.org/html/2609.02679#S4.F2)shows that low\-FPR performance is strongly dataset\-dependent\. At strict 1–3% FPR budgets, the leading methods are predominantly token\-based or supervised combinations: TopK leads on AmbigQA and HotpotQA, Stacked on AA Omni Finance and SQuAD, and Gated on Cheque Generation\. At higher budgets, CoCoA becomes strongest on AmbigQA, while the leading family also changes on HotpotQA and Cheque Generation\. Stacked remains strongest on AA Omni Finance and SQuAD\. Increasing the FPR budget from 5% to 15% substantially improves hallucination coverage on AA Omni Finance, SQuAD, and Cheque Generation, but produces little improvement on Financial Summaries or Long\-Text QA\. Thus, both the preferred method family and the benefit of relaxing the threshold depend on the dataset\.

Across the 26 comparisons, Stacked leads or shares the lead in 11, CoCoA in seven, TopK in five, Gated in three, and semantic or spectral scores in one; the counts sum to 27 because Stacked and CoCoA tie in one setting\. Because the leading method changes across models, datasets, and FPR budgets, win counts alone do not show how consistently a method performs\. We therefore also measure how far each family falls below the best method in the same model–dataset comparison\. Figure[3](https://arxiv.org/html/2609.02679#S4.F3)shows that Stacked remains consistently close to the leader: its strongest variant is within 0\.05 AUROC of the best method in 20 of 26 comparisons and within 0\.02 in 16\. TopK and CoCoA have the next\-smallest median shortfalls, indicating that they remain competitive in more specific settings even when they do not lead \(see Appendix[E\.1](https://arxiv.org/html/2609.02679#A5.SS1)for the model\-family ablation results\)\.

Figure 2:TPR at fixed FPR budgets\.Boxes pool five\-fold TPR values across available model runs at fixed calibration\. Text runs useN=10N=10andT=1\.0T=1\.0; Cheque Generation usesN=20N=20and includes GPT\-4\.1\-mini, GPT\-5\.1, and GPT\-5\.4\. Long\-Text QA hasn=30n=30\.Figure 3:Method\-family AUROC shortfall\.Shortfall is measured from the best method evaluated on the same dataset and model; lower is better\. Boxes summarise the available model runs, whose coverage differs because of missing labels and modality support\.
### 4\.2Why performance varies across datasets

Figure[4](https://arxiv.org/html/2609.02679#S4.F4)compares how well the four feature groups used by the Stacked classifier \(defined in Section[2\.2](https://arxiv.org/html/2609.02679#S2.SS2.SSS0.Px1)\) separate hallucinated from non\-hallucinated responses\. Separation is measured using absolute Cohen’sdd, the standardised difference between the two groups’ feature means\. Here, the 23 settings are the canonical configuration plus response\-count, temperature, generating\-model, top\-kk, and clustering\-threshold ablations; they are distinct from the 26 model–dataset comparisons above\. Token\-confidence or response\-dynamics features provide the strongest univariate separation in 121 of 148 ablations, across\-response aggregation in 20, and semantic\-diversity features in seven\. In AA Omni Finance, token confidence leads 15 of 23 settings: one evaluated question expects “December 15, 2022,” while the hallucinated response changes only the year to 2021\. Such exact\-value errors can provide informative token\-probability\-based features\. In AmbigQA, for “How many jury members \[are\] in a criminal trial?”, the accepted answers include 6, 7, 12, and 15 because the answer depends on jurisdiction\. Variation among sampled responses is therefore meaningful rather than automatically erroneous, and the strong results for semantic diversity features \(used in CoCoA and SE Von Neumann\) show the value of retaining graded similarities\. Financial Summaries combines similar fictional names and nearby figures, and the substitutions were generated confidently across samples; therefore, both token and semantic features provided weak separation\. In SQuAD, response\-dynamics features lead 22 of 23 settings, with response length providing the strongest individual separation\. The benchmark expects short extractive spans\. For example, the correct answer to “When did Beyoncé start becoming popular?” is “in the late 1990s,” while a longer multi\-clause response differs in length even when it contains the correct fact\. Unlike most other datasets, where the separating gap in Figure[5](https://arxiv.org/html/2609.02679#S4.F5)only emerges later in the response, SQuAD’s gap is concentrated near the opening\. Response length and where confidence changes occur therefore provide more separation than token\-confidence features on this dataset\.

Figure[5](https://arxiv.org/html/2609.02679#S4.F5)compares how token uncertainty develops across the response rather than reducing each response to a single average\. In AA Omni Finance, hallucinated responses have higher entropy across most token positions, indicating that the separation is distributed throughout the answer\. In HotpotQA, the gap widens later: for a question asking when the university where Sergei Aleksandrovich Tokarev taught was founded, the model must first identify the university before producing “1755\.” Long\-Text QA shows a similar late\-response pattern, although this result is uncertain because the dataset contains only 30 question sets\. SQuAD differs from both: its entropy ordering is inverted near the opening and largely converges later\. So, averaging entropy across the full response obscures this early distinction\. Response\-dynamics features instead retain information about response length and how confidence changes across token positions, which provides stronger separation on SQuAD\. Cheque Generation and Financial Summaries show substantial overlap across most positions, demonstrating how whole\-response averages can obscure local uncertainty\. For cheque amounts, for example, uncertainty may be concentrated at one visually confusable digit and is better represented by the smallest difference between the two most probable token candidates \(see Appendix[D\.2](https://arxiv.org/html/2609.02679#A4.SS2)for the complete token\-level analysis\)\.

### 4\.3Conditional method and threshold selection

The results support selecting both the method and its operating point for the target setting\. With labelled target\-domain data, Stacked is a strong starting point because it uses semantic and token evidence for every query and usually remains close to the leading method; without supervised training labels, TopK and CoCoA are useful alternatives, but their thresholds still require calibration on representative data\. The ablations show where additional API information is worth its cost\. On AA Omni Finance, increasing the retained token candidates from one to three raises TopK’s performance from 0\.621 to 0\.721 AUROC and average Stacked AUROC from 0\.634 to 0\.741, consistent with alternatives helping around precise dates, amounts, and entities\. The same change leaves average Stacked essentially unchanged on SQuAD \(0\.767 to 0\.768\) and is non\-monotonic on Long\-Text QA\. Sampling ten responses gives the highest mean AUROC across the six shared text datasets, but individual optima vary\. The temperature sweep also shows opposite dataset\-level preferences\. For average Stacked, AmbigQA peaks at 0\.700 AUROC with a temperature of 0\.5 and falls to 0\.461 at 1\.2, whereas HotpotQA reaches its highest AUROC of 0\.706 at 1\.2, compared with 0\.634 at 0\.5\. This is consistent with HotpotQA’s single\-cluster rate falling from 0\.89 to 0\.79 as temperature rises, giving semantic\-diversity features more multi\-cluster cases; AmbigQA does not follow the same pattern, since its single\-cluster rate is highest, not lowest, at the temperature where its AUROC peaks\. Thus, increasing sampling randomness does not consistently improve separation: the useful degree of response variation depends on the dataset\. Overall calibration is stable for Stacked on AA Omni Finance and SQuAD but much more variable on the small Long\-Text QA dataset\. These results motivate choosing response count, token\-candidate count, temperature, and threshold jointly with the dataset, model, available labels, false\-positive cost, and review capacity \(see Appendix[E](https://arxiv.org/html/2609.02679#A5)for the complete generation and calibration sweeps\)\.

Figure 4:Discriminative associations of feature groups by dataset\.Bars show the median, across ablation runs, of the largest absolute Cohen’sddwithin each group; whiskers show the interquartile range\.Figure 5:Positional top\-kkentropy profiles\.Mean top\-kkentropy over normalised token position with±1\\pm 1standard\-error bands\. Separation is persistent for AA Omni Finance, emerges later for several QA tasks, is weak for Cheque and Financial Summaries, and is inverted near the opening of SQuAD responses\.

## 5Limitations

Gated and Stacked require labelled target\-domain hallucination data; zero\-shot transfer is not evaluated\. Long\-Text QA has only 30 questions, and leading confidence intervals overlap across all datasets\. Token methods require API\-exposed log\-probabilities; GPT\-5\.4 returned one candidate per token \(Appendix[F](https://arxiv.org/html/2609.02679#A6)further reports significance tests and bootstrap confidence intervals\)\.

## 6Conclusion

We studied black\-box hallucination detection when no reference evidence is available, using semantic entropy and token log\-probabilities from sampled responses\. For institutional deployments, our FPR analysis across seven datasets and four models shows that the two signal families provided complementary information, although both failed when hallucinations were semantically consistent and generated with high token confidence\. Stacked led or shared the lead in 11 comparisons and remained within 0\.05 AUROC of the best method in 20, making it a strong starting point when labelled target\-domain data are available\. Without labels, TopK and CoCoA were competitive alternatives, subject to careful threshold calibration\. Performance nevertheless varied across datasets, generating models, and false\-positive\-rate budgets, while response count, temperature, and token\-candidate count did not produce uniform improvements\. These findings support selecting the method, generation configuration, and operating threshold jointly for the target setting\. Future work should examine transfer to unseen domains\.

## Acknowledgments and Disclosure of Funding

## References

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## Appendix

The appendix is organised into six parts\. Appendix[A](https://arxiv.org/html/2609.02679#A1)provides further details on the evaluated uncertainty estimation methods, including CoCoA and the Gated and Stacked algorithms\. Appendix[B](https://arxiv.org/html/2609.02679#A2)describes the experimental implementation, including the embedding model, clustering configuration, and classifier settings\. Appendix[C](https://arxiv.org/html/2609.02679#A3)documents the datasets, token\-level features, dataset\-construction procedures, and prompts used to obtain hallucination labels\. Appendix[D](https://arxiv.org/html/2609.02679#A4)presents extended diagnostic results, beginning with the complementary failure modes of semantic and token uncertainty and followed by ROC and cross\-model analyses\. Appendix[E](https://arxiv.org/html/2609.02679#A5)reports the response\-count, temperature, top\-kk, model, and calibration ablations, while Appendix[F](https://arxiv.org/html/2609.02679#A6)provides statistical significance tests and bootstrap confidence intervals\. CoCoA is included in direct comparisons, ROC curves, significance tests, bootstrap intervals, and the top\-kktable, and appears as an invariant reference in calibration because it uses neitherτ\\taunorCC\. It is omitted from the response\-count and temperature sweeps because the required main\-answer embeddings are unavailable, and from the externally scored Cheque sweep because no CoCoA scores were produced\.

## Appendix AUncertainty detection methods

This section provides the methodological details omitted from the main text for brevity\. We first describe the semantic and spectral uncertainty measures, followed by TopK, CoCoA, and the Gated and Stacked methods\.

### A\.1Semantic and spectral uncertainty measures

All semantic baselines are computed from the sameNNsampled responses and their pairwise cosine\-similarity matrix\. Standard SE uses the empirical semantic\-cluster distribution defined in Eq\.[1](https://arxiv.org/html/2609.02679#S2.E1)\. The remaining variants either correct for semantic classes that may be unobserved in a finite sample or retain graded similarities instead of reducing each response pair to a binary clustering decision\.

The Good–Turing correction estimates the semantic alphabet size from the number of observed clustersKKand singleton clustersn1n\_\{1\}:

K^GT=K​NN−n1\.\\widehat\{K\}\_\{\\mathrm\{GT\}\}=\\frac\{KN\}\{N\-n\_\{1\}\}\.UEigV instead estimates the effective number of clusters by counting eigenvalues of the normalised similarity\-graph Laplacian below a thresholdϵ\\epsilon\. Hybrid uses the larger of the Good–Turing and UEigV estimates, falling back to UEigV when every observed cluster is a singleton, and recomputes entropy using the corrected alphabet size\[[McCabe et al\., 2026](https://arxiv.org/html/2609.02679#bib.bib7)\]\. Von Neumann entropy avoids a discrete alphabet\-size estimate and computes uncertainty directly from the spectrum of the response\-similarity matrix\[[Walha et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib9)\]\.

### A\.2TopK

For responseii, letHi\(k\)H\_\{i\}^\{\(k\)\}denote its mean token entropy after renormalising the returned candidate probabilities over the availablekkcandidates at each token position, and letℓ¯i\\bar\{\\ell\}\_\{i\}denote its mean chosen\-token log\-probability\. The TopK score is

UTopK​\(x\)=1N​∑i=1NHi\(k\)\+Vari=1,…,N⁡\(ℓ¯i\)\.U\_\{\\mathrm\{TopK\}\}\(x\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}H\_\{i\}^\{\(k\)\}\+\\operatorname\{Var\}\_\{i=1,\\ldots,N\}\\\!\\left\(\\bar\{\\ell\}\_\{i\}\\right\)\.Higher values indicate greater uncertainty\. When only one candidate is returned,Hi\(1\)=0H\_\{i\}^\{\(1\)\}=0, so the score retains only the across\-response confidence\-variation term\.

### A\.3CoCoA

CoCoA combines the confidence of a target responsey∗y^\{\*\}with its semantic disagreement fromNNsampled alternatives\[[Vashurin et al\., 2025](https://arxiv.org/html/2609.02679#bib.bib8)\]:

UCoCoA​\(y∗∣x\)=u⁡\(y∗∣x\)​1N​∑i=1N\[1−s⁡\(y∗,y\(i\)\)\],U\_\{\\mathrm\{CoCoA\}\}\(y^\{\*\}\\mid x\)=u\(y^\{\*\}\\mid x\)\\,\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\left\[1\-s\\\!\\left\(y^\{\*\},y^\{\(i\)\}\\right\)\\right\],wheres⁡\(⋅,⋅\)s\(\\cdot,\\cdot\)is cosine similarity between response embeddings\. We evaluate two variants: CoCoA SP uses sequence\-level uncertainty,uSP=−∑t=1Llogp\(yt∗∣y<t∗,x\)u\_\{\\mathrm\{SP\}\}=\-\\sum\_\{t=1\}^\{L\}\\log p\(y\_\{t\}^\{\*\}\\mid y\_\{<t\}^\{\*\},x\), whereas CoCoA PPL uses its length\-normalised form,uPPL=−L−1∑t=1Llogp\(yt∗∣y<t∗,x\)u\_\{\\mathrm\{PPL\}\}=\-L^\{\-1\}\\sum\_\{t=1\}^\{L\}\\log p\(y\_\{t\}^\{\*\}\\mid y\_\{<t\}^\{\*\},x\)\. Neither uses supervised training labels, and both use the same sampled responses available to the other methods\. Because SP sums log\-probabilities over allLLtokens while PPL divides byLL, SP is sensitive to response length whereas PPL is not\.

### A\.4Gated and Stacked methods

Gated and Stacked combine semantic and token\-level uncertainty in different ways\. Gated uses a semantic score when the sampled responses form multiple clusters and a classifier over aggregated token features when they collapse into one cluster\. Stacked removes this routing rule and learns jointly from the semantic representation, cluster count, and aggregated token features for every query\. Algorithm[1](https://arxiv.org/html/2609.02679#alg1)summarises both procedures\.

Algorithm 1Gated and Stacked uncertainty detection0:Query

xx, language model

pθp\_\{\\theta\}, response count

NN, semantic variant

mm, fitted Gated classifier

ℱG\\mathcal\{F\}\_\{G\}, fitted Stacked pipeline

ℱS\\mathcal\{F\}\_\{S\}, method

dd
0:Uncertainty score

ss
1:Sample responses

\{y\(i\)\}i=1N∼pθ\(⋅∣x\)\\\{y^\{\(i\)\}\\\}\_\{i=1\}^\{N\}\\sim p\_\{\\theta\}\(\\cdot\\mid x\)and retain their token log\-probabilities

2:Embed the responses and compute their pairwise cosine similarities

3:Form semantic clusters

𝒞=\{C1,…,CK\}\\mathcal\{C\}=\\\{C\_\{1\},\\ldots,C\_\{K\}\\\}
4:Compute semantic score

umu\_\{m\}, semantic feature block

𝐟semm\\mathbf\{f\}\_\{\\mathrm\{sem\}\}^\{m\}, and aggregated token features

𝐟tok\\mathbf\{f\}\_\{\\mathrm\{tok\}\}
5:if

d=Gatedd=\\textsc\{Gated\}then

6:if

K≥2K\\geq 2then

7:

s←ums\\leftarrow u\_\{m\}
8:else

9:

s←ℱG​\(𝐟tok\)s\\leftarrow\\mathcal\{F\}\_\{G\}\(\\mathbf\{f\}\_\{\\mathrm\{tok\}\}\)
10:endif

11:elseif

d=Stackedd=\\textsc\{Stacked\}then

12:

𝐟←\[𝐟semm,K,𝐟tok\]\\mathbf\{f\}\\leftarrow\[\\mathbf\{f\}\_\{\\mathrm\{sem\}\}^\{m\},K,\\mathbf\{f\}\_\{\\mathrm\{tok\}\}\]
13:

s←ℱS​\(𝐟\)s\\leftarrow\\mathcal\{F\}\_\{S\}\(\\mathbf\{f\}\)
14:endif

15:return

ss

## Appendix BAdditional experimental details

This section records the implementation choices shared across the evaluated methods, including the embedding model, clustering configuration, and supervised\-classifier settings\. Unless otherwise stated, these settings are fixed within each dataset and applied consistently across method variants\.

##### Response embeddings\.

We embed every sampled response usingtext\-embedding\-3\-largeand compute cosine similarity between the resulting 3,072\-dimensional vectors\. No dimensionality reduction is applied before semantic clustering or construction of the response\-similarity matrix\.

##### Semantic clustering\.

The reported results use greedy representative\-based clustering\. Responses are processed in sampling order and assigned to the first existing cluster whose representative has cosine similarity at leastτ\\tau; otherwise, a new cluster is created\. The dataset\-specific thresholds in Table[2](https://arxiv.org/html/2609.02679#S3.T2)are selected on held\-out validation data and then fixed before cross\-validation\. UEigV uses a Laplacian\-eigenvalue threshold ofϵ=0\.1\\epsilon=0\.1, whereas the Von Neumann and spectral measures operate directly on the continuous similarity matrix\.

##### Supervised classifiers\.

Both supervised methods useL2L\_\{2\}\-regularised logistic regression withmax\_iter=1000\\texttt\{max\\\_iter\}=1000\. Gated standardises the aggregated token features and fits its classifier only on single\-cluster training examples\. Stacked standardises the combined semantic and token feature vector, reduces it to at most 15 principal components, and then fits the classifier\. Scaling, principal\-component analysis, and logistic regression are fitted within each training fold to prevent information leakage\. The dataset\-specific regularisation strengthsCCare reported in Table[2](https://arxiv.org/html/2609.02679#S3.T2)\.

## Appendix CDatasets, features, and labelling resources

This section provides the supporting details required to reproduce the evaluation data and labels\. We first document the public and custom datasets, then list the token\-level features used by the supervised methods, and finally provide the dataset\-construction and hallucination\-judging prompts\.

### C\.1Public datasets

Our public benchmarks cover domain knowledge, ambiguous questions, multi\-hop reasoning, extractive question answering, and multimodal text recognition\. They comprise AA Omni Finance, AmbigQA, HotpotQA, SQuAD, and Cheque Generation\.

1. 1\.AA Omni Finance\[[Team, 2025](https://arxiv.org/html/2609.02679#bib.bib4)\]: the finance and law subset \(∼200\{\\sim\}200questions\) of the AA\-Omniscience cross\-domain knowledge evaluation benchmark \(∼6,000\{\\sim\}6\{,\}000questions total, Apache\-2\.0\), which measures factual accuracy of LLMs across diverse domains\. Each question includes domain, topic, ground\-truth answer, and metadata; answers are graded on a four\-point scale \(Correct / Incorrect / Partial / Not Attempted\)\.
2. 2\.AmbigQA\[[Min et al\., 2020](https://arxiv.org/html/2609.02679#bib.bib1)\]: derived from Google Natural Questions\[[Kwiatkowski et al\., 2019](https://arxiv.org/html/2609.02679#bib.bib6)\], this dataset \(∼12,000\{\\sim\}12\{,\}000questions, CC BY\-SA 3\.0\) targets inherently ambiguous open\-domain questions that admit multiple valid answers depending on interpretation\. Crowdworkers annotated each question with disambiguated question–answer pairs; we retain the original ambiguous question and evaluate against all plausible gold answers\.
3. 3\.HotpotQA\[[Yang et al\., 2018](https://arxiv.org/html/2609.02679#bib.bib2)\]: a multi\-hop QA dataset \(∼7,400\{\\sim\}7\{,\}400dev questions, CC BY\-SA 4\.0\) where answering requires reasoning over multiple Wikipedia paragraphs\. No single paragraph contains the full answer\. Sentence\-level supporting facts are provided for explainability\.
4. 4\.Cheque Generation\[[Verma, 2024](https://arxiv.org/html/2609.02679#bib.bib5)\]: a visual question answering dataset of handwritten cheque images \(Apache\-2\.0\) containing ground\-truth Courtesy Amount \(numeric\) and Legal Amount \(written\-out\) fields\. Models receive cheque images as base64\-encoded multimodal inputs and must perform precise numeric extraction, testing hallucination detection in vision\-language settings\.
5. 5\.SQuAD\[[Rajpurkar et al\., 2016](https://arxiv.org/html/2609.02679#bib.bib3)\]: the Stanford Question Answering Dataset \(100,000\+100\{,\}000\{\+\}questions over500\+500\{\+\}Wikipedia articles, CC BY\-SA 4\.0\), an extractive reading comprehension benchmark where the answer is a contiguous span within the source paragraph\. Provides a context\-grounded baseline where the answer is explicitly present in the input\.

### C\.2Custom datasets

We additionally evaluate Financial Summaries and Long\-Text QA, two custom benchmarks designed to test hallucinations in finance\-related summarisation and document\-grounded question answering\. Their construction procedures and prompts are reported in Appendix[C\.5](https://arxiv.org/html/2609.02679#A3.SS5)\.

1. 1\.Financial Summaries\(finance domain\-specific, 200 sampled,n=171n\{=\}171with a valid hallucination label\): an LLM\-generated summarisation dataset of financial texts covering AML enforcement actions, cross\-border M&A activity, and regulatory proceedings\. Passages are crafted with deliberately confusable near\-duplicate entity names and similar numeric figures to probe whether models introduce cross\-contamination errors during summarisation\.
2. 2\.Long\-Text QA\(finance domain\-specific,n=150n\{=\}150sub\-questions,n=30n\{=\}30question sets used for evaluation\): 30 multi\-part question sets \(5 sub\-questions each\) over regulatory and financial filings, with one deliberately unanswerable sub\-question per set to provide a controlled hallucination signal\. Tests whether models fabricate answers when the source document does not contain the required information\.

Table[4](https://arxiv.org/html/2609.02679#A3.T4)summarises every dataset used in this work together with its license, source URL, sample size, and the modifications we applied\.

Table 4:Datasets used in this paper: licenses, sources, and modifications\.nnhere is the raw number of items sampled/generated before hallucination labelling; Table[2](https://arxiv.org/html/2609.02679#S3.T2)reports the number with a valid label actually used in cross\-validation \(lower for AmbigQA, HotpotQA, Cheque Generation, and Financial Summaries, where some queries failed judging\)\.DatasetLicensennSourceModificationsAmbigQA[Min et al\. \[2020\]](https://arxiv.org/html/2609.02679#bib.bib1)CC BY\-SA 3\.0200[https://huggingface\.co/datasets/sewon/ambig\_qa](https://huggingface.co/datasets/sewon/ambig_qa)Randomly sampled 200 questions; reformatted intoQuestion/Answercolumns\.SQuAD[Rajpurkar et al\. \[2016\]](https://arxiv.org/html/2609.02679#bib.bib3)CC BY\-SA 4\.0200[https://huggingface\.co/datasets/rajpurkar/squad](https://huggingface.co/datasets/rajpurkar/squad)Sampled 200 questions; reformatted with context passages\.HotpotQA[Yang et al\. \[2018\]](https://arxiv.org/html/2609.02679#bib.bib2)CC BY\-SA 4\.0200[https://huggingface\.co/datasets/hotpotqa/hotpot\_qa](https://huggingface.co/datasets/hotpotqa/hotpot_qa)Sampled 200 questions from the dev split; reformatted\.AA Omni Finance[Team \[2025\]](https://arxiv.org/html/2609.02679#bib.bib4)Apache 2\.0∼200\{\\sim\}200[https://huggingface\.co/datasets/ArtificialAnalysis/AA\-Omniscience\-Public](https://huggingface.co/datasets/ArtificialAnalysis/AA-Omniscience-Public)Extracted finance subset; wrapped with domain\-specific prompt\.Cheque Generation[Verma \[2024\]](https://arxiv.org/html/2609.02679#bib.bib5)Apache 2\.0200[https://huggingface\.co/datasets/aniketVerma07/handwritten\_cheque\_vqa\_dataset](https://huggingface.co/datasets/aniketVerma07/handwritten_cheque_vqa_dataset)Sampled 200 cheque images; sent as base64\-encoded inputs\.Financial SummariesInternal200newly generatedLLM\-synthesised financial passages with deliberate confusables\.Long\-Text QAInternal150newly generated30 multi\-part question sets over regulatory/financial documents\.
### C\.3Token\-level feature catalogue

Table[5](https://arxiv.org/html/2609.02679#A3.T5)lists the features extracted from each generated response\. These measurements capture sequence confidence, token uncertainty, candidate margins, positional changes, and response length\. Query\-level token representations are formed by aggregating these features across theNNsampled responses; the exact subsets used by Gated and Stacked are specified in Appendix[C\.4](https://arxiv.org/html/2609.02679#A3.SS4)\.

Table 5:Complete per\-response token\-level features\. Features marked⋆\\starare highlighted in §[2\.2](https://arxiv.org/html/2609.02679#S2.SS2.SSS0.Px1)\.FeatureSourceDescription*Token log\-probability features*mean\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Mean chosen\-token log\-probmin\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Min chosen\-token log\-probmax\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Max chosen\-token log\-probstd\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Std of chosen\-token log\-probsp90\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}10th percentile \(= 90th uncertainty pctl\)p95\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}5th percentile \(= 95th uncertainty pctl\)logprob\_varianceℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Variance of chosen\-token log\-probsperplexity⋆ℓt\(1\)\\ell\_\{t\}^\{\(1\)\}exp\(−1T∑tℓt\(1\)\)\\exp\(\-\\frac\{1\}\{T\}\\sum\_\{t\}\\ell\_\{t\}^\{\(1\)\}\)length\_norm\_seqlogprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Length\-normalised sequence log\-probseq\_lengthN/ANumber of tokensTiT\_\{i\}last1\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Log\-prob of last tokenlast3\_mean\_logprobℓt\(1\)\\ell\_\{t\}^\{\(1\)\}Mean log\-prob of last 3 tokensmean\_lp\_deltaΔ​ℓt\\Delta\\ell\_\{t\}Mean\|ℓt\+1\(1\)−ℓt\(1\)\|\|\\ell\_\{t\+1\}^\{\(1\)\}\-\\ell\_\{t\}^\{\(1\)\}\|max\_abs\_lp\_deltaΔ​ℓt\\Delta\\ell\_\{t\}Max\|ℓt\+1\(1\)−ℓt\(1\)\|\|\\ell\_\{t\+1\}^\{\(1\)\}\-\\ell\_\{t\}^\{\(1\)\}\|std\_lp\_deltaΔ​ℓt\\Delta\\ell\_\{t\}Std of log\-prob deltas*Top\-kkfeatures \(require top\-kklog\-probs\)*mean\_topk\_entropy⋆Htopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Mean top\-kkentropy \(§[2\.1](https://arxiv.org/html/2609.02679#S2.SS1.SSS0.Px2)\)max\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Max top\-kkentropymin\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Min top\-kkentropystd\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Std of top\-kkentropyp90\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)90th percentile of top\-kkentropyp95\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)95th percentile of top\-kkentropylast1\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Top\-kkentropy of last tokenlast3\_mean\_topk\_entropyHtopk​\(t\)H\_\{\\text\{topk\}\}\(t\)Mean top\-kkentropy of last 3 tokensmean\_ent\_deltaΔ​Ht\\Delta H\_\{t\}Mean\|Htopk​\(t\+1\)−Htopk​\(t\)\|\|H\_\{\\text\{topk\}\}\(t\\\!\+\\\!1\)\-H\_\{\\text\{topk\}\}\(t\)\|max\_abs\_ent\_deltaΔ​Ht\\Delta H\_\{t\}Max entropy delta \(spike signal\)std\_ent\_deltaΔ​Ht\\Delta H\_\{t\}Std of entropy deltasmean\_entropy\_alts⋆ranks 2–kkMean entropy over non\-greedy tokensmax\_entropy\_altsranks 2–kkMax entropy over non\-greedy tokensmean\_margin⋆ℓt\(1​–​2\)\\ell\_\{t\}^\{\(1\\text\{\-\-\}2\)\}Mean\(ℓt\(1\)−ℓt\(2\)\)\(\\ell\_\{t\}^\{\(1\)\}\-\\ell\_\{t\}^\{\(2\)\}\)min\_marginℓt\(1​–​2\)\\ell\_\{t\}^\{\(1\\text\{\-\-\}2\)\}Min margin \(least decisive position\)std\_marginℓt\(1​–​2\)\\ell\_\{t\}^\{\(1\\text\{\-\-\}2\)\}Std of marginsp10\_marginℓt\(1​–​2\)\\ell\_\{t\}^\{\(1\\text\{\-\-\}2\)\}10th percentile marginfrac\_non\_greedytop\-kkFraction of positions where chosen≠\\neqargmax
### C\.4Exact classifier input features

Since our code is not released under an unrestricted public licence because of institutional data\-governance constraints, we specify the exact feature subsets fed to each trained component here rather than leaving them implicit\.

Gated cascade,\|S\|=1\|S\|=1branch\.Features are aggregated across theNNsampled responses as described in §[2\.2](https://arxiv.org/html/2609.02679#S2.SS2.SSS0.Px1), standardised, and passed to an L2\-regularised logistic regression:

> mean\_max\_topk\_entropy, mean\_max\_abs\_ent\_delta, mean\_p95\_topk\_entropy, mean\_min\_margin, mean\_length\_norm\_seqlogprob, confidence\_spread, mean\_perplexity, std\_mean\_topk\_entropy, mean\_mid\_mean\_topk\_entropy, mean\_end\_mean\_topk\_entropy, mean\_mid\_max\_ent\_spike, entropy\_drift\_start\_to\_end, mid\_vs\_start\_spike\_ratio, margin\_decay\_start\_to\_end

When top\-kklog\-probabilities are unavailable \(has\_topk=False, e\.g\. some model and API configurations; see Appendix[E\.4](https://arxiv.org/html/2609.02679#A5.SS4)\), the following reduced, chosen\-token\-only feature set is used instead:

> mean\_max\_logprob, mean\_max\_abs\_lp\_delta, mean\_p95\_logprob, mean\_min\_logprob, mean\_length\_norm\_seqlogprob, confidence\_spread, mean\_perplexity, std\_mean\_logprob, mean\_mid\_mean\_logprob, mean\_end\_mean\_logprob, entropy\_drift\_start\_to\_end, margin\_decay\_start\_to\_end

Stacked classifier\.This model pools one SE\-variant feature block with the full aggregated token feature vector from Table[5](https://arxiv.org/html/2609.02679#A3.T5)\. It then applies StandardScaler, PCA with the top 15 components, and L2\-regularised logistic regression\. The SE\-variant block is one of:

> Hybrid: se\_hybrid, s\_hat\_hybrid, n\_clusters Spectral: spectral\_total, spectral\_erank, spectral\_epistemic, n\_clusters Von Neumann: se\_von\_neumann, n\_clusters

### C\.5Dataset construction

The following materials document the construction of the two custom benchmarks\. We retain the complete instructions and question sets to make the evaluation design reproducible\.

#### C\.5\.1Long\-Text QA

The Long\-Text QA dataset comprises 30 multi\-part question sets \(5 sub\-questions each, 150 sub\-questions total\) over three regulatory and financial source documents:*SR 11\-7: Guidance on Model Risk Management*\(Federal Reserve / OCC\), an anonymised 10\-K annual filing, and the*EU AI Act*\. Each question set is presented to the model together with the full source document as context, using the shared prompt template below\. Exactly one sub\-question per set \(always Q5\) is*deliberately unanswerable*from the source document; the gold answer for that question is “Not provided in context\.” This design provides a controlled hallucination signal: a model that fabricates an answer to Q5 produces a detectable hallucination\.

##### Prompt template\.

All 30 question sets share the following instruction prefix \(question content varies per set\):

> Answer the questions based on the document attached\. Use only the context provided in the document and do not use your training data for this\. There are five questions\. Return the output in the provided output format\. Strictly follow the output format\. Do not add anything else\. If you do not know the answer or the answer is not present in the attached document, output ‘‘Not provided in context\.’’ INPUT Format: \[’question 1’, ’question 2’, \.\.\., ’question 5’\] OUTPUT format: \{’question 1’: ’answer’, ’question 2’: ’answer’, \.\.\., ’question 5’: ’answer’\}

##### Complete question listing\.

Table[6](https://arxiv.org/html/2609.02679#A3.T6)lists all 30 question sets\. The unanswerable sub\-question \(Q5\) in each set is marked with†\\dagger\.

Table 6:Long\-Text QA: all 30 question sets\.Each set contains five sub\-questions posed over the indicated source document\. Q5 \(marked†\\dagger\) is deliberately unanswerable from the source; the expected answer is “Not provided in context\.”SetSourceSub\-questions1SR 11\-7Q1: Explain the core elements of an effective validation framework\.
Q2: Explain the processes and activities associated with model validation\.
Q3: What is the definition of a Model according to SR 11\-7?
Q4: What is the exact definition of model validation according to SR 11\-7?
Q5†\\dagger: What are the five key elements of comprehensive validation?2SR 11\-7Q1: What should a bank’s board of directors and senior management demand from model validation?
Q2: What are the three core elements of effective model risk management?
Q3: What are the key components of a robust model validation framework?
Q4: What types of outcomes analysis should be performed during model validation?
Q5†\\dagger: What are the regulatory penalties for non\-compliance with SR 11\-7?3SR 11\-7Q1: What is the role of conceptual soundness evaluation in model validation?
Q2: What does SR 11\-7 say about the use of vendor models?
Q3: How should model limitations be documented according to SR 11\-7?
Q4: What are the key elements of an effective model inventory?
Q5†\\dagger: What specific software tools does SR 11\-7 recommend for model validation?4SR 11\-7Q1: What is the definition of model risk according to SR 11\-7?
Q2: What are the two main sources of model risk identified in SR 11\-7?
Q3: Why might models with good performance become inaccurate over time?
Q4: How does SR 11\-7 define an effective challenge?
Q5†\\dagger: What penalties can be imposed for failing to comply with SR 11\-7?5SR 11\-7Q1: What is the suggested frequency of model validation reviews?
Q2: What role does internal audit play in model risk management?
Q3: How should banks manage model risk for models developed by vendors or third parties?
Q4: What are the key considerations for using sensitivity analysis in model validation?
Q5†\\dagger: What machine learning methods does SR 11\-7 recommend for model validation?6SR 11\-7Q1: What is the role of the board of directors in model risk management according to SR 11\-7?
Q2: What does SR 11\-7 say about the documentation requirements for models?
Q3: How should banks handle models that show declining performance?
Q4: What are the key elements of an effective model development process?
Q5†\\dagger: What cloud computing requirements does SR 11\-7 specify?7SR 11\-7Q1: What does SR 11\-7 say about model governance policies?
Q2: How should banks assess model limitations?
Q3: What is the role of outcomes analysis in model validation?
Q4: What are the key considerations for back\-testing models?
Q5†\\dagger: What specific stress testing scenarios does SR 11\-7 require banks to run?810\-KQ1: How many people does \[ANON\] employ globally? Give time as well\.
Q2: What are the MD&A\-risk factors? What happens if they are not adhered to?
Q3: Summarise the Description of Securities Registered Pursuant to Section 12 of the Securities Exchange Act of 1934 as provided in Exhibit 4\.2\.
Q4: Describe the \[X\.XXX\]% Fixed\-to\-Floating Rate Preferred Capital Securities\.
Q5†\\dagger: Describe the deliquidification process?910\-KQ1: What is the position of \[EXEC\_A\]?
Q2: What is the position of \[EXEC\_B\]?
Q3: What does the Vesting Schedule indicate?
Q4: What does Risk Adjustment Process indicate?
Q5†\\dagger: What is the position of \[EXEC\_C\]?1010\-KQ1: Is the registration statement \[REG\_NO\_1\] included?
Q2: Is the registration statement \[REG\_NO\_2\] included?
Q3: What does TSR mean?
Q4: What does Transition Period mean?
Q5†\\dagger: What does TSP mean?1110\-KQ1: What is the definition of Firm in this document?
Q2: What Financial Reporting Measures are used?
Q3: What is the par value of the common stock listed under securities registered?
Q4: How much do the preferred securities initially pay?
Q5†\\dagger: How does the vesting schedule differ from the non\-vesting schedule?1210\-KQ1: Who is the issuing entity?
Q2: Are there any Securities registered pursuant to Section 12\(g\) of the Act?
Q3: Who are the current Executive Officers?
Q4: How many Executive Officers are there at \[ANON\] currently?
Q5†\\dagger: How many shares does the company need to issue as per the reinstated certificate of incorporation?1310\-KQ1: What is the main banking subsidiary of \[ANON\] in continental Europe?
Q2: At the end of 2023, what % of \[ANON\]’s global workforce were women?
Q3: At the end of 2023, what % of \[ANON\]’s executive committee were women?
Q4: At Dec\. 31, 2023, approximately what % of \[ANON\]’s total employees were based outside the U\.S\.?
Q5†\\dagger: At Dec\. 31, 2023, approximately what % of \[ANON\]’s total employees were based in India?1410\-KQ1: Where are the corporate headquarters of \[ANON\] located?
Q2: Who supervised the evaluation of the effectiveness of \[ANON\]’s disclosure controls and procedures?
Q3: Who is the oldest Executive Officer?
Q4: Do the holders of Common Stock have cumulative voting rights?
Q5†\\dagger: Do the holders of Dispersed Stock have cumulative voting rights?1510\-KQ1: Who has right to request that the Secretary call a special meeting of stockholders?
Q2: Approximately how many participants were covered by the frozen U\.S\. defined benefit pension plan at December 31, 2023?
Q3: Approximately how many participants were covered by non\-U\.S\. defined benefit plans at December 31, 2023?
Q4: What is the approximate size of the company’s headquarters building?
Q5†\\dagger: What is the approximate size of the company’s office building in \[CITY\_A\]?1610\-KQ1: What is included in the Other segment?
Q2: What were assets under custody and/or administration at December 31, 2023?
Q3: What were assets under management at December 31, 2023?
Q4: Which principal U\.S\. banking subsidiary houses \[SEGMENT\_A\] businesses?
Q5†\\dagger: Who is the Chief AI Officer of \[ANON\]?1710\-KQ1: How many employees were in EMEA at the end of 2023?
Q2: How many employees were in APAC at the end of 2023?
Q3: What percentage of the global workforce were women at the end of 2023?
Q4: What percentage of the U\.S\. workforce were from underrepresented ethnic and/or racial backgrounds at the end of 2023?
Q5†\\dagger: How many employees were in APAC at the end of 2025?1810\-KQ1: What equity award was given to eligible employees as described in Human Capital Management?
Q2: How many participants were in the 401\(k\) plan at December 31, 2023?
Q3: How many U\.S\. participants were covered by the frozen defined benefit pension plan at December 31, 2023?
Q4: How many non\-U\.S\. participants were covered by non\-U\.S\. defined benefit plans at December 31, 2023?
Q5†\\dagger: How many participants were in the dental insurance plan at December 31, 2023?1910\-KQ1: How much space does the company have in EMEA?
Q2: How much leased space does the company have in APAC?
Q3: How many holders of record of common stock were there as of January 31, 2024?
Q4: Were the company’s disclosure controls and procedures effective as of December 31, 2023?
Q5†\\dagger: How much leased space does the company have in Germany?2010\-KQ1: What PCAOB firm identification number is listed for \[AUDIT\_FIRM\]?
Q2: What was fee and other revenue in 2023?
Q3: What was net interest revenue in 2023?
Q4: What was total revenue in 2023?
Q5†\\dagger: What was total revenue in 2003?2110\-KQ1: What is the registrant’s legal name in the filing?
Q2: What is the company’s stock ticker symbol?
Q3: On which exchange is the common stock listed?
Q4: What was the aggregate market value of common stock held by non\-affiliates as of June 30, 2023?
Q5†\\dagger: What is the company’s stock ticker symbol for NSE?2210\-KQ1: What was net income applicable to common shareholders in 2023?
Q2: What were diluted earnings per common share in 2023?
Q3: What was return on common equity in 2023?
Q4: What was return on tangible common equity in 2023?
Q5†\\dagger: What was return on tangible common equity in 2003?2310\-KQ1: What was the common cash dividend per share in 2023?
Q2: What was the common dividend payout ratio in 2023?
Q3: What was the closing stock price at December 31, 2023?
Q4: What was market capitalisation at December 31, 2023?
Q5†\\dagger: What was market capitalisation at December 31, 2003?2410\-KQ1: What was book value per common share at December 31, 2023?
Q2: What was tangible book value per common share at December 31, 2023?
Q3: What was the CET1 ratio at December 31, 2023?
Q4: What was the Tier 1 capital ratio at December 31, 2023?
Q5†\\dagger: What was the Tier 1 capital ratio at December 31, 2003?2510\-KQ1: What was the total capital ratio at December 31, 2023?
Q2: What was the Tier 1 leverage ratio at December 31, 2023?
Q3: What was the supplementary leverage ratio at December 31, 2023?
Q4: What was the effective tax rate in 2023?
Q5†\\dagger: What was the effective tax rate in 2003?2610\-KQ1: What was the FDIC special assessment accrual recorded in 2023 noninterest expense?
Q2: What adjustment was made in February 2024 related to the FDIC special assessment?
Q3: What was the average common shares outstanding on a diluted basis in 2023?
Q4: What Financial Reporting Measures are used under the Recovery of Erroneously Awarded Incentive\-Based Compensation Policy?
Q5†\\dagger: What was the average common shares outstanding on a diluted basis in 2003?27EU AI ActQ1: What is the main purpose of the EU AI Act?
Q2: What kinds of rules does the EU AI Act lay down?
Q3: To whom does the EU AI Act apply?
Q4: Does the EU AI Act apply to AI used exclusively for military, defence or national security purposes?
Q5†\\dagger: What special provisions are made for AI in financial services?28EU AI ActQ1: Does the EU AI Act apply to AI systems developed solely for scientific research and development?
Q2: What is an ‘AI system’ under the EU AI Act?
Q3: What does the EU AI Act mean by ‘provider’?
Q4: What is a ‘deployer’ under the Act?
Q5†\\dagger: What is meant by ‘subscriber’?29EU AI ActQ1: What does the EU AI Act mean by ‘provider’?
Q2: What is a ‘deployer’ under the Act?
Q3: What is meant by ‘intended purpose’ in the EU AI Act?
Q4: What is ‘reasonably foreseeable misuse’?
Q5†\\dagger: Is the Act applicable to the United Kingdom of Great Britain and Ireland?30EU AI ActQ1: What is ‘AI literacy’ under the Act?
Q2: What obligation does the EU AI Act impose regarding AI literacy?
Q3: What overall regulatory approach does the EU AI Act use?
Q4: What AI practice involving subliminal or manipulative techniques is prohibited?
Q5†\\dagger: What exemptions are provided for BigTech companies under the Act?

#### C\.5\.2Financial Summaries

The following prompt template is used to generate the synthetic finance texts in our summarisation hallucination dataset \(Section[3\.2](https://arxiv.org/html/2609.02679#S3.SS2)\)\. Placeholders in braces are filled programmatically for each sample\.

> You are constructing a research dataset for studying LLM hallucination in financial text summarisation\. Your task is to generate a synthetic finance text that is designed to trigger a specific type of hallucination when an LLM attempts to summarise it\. HALLUCINATION TYPE: \{type\_code\} \-\- \{type\_name\} TYPE DESCRIPTION: \{type\_description\} \{trigger\_instructions\} FINANCE SUB\-DOMAIN: \{subdomain\} CONSTRAINTS: 1. 1\.The text must be approximately 500 words \(450\-\-550 acceptable range\)\. 2. 2\.The text must read like a realistic finance document \(news article, analyst note, regulatory filing excerpt, earnings summary, etc\.\) in the specified sub\-domain\. 3. 3\.All entities, figures, and events in the text are FICTIONAL but must be plausible\. Do not use real company names, real people, or real events\. 4. 4\.The text must be self\-contained \(no references to external documents\)\. 5. 5\.Do NOT include any meta\-commentary about the hallucination design\. The text should look completely natural\. VARIATION SEED: \{variation\_seed\} Use this seed to ensure this text is distinct from others in the same category\. Vary the entities, scenario, writing style, and structure\. Return your response as a JSON object with these exact fields: \{ "text": "<the ~500\-word finance text\>", "ground\_truth\_summary": "<a correct 3\-\-4 sentence summary that faithfully captures the key facts without any distortion\>", "key\_facts": \["<fact 1\>", "<fact 2\>", …\], "hallucination\_traps": \["<description of trap 1\>", "<description of trap 2\>", …\], "design\_notes": "<brief explanation of what structural features were embedded\>" \} key\_facts: An exhaustive list of every verifiable claim in the text \(aim for 8\-\-15 facts\)\. hallucination\_traps: Specific descriptions of what a summariser is likely to get wrong \(e\.g\. "May attribute Meridian’s $2\.1B revenue to Apex Corp"\)\. Aim for 3\-\-5 traps\. For CTRL texts, set hallucination\_traps to an empty list\. Return ONLY the JSON object, no other text\.

##### Per\-type trigger instructions

The\{trigger\_instructions\}placeholder in the prompt above is filled with one of the following type\-specific structural requirement blocks\. Each block steers the generator toward text structures that are known to elicit the corresponding hallucination type during summarisation\.

ES– Entity Swap\.> STRUCTURAL REQUIREMENTS for Entity Swap triggers: 1. 1\.Include at least 3 named entities of the same type \(e\.g\. 3 banks, 3 CEOs, 3 tickers\) in close proximity within the text\. 2. 2\.Interleave metrics and attributes across entities so that a careless reader \(or LLM\) could easily attribute Entity A’s figures to Entity B\. 3. 3\.Use similar\-sounding or related entity names \(e\.g\. Apex Securities and Meridian Capital, or Northern Bank and Pacific Trust\) to increase confusion potential\. 4. 4\.Ensure each entity has distinct, verifiable facts that must not be swapped\.

NE– Numerical Error\.> STRUCTURAL REQUIREMENTS for Numerical Error triggers: 1. 1\.Pack the text with at least 8\-\-10 specific numbers: revenue figures, percentages, basis points, dates, headcounts, or dollar amounts\. 2. 2\.Include numbers that differ by small amounts \(e\.g\. 3\.2% vs 3\.7%, $14\.2B vs $14\.8B\)\. 3. 3\.Mix absolute values with percentages and year\-over\-year changes in the same paragraph\. 4. 4\.Include at least one number that contradicts an intuitive expectation\.

NF– Negation Flip\.> STRUCTURAL REQUIREMENTS for Negation Flip triggers: 1. 1\.Include at least 2\-\-3 negated statements that carry critical meaning \(e\.g\. ‘‘the board did NOT approve’’, ‘‘excluding derivatives exposure’’\)\. 2. 2\.Bury negations in subordinate clauses or complex sentence structures\. 3. 3\.Use double negatives, ‘‘except’’/‘‘excluding’’/‘‘unless’’ constructions\. 4. 4\.Place an affirmative statement near a negated one about a related topic so a summariser might conflate them\.

TC– Temporal Confusion\.> STRUCTURAL REQUIREMENTS for Temporal Confusion triggers: 1. 1\.Present events in non\-chronological order \(e\.g\. mention Q4 results before Q2 events\)\. 2. 2\.Include at least 4\-\-5 distinct date references \(specific months, quarters, years\)\. 3. 3\.Describe actions that were planned, deferred, completed, or reversed across different time periods\. 4. 4\.Use ambiguous temporal language \(‘‘previously’’, ‘‘earlier this year’’, ‘‘following the announcement’’\) alongside specific dates\.

CF– Causal Fabrication\.> STRUCTURAL REQUIREMENTS for Causal Fabrication triggers: 1. 1\.Juxtapose two events or facts that are temporally or thematically related but have NO stated causal connection\. 2. 2\.A summariser is likely to infer ‘‘A caused B’’ or ‘‘A led to B’’ when the text only says ‘‘A happened\. Separately, B happened\.’’ 3. 3\.Include at least 2 such juxtaposition pairs\. 4. 4\.Explicitly avoid causal language \(‘‘because’’, ‘‘therefore’’, ‘‘as a result’’\) between the juxtaposed events\.

UI– Unsupported Inference\.> STRUCTURAL REQUIREMENTS for Unsupported Inference triggers: 1. 1\.Present raw data, metrics, or observations WITHOUT drawing conclusions\. 2. 2\.A summariser is likely to add interpretive statements like ‘‘this suggests’’, ‘‘indicating strong performance’’, or ‘‘raising concerns about’’\. 3. 3\.Include data that could support multiple contradictory interpretations\. 4. 4\.Avoid any evaluative or interpretive language in the source text\.

DF– Detail Fabrication\.> STRUCTURAL REQUIREMENTS for Detail Fabrication triggers: 1. 1\.Leave deliberate gaps: mention ‘‘a major US bank’’ without naming it, refer to ‘‘the acquiring firm’’ without specifics, cite ‘‘an undisclosed sum’’\. 2. 2\.Include partial information that a summariser might ‘‘complete’’ by fabricating plausible details \(names, amounts, locations\)\. 3. 3\.Reference unnamed parties, unspecified contract terms, or redacted figures\. 4. 4\.Include at least 3 such deliberate vagueness points\.

MC– Merging Claims\.> STRUCTURAL REQUIREMENTS for Merging Claims triggers: 1. 1\.Include at least 2 pairs of related but distinct claims in adjacent sentences\. 2. 2\.For example: ‘‘Bank A expanded its equities desk in London’’ followed by ‘‘Bank A also opened a new fixed\-income operation in Frankfurt’’ \-\- a summariser might merge these into ‘‘Bank A expanded its equities and fixed\-income operations in London and Frankfurt’’ \(incorrectly linking equities to Frankfurt\)\. 3. 3\.Make the claims similar enough to invite merging but with distinct specifics\.

OQ– Omission of Qualifiers\.> STRUCTURAL REQUIREMENTS for Omission of Qualifiers triggers: 1. 1\.Saturate the text with hedging language: ‘‘preliminary estimates suggest’’, ‘‘subject to regulatory approval’’, ‘‘under certain market conditions’’, ‘‘the committee may consider’’, ‘‘pending final review’’\. 2. 2\.At least 4\-\-5 key claims should be heavily qualified\. 3. 3\.Make the underlying claims significant enough that dropping the qualifier materially changes the meaning \(e\.g\. ‘‘may acquire’’ vs ‘‘will acquire’’\)\.

OG– Over\-generalisation\.> STRUCTURAL REQUIREMENTS for Over\-generalisation triggers: 1. 1\.Make every claim narrowly scoped: specify region, time period, client segment, product line, or division\. 2. 2\.For example: ‘‘Among retail clients in the APAC region during Q3 2025, adoption of digital advisory tools rose 12%\.’’ A summariser might drop the qualifiers and report ‘‘digital advisory adoption rose 12%’’ as if it were global/universal\. 3. 3\.Include at least 3\-\-4 such narrowly scoped claims\.

CTRL– Control \(easy\)\.> STRUCTURAL REQUIREMENTS for Control \(easy\) texts: 1. 1\.Write a clear, well\-structured finance text with unambiguous facts\. 2. 2\.Use straightforward sentence structure with no buried negations or tricky phrasing\. 3. 3\.Include a reasonable number of facts \(4\-\-6\) that are easy to verify in a summary\. 4. 4\.Avoid any adversarial structural patterns\. This text should be easy to summarise correctly\.

### C\.6LLM judge prompts for hallucination labels

We use GPT\-4\.1 for four judging protocols according to the reference information available for each task\. The blind judge evaluates an answer without a gold reference, the binary and scalar judges compare it with an expected answer, and the summarisation judge checks generated claims against the source text\. The complete prompts are reproduced below\.

#### C\.6\.1Blind judge

The blind judge receives only the question and the model’s answer; no reference material is provided\. It returns a continuous hallucination probability, a confidence score, and free\-text reasoning\. The prompt instructs the judge to look for suspiciously specific details, contradiction of widely known facts, false confidence, internal inconsistency, and confabulation\.

> You are an expert evaluator\. Given a question and the model’s answer, assess whether the answer is likely to contain hallucinated or fabricated information\. You do NOT have access to the correct answer or any reference material\. Use only your knowledge and reasoning to judge the answer\. Look for the following signals of hallucination: 1. 1\.Suspiciously specific details \(e\.g\. invented names, dates, statistics\) 2. 2\.Contradiction of widely known facts 3. 3\.False confidence about uncertain or obscure topics 4. 4\.Internal inconsistency within the answer 5. 5\.Confabulation: plausible\-sounding but fabricated information Return: \{"hallucination\_probability": <0\.0\-\-1\.0\>, "confidence": <0\.0\-\-1\.0\>, "reasoning": "\.\.\."\}

#### C\.6\.2Binary judge

The binary judge compares the proposed answer against an expected answer and returns a binary verdict \(1 = correct, 0 = incorrect\)\. Inspired by[Kuhn et al\. \[2023\]](https://arxiv.org/html/2609.02679#bib.bib11)and[Farquhar et al\. \[2024\]](https://arxiv.org/html/2609.02679#bib.bib10), the prompt explicitly permits additional detail beyond the expected answer while requiring that the core claim not be wrong\.

> We are assessing the quality of answers to the following question\. QUESTION: \{question\} EXPECTED ANSWER: \{expected\_answer\} PROPOSED ANSWER: \{proposed\_answer\} Important: It is acceptable for the proposed answer to include additional details beyond the expected answer\. The core of the proposed answer must not be incorrect, it may be incomplete, but it must not be wrong\. Within the context of the question, does the proposed answer mean the same as the expected answer? Respond ONLY with "yes" or "no"\.

#### C\.6\.3Scalar judge

The scalar judge uses the same inputs as the binary judge but returns a continuous consistency score on a 0–100 scale together with free\-text reasoning\.

> Rate the level of consistency between the proposed answer and the expected answer to the question, from 0 to 100\. Important: It is acceptable for the proposed answer to include additional details beyond the expected answer\. The core of the proposed answer must not be incorrect\-\-it may be incomplete, but it must not be wrong\. A correct but incomplete answer should still receive a high score\. Question: \{question\} Expected answer: \{expected\_answer\} Proposed answer: \{proposed\_answer\} Return: \{"rating": <0\-\-100\>, "reasoning": "\.\.\."\}

#### C\.6\.4Summarisation judge

The summarisation judge evaluates whether an LLM\-generated summary of a financial text contains hallucinations\. It receives the source text, a list of key facts extracted from the source, and the summary to evaluate\. The prompt defines a ten\-category hallucination taxonomy, Entity Swap \(ES\), Numerical Error \(NE\), Negation Flip \(NF\), Temporal Confusion \(TC\), Causal Fabrication \(CF\), Unsupported Inference \(UI\), Detail Fabrication \(DF\), Merging Claims \(MC\), Omission of Qualifiers \(OQ\), and Over\-generalisation \(OG\)–and asks the judge to return structured JSON with a binary flag, severity rating, and per\-claim explanations\. In practice, two independent GPT\-4\.1 judgements are combined using the union verdict\.

> You are an expert evaluator assessing whether an LLM\-generated summary of a financial text contains hallucinations\. A hallucination is any claim in the summary that is not supported by, or contradicts, the source text\. HALLUCINATION TAXONOMY: \- ES \(Entity Swap\): Attributes facts to the wrong entity \- NE \(Numerical Error\): Incorrect numbers, percentages, or dates \- NF \(Negation Flip\): Inverts or drops a negation \- TC \(Temporal Confusion\): Wrong ordering or dating of events \- CF \(Causal Fabrication\): Invents cause\-effect links not in the source \- UI \(Unsupported Inference\): Draws conclusions beyond the text \- DF \(Detail Fabrication\): Invents specific details absent from the source \- MC \(Merging Claims\): Fuses distinct statements into a false composite \- OQ \(Omission of Qualifiers\): Drops hedging, making tentative claims definitive \- OG \(Over\-generalisation\): Broadens a narrowly scoped claim SOURCE TEXT: \{source\_text\} KEY FACTS \(exhaustive list of verifiable claims in the source\): \{key\_facts\} SUMMARY TO EVALUATE: \{summary\} Evaluate the summary carefully\. Return a JSON object: \{\{ "has\_hallucination": <true or false\>, "hallucination\_types\_found": \["<code\>", \.\.\.\], "hallucinated\_claims": \[\{\{ "claim": "<the hallucinated statement from the summary\>", "type": "<hallucination type code\>", "explanation": "<why this is a hallucination, referencing the source text\>" \}\}\], "severity": "<none\|low\|medium\|high\>", "overall\_reasoning": "<2\-3 sentence assessment\>" \}\} If the summary is faithful, set has\_hallucination to false, hallucination\_types\_found and hallucinated\_claims to empty lists, severity to "none"\. Return ONLY the JSON object\.

## Appendix DExtended diagnostic results

This section provides additional evidence supporting the main results and discussion\. We first examine the complementary failure modes of semantic and token uncertainty, then report ROC curves and cross\-model comparisons, including CoCoA\.

### D\.1Complementary failure modes

Figures[6](https://arxiv.org/html/2609.02679#A4.F6)and[7](https://arxiv.org/html/2609.02679#A4.F7)examine the principal blind spot of semantic entropy\. Among hallucinated queries, the proportion whoseNNresponses collapse into one semantic cluster ranges from 39% on AmbigQA to 99% on Financial Summaries, forcing Standard SE to zero\. Within these single\-cluster cases, median TopK uncertainty remains higher for hallucinated queries on six of seven datasets, showing that token probabilities often retain information after semantic disagreement disappears\. The exception is SQuAD, and substantial distributional overlap remains elsewhere: between 21% and 56% of hallucinated single\-cluster cases also fall below the corresponding non\-hallucinated TopK median\. Thus, the two signals are complementary, but neither guarantees detection when the model is both semantically consistent and token\-confident\.

Figure 6:Single\-cluster collapse among hallucinated queries\.Hallucinated queries are divided according to whether theirNNsampled responses form one semantic cluster \(K=1K=1\) or multiple clusters \(K≥2K\\geq 2\)\. Standard SE is necessarily zero in the single\-cluster group\. Percentages above the bars report theK=1K=1share for each dataset\.Figure 7:TopK uncertainty within the semantic\-entropy blind spot\.The analysis is restricted toK=1K=1queries, for which Standard SE is zero\. Bars compare median TopK uncertainty for hallucinated and non\-hallucinated queries\. The annotation for each dataset reports the proportion of hallucinated queries whose TopK score also falls below the non\-hallucinated median\.
### D\.2Token\-feature discriminability

Figure[8](https://arxiv.org/html/2609.02679#A4.F8)relates the strongest token\-level effect size on each dataset to the performance of the Stacked classifier\. Datasets with clearer token\-feature separation generally yield stronger Stacked performance, whereas Financial Summaries remains weak on both measures\. This relationship is descriptive rather than causal: it is based on seven datasets, and Stacked also uses semantic features rather than the displayed token feature alone\.

Figure 8:Token\-feature discriminability and Stacked performance\.Stacked AUROC plotted against the largest absolute Cohen’sddamong token\-level features for each dataset\. Stronger token\-feature separation is generally associated with higher Stacked AUROC, while Financial Summaries is the clearest low\-separation outlier\.
### D\.3ROC curves across datasets

Figure[9](https://arxiv.org/html/2609.02679#A4.F9)shows how method sensitivity changes across false\-positive\-rate thresholds, including CoCoA SP and CoCoA PPL\. The curves reinforce the dataset\-dependent rankings reported in the main text: relative performance varies across both datasets and operating points\.

Figure 9:ROC curves across datasets, with CoCoA\.Each panel shows the representative held\-out\-fold ROC curves from the five\-fold evaluation of the canonical model run, including CoCoA SP and CoCoA PPL computed from the same generations; the diagonal denotes chance performance\.
### D\.4Cheque Generation robustness

Figure[10](https://arxiv.org/html/2609.02679#A4.F10)compares method categories across response counts for the three vision\-capable models\. IncreasingNNgenerally improves the strongest semantic method for GPT\-5\.1 and GPT\-5\.4, but TopK, Gated, and Stacked vary comparatively little\. Thus, additional responses do not produce a consistent gain for the combined methods\. These matched sweeps were scored externally and are reported separately from the canonical pooled out\-of\-fold evaluation because their scoring pipeline could not be fully verified against our evaluator\.

Figure 10:Cheque Generation robustness across models and response counts\.AUROC is shown for the strongest semantic method, TopK, the mean of three Stacked variants, and the mean of two Gated variants overN∈\{3,5,7,10,15,20\}N\\in\\\{3,5,7,10,15,20\\\}\. The externally scored sweeps are kept separate from the primary pooled out\-of\-fold results\. CoCoA is omitted because this sweep contains no CoCoA scores\.

## Appendix EAblation studies

This section examines sensitivity to model family, response countNN, sampling temperature, the number of returned token candidateskk, and classifier calibration\. We report each factor separately to distinguish changes in the available uncertainty evidence from changes in the fitted method\.

### E\.1Model\-family ablation

Table[7](https://arxiv.org/html/2609.02679#A5.T7)reports the primary GPT\-4\.1\-mini fold\-mean results discussed in Section[4\.1](https://arxiv.org/html/2609.02679#S4.SS1)\. Tables[8](https://arxiv.org/html/2609.02679#A5.T8)–[11](https://arxiv.org/html/2609.02679#A5.T11)report pooled out\-of\-fold results for the 23 available text model–dataset settings using GPT\-5\.1, GPT\-5\.4, Llama 3\.3 70B, and a regenerated GPT\-4\.1\-mini run\. Appendix[D\.4](https://arxiv.org/html/2609.02679#A4.SS4)reports the three vision\-capable Cheque Generation settings, giving 26 available settings overall\. CoCoA SP and CoCoA PPL are computed from the same generated responses as the other methods that do not use supervised training labels\. The dataset\-specificτ\\tauandCCvalues are held fixed across models rather than retuned, and the single\-class Llama Long\-Text QA setting is marked as N/A\.

Table 7:GPT\-4\.1\-mini AUROC by method and dataset on the held\-out folds of 5\-fold cross\-validation\. Generation usesN=10N\{=\}10, temperature 1\.0, and five returned log\-probability candidates per token \(k=5k\{=\}5\)\. CoCoA SP and PPL use their respective confidence variants with semantic consistency computed from the same generations\. Each value is the mean of the five fold\-level test AUROCs\. Bold marks the highest AUROC in each dataset\.Pooled out\-of\-fold AUROC differs by 0\.01 to 0\.02 on most cells\. It ranks TopK marginally above Gated Hybrid on HotpotQA and Long Text QA\. Appendix[F\.2](https://arxiv.org/html/2609.02679#A6.SS2)reports pooled estimates and 95% bootstrap intervals for every cell\.

Table 8:Complete pooled out\-of\-fold AUROC for GPT\-4\.1\-mini\-new\. Missing generation, judging, or modality combinations are shown as N/A; Cheque Generation is reported separately because these agents are text\-only\.Table 9:Complete pooled out\-of\-fold AUROC for Llama\-70B\. Missing generation, judging, or modality combinations are shown as N/A; Cheque Generation is reported separately because these agents are text\-only\.Table 10:Complete pooled out\-of\-fold AUROC for GPT\-5\.1\. Missing generation, judging, or modality combinations are shown as N/A; Cheque Generation is reported separately because these agents are text\-only\.Table 11:Complete pooled out\-of\-fold AUROC for GPT\-5\.4\. Missing generation, judging, or modality combinations are shown as N/A; Cheque Generation is reported separately because these agents are text\-only\.Table 12:Cheque Generation is a multimodal vision\-language task\. Text\-only model\-family agents are structurally unavailable rather than failed or imputed\. The primary GPT\-4\.1\-mini result and the externally scored N\-sweep are reported separately\.
### E\.2Response\-count ablation

Figure[11](https://arxiv.org/html/2609.02679#A5.F11)examines sensitivity to the number of sampled responses\. IncreasingNNcan expose additional semantic variation and stabilise multi\-response token statistics, but the resulting AUROC changes are not uniformly positive across datasets or method categories\. The absence of a consistent monotonic improvement indicates that additional samples alone do not resolve the dataset\-dependent weaknesses of either signal family\.

Figure 11:AUROC versus response count\.Performance is shown overN∈\{3,5,7,10,15,20\}N\\in\\\{3,5,7,10,15,20\\\}for the strongest semantic method, TopK, and the averaged Gated and Stacked variants\. The effect of increasingNNvaries across datasets and methods\. CoCoA is omitted because the required main\-answer embeddings are unavailable for this sweep\.
### E\.3Sampling\-temperature ablation

Sampling temperature changes both response diversity and token confidence\. Figure[12](https://arxiv.org/html/2609.02679#A5.F12)shows that higher temperatures generally reduce the single\-cluster rate, but do not produce a consistent AUROC improvement\. Greater diversity can expose alternative meanings, but it can also introduce variation unrelated to hallucination; the preferred temperature is therefore dataset\-dependent\.

Figure 12:Sensitivity to sampling temperature\.\(a\) Detector AUROC and \(b\) the proportion of queries forming a single semantic cluster overT∈\{0\.3,0\.5,0\.7,1\.0,1\.2\}T\\in\\\{0\.3,0\.5,0\.7,1\.0,1\.2\\\}\. Higher response diversity does not consistently improve hallucination separation\. CoCoA is omitted because the required main\-answer embeddings are unavailable for this sweep\.
### E\.4Top\-kkcandidate ablation

Table[13](https://arxiv.org/html/2609.02679#A5.T13)evaluates the number of token candidates retained at each generation step\. Most improvements occur when moving fromk=1k=1tok=2k=2ork=3k=3; additional candidates produce smaller and less consistent changes\. Semantic\-only performance is unchanged because it does not use token alternatives, and CoCoA SP/PPL are likewise flat acrosskk: both depend only on the generated answer’s own token log\-probabilities, never the truncated alternative list\. The available responses store at most five candidates, so values abovek=5k=5cannot be reconstructed without new generations\.

Table 13:Top\-kklog\-probability sensitivity, by detector category\.AUROC vs\. the number of top\-kklog\-probability alternatives requested per token, recomputed from existing saved responses \(no new generation;k\>5k\>5not recoverable from already\-stored data\)\. Best SE is the strongest of the 4 SE variants per cell; Avg Stacked/Gated average their variants\. CoCoA SP/PPL are flat acrosskkbecause both depend only on the generated answer’s own token log\-probabilities, never the truncated alternatives\.
### E\.5Calibration sensitivity

We sweep five clustering thresholdsτ∈\{0\.80,0\.85,0\.90,0\.95,0\.98\}\\tau\\in\\\{0\.80,0\.85,0\.90,0\.95,0\.98\\\}and three regularisation strengthsC∈\{0\.1,1,10\}C\\in\\\{0\.1,1,10\\\}, giving 15 calibration settings\. Figure[13](https://arxiv.org/html/2609.02679#A5.F13)shows the method\-level minimum–maximum AUROC ranges, with colour indicating interval width; Table[14](https://arxiv.org/html/2609.02679#A5.T14)gives the corresponding method\-category summary\. Narrow intervals indicate that a method is comparatively insensitive to calibration, whereas wider intervals identify datasets and method families for which the selected configuration materially affects performance\. CoCoA SP and CoCoA PPL have zero\-width intervals throughout, alongside SE Von Neumann and TopK: CoCoA has neither a clustering threshold nor a regularisation strength to sweep\.

![Refer to caption](https://arxiv.org/html/2609.02679v1/F_GEN1_sensitivity_with_cocoa.png)Figure 13:Calibration sensitivity, with CoCoA\.Cells report the minimum–maximum AUROC over 15\(τ,C\)\(\\tau,C\)settings; colour encodes the width of that range\. CoCoA SP/PPL are included as zero\-width reference rows\.Table 14:Calibration\-threshold sensitivity, by detector category\.AUROC sensitivity to clustering thresholdτ\\tauand regularisationCC, swept overτ∈\{0\.80,…,0\.98\}\\tau\\in\\\{0\.80,\\ldots,0\.98\\\},C∈\{0\.1,1,10\}C\\in\\\{0\.1,1,10\\\}, for each detector category \(Best SE = strongest of the 4 SE variants at that grid point; Avg Stacked/Gated average their variants\)\. The reported interval is the minimum–maximum AUROC over all 15 grid points\. CoCoA SP/PPL have zero\-width ranges because CoCoA has neither a clustering threshold nor a regularisation strength to sweep\.

## Appendix FStatistical significance and uncertainty

This section complements the point\-estimate results with significance tests and bootstrap confidence intervals\. The significance tests assess whether method scores differ between hallucinated and non\-hallucinated queries, while the bootstrap intervals quantify uncertainty in the resulting AUROC estimates\.

### F\.1Significance tests

Table 15:Mann–Whitney U testpp\-values for separation between hallucinated and non\-hallucinated out\-of\-fold scores\. The available artifact does not contain a Gated Spectral significance row\.∗p<0\.05\{\}^\{\*\}p<0\.05,p∗⁣∗<0\.01\{\}^\{\*\*\}p<0\.01, and∗∗∗p<0\.001\{\}^\{\*\*\*\}p<0\.001\.Table[15](https://arxiv.org/html/2609.02679#A6.T15)reports Mann–WhitneyUUtests on the pooled out\-of\-fold method scores\. All three Stacked variants significantly separate the labels on AA Omni Finance, Cheque Generation, and SQuAD, but not on AmbigQA, HotpotQA, Long\-Text QA, or Financial Summaries atp<0\.05p<0\.05\. Gated Hybrid is significant on five datasets, with Long\-Text QA and Financial Summaries as the exceptions\. AmbigQA shows the clearest significance for semantic\-only methods\. CoCoA SP and CoCoA PPL are significant on AA Omni Finance, AmbigQA, Cheque Generation, and HotpotQA; CoCoA SP is additionally significant on Long\-Text QA\. Neither CoCoA variant reaches significance on SQuAD or Financial Summaries\. On Financial Summaries this mirrors the weak, non\-significant separation semantic\-only methods also show there\. On SQuAD, however, four of the five semantic\-only methods do reach significance \(Standard SE, UEigV, Hybrid, Spectral Epistemic; allp<0\.03p<0\.03\) despite sub\-chance AUROC \(≈\\approx0\.48\) — indicating a significant but inverted relationship, consistent with the positional\-entropy inversion for SQuAD discussed in Section[4\.2](https://arxiv.org/html/2609.02679#S4.SS2), rather than an absence of signal\. These unadjusted tests are exploratory and should be interpreted alongside effect sizes and confidence intervals rather than as evidence that one method is universally superior\.

### F\.2Bootstrap confidence intervals and AUROC aggregation

Table[7](https://arxiv.org/html/2609.02679#A5.T7)reports the mean of five held\-out fold AUROCs, whereas Table[16](https://arxiv.org/html/2609.02679#A6.T16)pools the same out\-of\-fold predictions before computing AUROC\. For the pooled estimates, we obtain 95% confidence intervals from 2,000 query\-level percentile\-bootstrap resamples\. The two aggregation procedures produce similar values in most cells, although small differences can change the leading method when methods perform closely\.

##### Aggregation\-sensitive rankings\.

The leading method category is stable under both aggregation procedures on six of seven datasets\. Long\-Text QA is the exception: mean fold AUROC ranks Gated Hybrid first at 0\.731, whereas pooled AUROC ranks CoCoA SP first at 0\.728 and Gated Hybrid at 0\.688\. AmbigQA and SQuAD also exhibit small reversals between variants within the same method category, but the corresponding differences are approximately0\.0010\.001\. These changes show that close rankings should not be interpreted without their uncertainty intervals\.

##### Interpreting the intervals\.

The leading estimate overlaps at least one competing method on every dataset\. For example, Stacked Hybrid reaches 0\.746 on AA Omni Finance with a 95% interval of\[0\.67,0\.82\]\[0\.67,0\.82\], overlapping TopK at 0\.735\[0\.66,0\.81\]\[0\.66,0\.81\]\. Uncertainty is especially large for Long\-Text QA because it contains only 30 evaluated queries\. The results therefore support broad cross\-dataset patterns, but do not establish statistically distinct superiority for an individual method on any single benchmark\.

Table 16:Pooled out\-of\-fold AUROC with 95% bootstrap confidence intervals\(2000 resamples; percentile method\) for every \(dataset, method\) cell in Table[7](https://arxiv.org/html/2609.02679#A5.T7)\. Point estimates here are the pooled\-OOF AUROC, which can differ slightly from Table[7](https://arxiv.org/html/2609.02679#A5.T7)’s mean\-of\-fold AUROC \(see that table’s footnote and §[F\.2](https://arxiv.org/html/2609.02679#A6.SS2)for the two datasets where this changes the outright winner\)\.