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From Tokens to Semantics: Leveraging Complementary Signals for Hallucination Detection in Black-Box LLMs
Urja Pawar, Rajitha Ramanayake, Owen O'Neill, Nabeel Kemal, Abhishek Mandal, Houssem Chatbri, Christopher Martin
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Summary
This paper investigates hallucination detection in black-box Large Language Models (LLMs) using two complementary signals: semantic entropy (measuring disagreement among sampled responses) and token log-probability uncertainty. The authors propose and evaluate several methods, including TopK (aggregating token signals), CoCoA (hybrid score), Gated (routing based on semantic clusters), and Stacked (joint learning). Evaluations across seven benchmarks and four models show that no single method is universally best; supervised methods like Stacked perform well in many cases, while unsupervised methods like TopK and CoCoA remain competitive. The study highlights the complementary failure modes of semantic and token signals.
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Stacked → evaluatedon → SQuAD
confidence 95% · We evaluate seven benchmarks... including... SQuAD... using four language models
Stacked → evaluatedon → AA Omni Finance
confidence 95% · We evaluate seven benchmarks... including... AA Omni Finance... using four language models
Stacked → evaluatedon → Cheque Generation
confidence 95% · We evaluate seven benchmarks... including... Cheque Generation... using four language models
Stacked → evaluatedon → Financial Summaries
confidence 95% · We evaluate seven benchmarks... including... Financial Summaries... using four language models... Stacked gave the best performance
Stacked → evaluatedon → Long-Text QA
confidence 95% · We evaluate seven benchmarks... including... Long-Text QA... using four language models... Stacked gave the best performance
Stacked → evaluatedon → AmbigQA
confidence 95% · We evaluate seven benchmarks... including... AmbigQA... using four language models
Stacked → evaluatedon → HotpotQA
confidence 95% · We evaluate seven benchmarks... including... HotpotQA... using four language models
CoCoA → uses → Token Log-Probability
confidence 95% · CoCoA... combines target-response uncertainty with semantic dissimilarity
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Abstract
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 our TopK method, 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, and Stacked, 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), using 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.
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- Source: https://arxiv.org/abs/2609.02679v1
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From Tokens to Semantics: Leveraging Complementary Signals for Hallucination Detection in Black-Box LLMs Urja Pawar BNY urja.pawar@bny.com Rajitha Ramanayake BNY rajitha.ramanayake@bny.com Nabeel Kemal BNY nabeel.kemal@bny.com Owen O’Neill BNY owen.oneill@bny.com Abhishek Mandal BNY abhishek.mandal@bny.com Houssem Chatbri BNY houssem.chatbri@bny.com Christopher Martin BNY Christopher.Martin@bny.com 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 mea- sures 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 our TopK method, 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, and Stacked, which learns jointly from semantic uncer- tainty and broader token features. We evaluate seven benchmarks—including five public benchmarks (four text datasets and multimodal handwritten-cheque extrac- tion) 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. 1 Introduction Large language models increasingly support information workflows in regulated organizations and public-facing services [Sajadieh et al., 2026]. In these settings, fabricated claims or incorrect figures can mislead downstream reviewers and users, while false alarms consume limited human-review Preprint. arXiv:2609.02679v1 [cs.CL] 2 Sep 2026 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, Wang et al., 2025]. 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) samplesNresponses, groups them by meaning, and measures uncertainty from their distribution across semantic clus- ters [Kuhn et al., 2023, Farquhar et al., 2024]. Token-log-probability methods instead estimate confidence within a response using sequence likelihood, perplexity, top-kentropy, and candidate margins (the differences between the highest and second-highest token log-probabilities) [Kadavath et al., 2022, Duan et al., 2024, Shapiro et al., 2026]. Because SE already requires multiple responses, we compute token signals for each response and aggregate them across the same samples. Our scalar TopK score combines mean top-kentropy with cross-response confidence variation. We also evaluate CoCoA [Vashurin et al., 2025], 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). 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). 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. 2 Complementary Black-Box Hallucination Signals 2.1 Related work Semantic Entropy. Semantic entropy (SE) [Kuhn et al., 2023, Farquhar et al., 2024] drawsN responses to a promptx, groups them into semantic equivalence classesC = C 1 ,...,C K , and measures uncertainty over the empirical class distribution: SE(x) =− K X k=1 ˆp(C k ) log ˆp(C k ),ˆp(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]. 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, Nguyen et al., 2025]. 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]. Semantic Energy augments semantic classes with penultimate-layer logits but requires white-box access [Ma et al., 2025]. Semantically Diverse Language Generation (SDLG) elicits diverse alter- 2 (a) Semantic clustering C 1 C 2 C 3 high SE→ hallucination✓ (b) Token log-probabilities one generation r i high token uncertainty → hallucination✓ (c) Semantic-clustering failure C 1 (all wrong) r 1 r 2 r N SE=0✗ token uncertainty high✓ (d) Token-log-probability failure H topk low✗ |S|=2✓ (e) Multi-response TopK N resp. token log-probs aggregate across N U topk = mean(H topk ) + Var( logp) (f) Gated cascade N resp. |S|? SE score token-feature classifier > 1 = 1 P(hallu) (g) Stacked classifier semantic + spectral aggregated token combine features PCA + logistic P(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. natives from one model, while cross-model disagreement compares similarities within and across models [Aichberger et al., 2025, Hamidieh et al., 2026]. 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 to TRUE as confidence [Malinin and Gales, 2021, Kadavath et al., 2022]. Shifting Attention to Relevance (SAR) weights uncertainty associated with semantically important tokens and responses [Duan et al., 2024]. EAS accumulates entropy along a reasoning sequence, whereas Hallucination Assessment via Log-probs as Time Series (HALT) learns temporal patterns from top-ktoken log-probabilities [Zhu et al., 2025, Shapiro et al., 2026]. HaluNet also uses hidden-state features, placing it outside our black-box setting [Tong et al., 2025]. 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]. We instead aggregate broader token-probability features across allNresponses and combine them with query-level semantic signals. Other approaches require either additional context or hidden-state access [Wang et al., 2025, Chen et al., 2024, Sriramanan et al., 2024, Min et al., 2023, Es et al., 2024]. 2.2 Proposed methods The different failure modes of semantic and token signals motivate three methods using the sameN sampled 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 theNresponses using means, variances, quantiles, ranges, and extreme values. TopK uses only mean top-kentropy 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 as token confidence; positional confidence changes and response length 3 as response dynamics; cross-sample spreads and extremes as across-response aggregation; and response diversity and Von Neumann entropy as semantic diversity. Gated cascade.The Gated cascade uses the semantic-cluster countKfor routing. WhenK ≥ 2, it returns the selected semantic entropy score; whenK = 1, empirical SE is zero by construction, so it uses a logistic-regression classifier over the aggregated token features: score(x) = SE m (x),K ≥ 2, σ w ⊤ f tok (x) + b , K = 1. (2) Here,f tok (x)is the token-feature vector aggregated across theNresponses, andσconverts the classifier output into a hallucination probability. Gated (Hybrid) uses Hybrid SE on the semantic branch, whereas Gated (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, or Von Neumann. The features are standardised, reduced using principal component analysis (PCA), and passed toL 2 -regularised logistic regression, allowing both signals to influence every prediction (for details on feature inputs see Appendix C.4). 3 Methodology and Experimental Setup For each query, we compute several semantic uncertainty scores and the multi-response token representation from Section 2.2, 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. 3.1 Methods and Variants Table 1 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. MethodDescription Standard SEEmpirical entropy of the semantic-cluster frequencies in Eq. 1. SE UEigVAdjusts the finite-sample class estimate using eigenvalues of the response-similarity graph. SE HybridCombines UEigV with a Good–Turing correction based on singleton classes, using the larger estimate. SE Von NeumannVon Neumann entropy of a density matrix derived from graded response similarities. TopK Combines mean top-kentropy with confidence variation across responses; uses no supervised training labels but requires threshold calibration. Spectral EpistemicEpistemic component obtained by decomposing spectral uncertainty [Walha et al., 2025]. CoCoA SPMultiplies target sequence-level uncertainty by its average semantic dissimilarity to sampled alternatives; uses no supervised training labels but requires threshold calibration. CoCoA PPL Uses the length-normalised target uncertainty within the same CoCoA combination; uses no supervised training labels but requires threshold calibration. Gated HybridUses Hybrid SE for multiple-cluster queries and an aggregated-token classifier for one-cluster queries. Gated SpectralUses Von Neumann entropy for multiple-cluster queries and the same aggregated-token classifier for one-cluster queries. Stacked HybridClassifier combining the token vector with Hybrid SE, estimated semantic alphabet size, and cluster count. Stacked SpectralClassifier combining the token vector with total spectral entropy, effective rank, epistemic uncertainty, and cluster count. Stacked Von NeumannClassifier combining the token vector with Von Neumann entropy and cluster count. We embed theNresponses usingtext-embedding-3-largeand use pairwise cosine similarity to measure their semantic similarity. For Standard SE, UEigV, and Hybrid SE, a dataset-specific thresholdτdetermines whether a response is assigned to an existing semantic class. We selectτ using 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 contains full implementation details. 4 Table 2: Dataset profiles and hyperparameters.nis the number of valid labelled examples used in five-fold cross-validation. Complexity values are qualitative 0–5 ratings defined below.τis the cosine-similarity clustering threshold, andCis theL 2 regularisation strength for Gated (G) and Stacked (S). ComplexityC DatasetDescriptionnCtxAnsReasDom τGS A Omni FinanceFinance/law knowledge eval; 4-point grading [Team, 2025] 20001240.951.01.0 AmbigQA Ambiguous open-domain QA; multiple valid answers [Min et al., 2020] 19803100.950.10.1 HotpotQAMulti-hopreasoningover Wikipedia [Yang et al., 2018] 19921400.850.10.1 Cheque GenerationHandwritten cheque VQA; numeric ex- traction [Verma, 2024] 15452130.850.11.0 Financial Summaries Synthetic finance texts with confusable entities (author-constructed) 17114330.901.00.1 Long-Text QA Multi-part questions over regulatory fil- ings (author-constructed) 3035340.9010.01.0 SQuADExtractive reading comprehension [Ra- jpurkar et al., 2016] 20021100.800.10.1 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. 3.2 Datasets 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 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). 3.3 Models, 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 = 10responses per prompt, temperature 1.0, andk = 5requested log-probability candidates per token. GPT-5.4 returned only one candidate, giving effectivek = 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 5 Table 3: GPT-5.4 AUROC by method and dataset. Values are mean held-out AUROC over five folds. Text datasets useN = 10and Cheque Generation usesN = 20. The API returned one candidate per token, giving an effectivek = 1for token-based methods. Bold marks the highest value per dataset. MethodAA Fin. AmbigQA Summ. Fin. HotpotQA SQuAD Long QA Cheque Standard SE0.5070.6230.4920.6240.4810.6200.500 SE UEigV0.5060.6300.4920.6230.4810.5950.500 SE Hybrid0.5070.6340.4920.6280.4810.5950.500 SE Von Neumann0.5380.6290.7500.6680.5490.6200.774 TopK0.7210.6300.5600.7090.7450.6850.812 Spectral Epistemic0.5040.6150.4920.6310.4810.6200.500 CoCoA SP0.5800.6700.6940.6810.6210.7250.770 CoCoA PPL0.5860.6480.6470.7020.6080.6200.770 Gated Hybrid0.6110.6490.4090.6910.6550.4350.809 Gated Spectral0.5830.6380.4090.6500.6550.4350.809 Stacked Hybrid0.7130.6110.5160.7100.6780.7250.802 Stacked Spectral0.7140.6290.6300.7200.6600.6250.808 Stacked Von Neumann0.7140.6200.5410.7010.6590.6750.804 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 reports confidence intervals, and Appendix F.1 reports significance tests). Sensitivity analyses. We vary the response countN ∈ 3, 5, 7, 10, 15, 20, temperature in0.3, 0.5, 0.7, 1.0, 1.2, returned candidatesk ∈ 1,..., 5, clustering thresholdτ ∈ 0.80, 0.85, 0.90, 0.95, 0.98, and regularisation strengthC ∈ 0.1, 1, 10. Cross-model compar- isons reuse the selectedτandCvalues. Section 4 summarises the findings, and Appendix E reports the complete ablation results. 4 Results 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 for the failure mode analysis). 4.1 Performance 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 compar- isons, 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. Figure 2 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 6 1%2%3%5%10%15% 0.0 0.5 1.0 TPR (recall) A Omni Finance 1%2%3%5%10%15% 0.0 0.5 1.0 AmbigQA 1%2%3%5%10%15% 0.0 0.5 1.0 Cheque Generation 1%2%3%5%10%15% FPR budget 0.0 0.5 1.0 HotpotQA 1%2%3%5%10%15% FPR budget 0.0 0.5 1.0 TPR (recall) Long Text QA 1%2%3%5%10%15% FPR budget 0.0 0.5 1.0 SQuAD 1%2%3%5%10%15% FPR budget 0.0 0.5 1.0 Summarisation Finance Best SE TopK Avg Stacked Avg Gated CoCoA TPR @ FPR — pooled across model roster, with CoCoA (5-fold CV distribution) Figure 2: TPR at fixed FPR budgets. Boxes pool five-fold TPR values across available model runs at fixed calibration. Text runs useN = 10andT = 1.0; Cheque Generation usesN = 20and includes GPT-4.1-mini, GPT-5.1, and GPT-5.4. Long-Text QA has n = 30. A Finance AmbigQA Summ. Finance HotpotQA SQuAD Long Text QA Cheque 0.0 0.1 0.2 0.3 0.4 0.5 AUROC shortfall from within-pair best SE / spectralTopKGatedStackedCoCoA 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. on AmbigQA and HotpotQA, Stacked on A 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 A Omni Finance and SQuAD. Increasing the FPR budget from 5% to 15% substantially improves hallucination coverage on A 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 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 for the model-family ablation results). 7 4.2 Why performance varies across datasets Figure 4 compares how well the four feature groups used by the Stacked classifier (defined in Section 2.2) separate hallucinated from non-hallucinated responses. Separation is measured using absolute Cohen’sd, the standardised difference between the two groups’ feature means. Here, the 23 settings are the canonical configuration plus response-count, temperature, generating-model, top-k, 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 A 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 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 compares how token uncertainty develops across the response rather than reducing each response to a single average. In A 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 for the complete token-level analysis). 4.3 Conditional 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 A 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. 8 A Omni Finance AmbigQA Summarisation Finance HotpotQA SQuAD Long Text QA Cheque Generation 0.00 0.25 0.50 0.75 1.00 1.25 Maximum absolute Cohen's d Token confidenceResponse dynamicsAcross-response aggregationSemantic diversity Figure 4: Discriminative associations of feature groups by dataset. Bars show the median, across ablation runs, of the largest absolute Cohen’sdwithin each group; whiskers show the interquartile range. 0255075100 Token position (%) 0.2 0.4 0.6 Mean top- k entropy A Omni Finance 0255075100 Token position (%) 0.1 0.2 0.3 AmbigQA 0255075100 Token position (%) 0.1 0.2 HotpotQA 0255075100 Token position (%) 0.0 0.1 0.2 Long Text QA 0255075100 Token position (%) 0.0 0.2 0.4 0.6 Mean top- k entropy Cheque Generation 0255075100 Token position (%) 0.1 0.2 0.3 Summarisation Finance 0255075100 Token position (%) 0.0 0.1 0.2 0.3 SQuAD Non-Hallucination ±1 SE (Non-Hallucination) Hallucination ±1 SE (Hallucination) Figure 5: Positional top-kentropy profiles. Mean top-kentropy over normalised token position with ±1standard-error bands. Separation is persistent for A Omni Finance, emerges later for several QA tasks, is weak for Cheque and Financial Summaries, and is inverted near the opening of SQuAD responses. 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 A 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 for the complete generation and calibration sweeps). 5 Limitations Gated and Stacked require labelled target-domain hallucination data; zero-shot transfer is not eval- uated. 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 further reports significance tests and bootstrap confidence intervals). 6 Conclusion 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 9 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. 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Uncertainty under the curve: A sequence-level entropy area metric for reasoning LLM. arXiv preprint arXiv:2508.20384, 2025. URL https://arxiv.org/abs/2508.20384. Appendix The appendix is organised into six parts. Appendix A provides further details on the evaluated uncertainty estimation methods, including CoCoA and the Gated and Stacked algorithms. Appendix B describes the experimental implementation, including the embedding model, clustering configuration, and classifier settings. Appendix C documents the datasets, token-level features, dataset-construction procedures, and prompts used to obtain hallucination labels. Appendix D 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 reports the response-count, temperature, top-k, model, and calibration ablations, while Appendix F provides statistical significance tests and bootstrap confidence intervals. CoCoA is included in direct comparisons, ROC curves, significance tests, bootstrap intervals, and the top-ktable, and appears as an invariant reference in calibration because it uses neitherτnorC. 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. 12 A Uncertainty 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.1 Semantic and spectral uncertainty measures All semantic baselines are computed from the sameNsampled responses and their pairwise cosine- similarity matrix. Standard SE uses the empirical semantic-cluster distribution defined in Eq. 1. 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 clusters K and singleton clusters n 1 : b K GT = KN N − n 1 . UEigV instead estimates the effective number of clusters by counting eigenvalues of the normalised similarity-graph Laplacian below a thresholdε. 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]. 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]. A.2 TopK For responsei, letH (k) i denote its mean token entropy after renormalising the returned candidate probabilities over the availablekcandidates at each token position, and let ̄ ℓ i denote its mean chosen-token log-probability. The TopK score is U TopK (x) = 1 N N X i=1 H (k) i + Var i=1,...,N ̄ ℓ i . Higher values indicate greater uncertainty. When only one candidate is returned,H (1) i = 0, so the score retains only the across-response confidence-variation term. A.3 CoCoA CoCoA combines the confidence of a target responsey ∗ with its semantic disagreement fromN sampled alternatives [Vashurin et al., 2025]: U CoCoA (y ∗ | x) = u(y ∗ | x) 1 N N X i=1 h 1− s y ∗ ,y (i) i , wheres(·,·)is cosine similarity between response embeddings. We evaluate two variants: CoCoA SP uses sequence-level uncertainty,u SP =− P L t=1 logp(y ∗ t | y ∗ <t ,x) , whereas CoCoA PPL uses its length-normalised form,u PPL =−L −1 P L t=1 logp(y ∗ t | 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 allLtokens while PPL divides byL, SP is sensitive to response length whereas PPL is not. A.4 Gated 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 summarises both procedures. 13 Algorithm 1 Gated and Stacked uncertainty detection Require:Queryx, language modelp θ , response countN, semantic variantm, fitted Gated classifier F G , fitted Stacked pipelineF S , method d Ensure: Uncertainty score s 1: Sample responsesy (i) N i=1 ∼ p θ (·| x) and retain their token log-probabilities 2: Embed the responses and compute their pairwise cosine similarities 3: Form semantic clustersC =C 1 ,...,C K 4: Compute semantic score u m , semantic feature block f m sem , and aggregated token features f tok 5: if d = GATED then 6:if K ≥ 2 then 7: s← u m 8:else 9: s←F G (f tok ) 10:end if 11: else if d = STACKED then 12:f ← [f m sem ,K, f tok ] 13: s←F S (f ) 14: end if 15: return s B Additional 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-large and 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; otherwise, a new cluster is created. The dataset-specific thresholds in Table 2 are selected on held-out validation data and then fixed before cross-validation. UEigV uses a Laplacian-eigenvalue threshold ofε = 0.1, whereas the Von Neumann and spectral measures operate directly on the continuous similarity matrix. Supervised classifiers. Both supervised methods useL 2 -regularised logistic regression with 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 strengths C are reported in Table 2. C Datasets, 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.1 Public datasets Our public benchmarks cover domain knowledge, ambiguous questions, multi-hop reasoning, ex- tractive question answering, and multimodal text recognition. They comprise A Omni Finance, AmbigQA, HotpotQA, SQuAD, and Cheque Generation. 14 1.A Omni Finance [Team, 2025]: the finance and law subset (∼200questions) of the A-Omniscience cross-domain knowledge evaluation benchmark (∼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.AmbigQA [Min et al., 2020]: derived from Google Natural Questions [Kwiatkowski et al., 2019], this dataset (∼12,000questions, C 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.HotpotQA [Yang et al., 2018]: a multi-hop QA dataset (∼7,400dev questions, C 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.Cheque Generation [Verma, 2024]: 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. SQuAD [Rajpurkar et al., 2016]: the Stanford Question Answering Dataset (100,000+ques- tions over500+Wikipedia articles, C 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.2 Custom 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. 1.Financial Summaries (finance domain-specific, 200 sampled,n=171with a valid halluci- nation 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.Long-Text QA (finance domain-specific,n=150sub-questions,n=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 summarises every dataset used in this work together with its license, source URL, sample size, and the modifications we applied. C.3 Token-level feature catalogue Table 5 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 theNsampled responses; the exact subsets used by Gated and Stacked are specified in Appendix C.4. C.4 Exact 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. 15 Table 4: Datasets used in this paper: licenses, sources, and modifications.nhere is the raw number of items sampled/generated before hallucination labelling; Table 2 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). DatasetLicensenSourceModifications AmbigQA Min et al. [2020]C BY-SA 3.0200 https://huggingface. co/datasets/sewon/ ambig_qa Randomly sampled 200 questions; reformatted into Question/Answer columns. SQuAD Rajpurkar et al. [2016] C BY-SA 4.0200 https://huggingface. co/datasets/rajpur kar/squad Sampled 200 questions; reformatted with context passages. HotpotQA Yang et al. [2018]C BY-SA 4.0200 https://huggingface. co/datasets/hotpot qa/hotpot_qa Sampled 200 questions from the dev split; reformatted. A Omni Finance Team [2025] Apache 2.0 ∼200 https://huggingface. co/datasets/Artifi cialAnalysis/A-Omn iscience-Public Extracted finance subset; wrapped with domain-specific prompt. Cheque Generation Verma [2024] Apache 2.0200 https://huggingface. co/datasets/aniket Verma07/handwritte n_cheque_vqa_datas et 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. Gated cascade,|S| = 1branch. Features are aggregated across theNsampled responses as described in §2.2, 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-klog-probabilities are unavailable (has_topk=False, e.g. some model and API configu- rations; see Appendix E.4), 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. 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.5 Dataset 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. 16 Table 5: Complete per-response token-level features. Features marked ⋆ are highlighted in §2.2. FeatureSourceDescription Token log-probability features mean_logprobℓ (1) t Mean chosen-token log-prob min_logprobℓ (1) t Min chosen-token log-prob max_logprobℓ (1) t Max chosen-token log-prob std_logprobℓ (1) t Std of chosen-token log-probs p90_logprobℓ (1) t 10th percentile (= 90th uncertainty pctl) p95_logprobℓ (1) t 5th percentile (= 95th uncertainty pctl) logprob_varianceℓ (1) t Variance of chosen-token log-probs perplexity ⋆ ℓ (1) t exp(− 1 T P t ℓ (1) t ) length_norm_seqlogprobℓ (1) t Length-normalised sequence log-prob seq_lengthN/ANumber of tokens T i last1_logprobℓ (1) t Log-prob of last token last3_mean_logprobℓ (1) t Mean log-prob of last 3 tokens mean_lp_delta∆ℓ t Mean|ℓ (1) t+1 − ℓ (1) t | max_abs_lp_delta∆ℓ t Max|ℓ (1) t+1 − ℓ (1) t | std_lp_delta∆ℓ t Std of log-prob deltas Top-k features (require top-k log-probs) mean_topk_entropy ⋆ H topk (t)Mean top-k entropy (§2.1) max_topk_entropyH topk (t)Max top-k entropy min_topk_entropyH topk (t)Min top-k entropy std_topk_entropyH topk (t)Std of top-k entropy p90_topk_entropyH topk (t)90th percentile of top-k entropy p95_topk_entropyH topk (t)95th percentile of top-k entropy last1_topk_entropyH topk (t)Top-k entropy of last token last3_mean_topk_entropy H topk (t)Mean top-k entropy of last 3 tokens mean_ent_delta∆H t Mean|H topk (t+1)− H topk (t)| max_abs_ent_delta∆H t Max entropy delta (spike signal) std_ent_delta∆H t Std of entropy deltas mean_entropy_alts ⋆ ranks 2–kMean entropy over non-greedy tokens max_entropy_altsranks 2–kMax entropy over non-greedy tokens mean_margin ⋆ ℓ (1–2) t Mean (ℓ (1) t − ℓ (2) t ) min_marginℓ (1–2) t Min margin (least decisive position) std_marginℓ (1–2) t Std of margins p10_marginℓ (1–2) t 10th percentile margin frac_non_greedytop-kFraction of positions where chosen̸= argmax C.5.1 Long-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 17 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 lists all 30 question sets. The unanswerable sub-question (Q5) in each set is marked with†. Table 6: Long-Text QA: all 30 question sets. Each set contains five sub-questions posed over the indicated source document. Q5 (marked†) is deliberately unanswerable from the source; the expected answer is “Not provided in context.” SetSourceSub-questions 1SR 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†: 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†: 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†: What specific software tools does SR 11-7 recommend for model valida- tion? 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†: 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†: 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†: What cloud computing requirements does SR 11-7 specify? Continued on next page 18 SetSourceSub-questions 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†: 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.X]% Fixed-to-Floating Rate Preferred Capital Securi- ties. Q5†: 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†: 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†: 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 regis- tered? Q4: How much do the preferred securities initially pay? Q5†: 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†: 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†: 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 disclo- sure controls and procedures? Q3: Who is the oldest Executive Officer? Q4: Do the holders of Common Stock have cumulative voting rights? Q5†: Do the holders of Dispersed Stock have cumulative voting rights? Continued on next page 19 SetSourceSub-questions 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†: 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] busi- nesses? Q5†: 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†: 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†: 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†: 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†: 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†: What is the company’s stock ticker symbol for NSE? Continued on next page 20 SetSourceSub-questions 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†: 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†: 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†: 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†: What was the effective tax rate in 2003? 2610-KQ1: What was the FDIC special assessment accrual recorded in 2023 noninter- est 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 Erro- neously Awarded Incentive-Based Compensation Policy? Q5†: 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†: 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†: 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†: Is the Act applicable to the United Kingdom of Great Britain and Ireland? Continued on next page 21 SetSourceSub-questions 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†: What exemptions are provided for BigTech companies under the Act? C.5.2 Financial Summaries The following prompt template is used to generate the synthetic finance texts in our summarisation hallucination dataset (Section 3.2). 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. The text must be approximately 500 words (450–550 acceptable range). 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. 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. The text must be self-contained (no references to external documents). 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. 22 Per-type trigger instructions Thetrigger_instructionsplaceholder 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. 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. Interleave metrics and attributes across entities so that a careless reader (or LLM) could easily attribute Entity A’s figures to Entity B. 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. Ensure each entity has distinct, verifiable facts that must not be swapped. NE – Numerical Error. STRUCTURAL REQUIREMENTS for Numerical Error triggers: 1. Pack the text with at least 8–10 specific numbers: revenue figures, percentages, basis points, dates, headcounts, or dollar amounts. 2. Include numbers that differ by small amounts (e.g. 3.2% vs 3.7%, $14.2B vs $14.8B). 3. Mix absolute values with percentages and year-over-year changes in the same paragraph. 4. Include at least one number that contradicts an intuitive expectation. NF – Negation Flip. STRUCTURAL REQUIREMENTS for Negation Flip triggers: 1. Include at least 2–3 negated statements that carry critical meaning (e.g. “the board did NOT approve”, “excluding derivatives exposure”). 2. Bury negations in subordinate clauses or complex sentence structures. 3. Use double negatives, “except”/“excluding”/“unless” constructions. 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. Present events in non-chronological order (e.g. mention Q4 results before Q2 events). 2. Include at least 4–5 distinct date references (specific months, quarters, years). 3. Describe actions that were planned, deferred, completed, or reversed across different time periods. 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. Juxtapose two events or facts that are temporally or thematically related but have NO stated causal connection. 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. Include at least 2 such juxtaposition pairs. 4. Explicitly avoid causal language (“because”, “therefore”, “as a result”) between the juxtaposed events. UI – Unsupported Inference. STRUCTURAL REQUIREMENTS for Unsupported Inference triggers: 1. Present raw data, metrics, or observations WITHOUT drawing conclusions. 2. A summariser is likely to add interpretive statements like “this suggests”, “indicating strong performance”, or “raising concerns about”. 3. Include data that could support multiple contradictory interpretations. 4. Avoid any evaluative or interpretive language in the source text. 23 DF – Detail Fabrication. STRUCTURAL REQUIREMENTS for Detail Fabrication triggers: 1. Leave deliberate gaps: mention “a major US bank” without naming it, refer to “the acquiring firm” without specifics, cite “an undisclosed sum”. 2. Include partial information that a summariser might “complete” by fabricating plausible details (names, amounts, locations). 3. Reference unnamed parties, unspecified contract terms, or redacted figures. 4. Include at least 3 such deliberate vagueness points. MC – Merging Claims. STRUCTURAL REQUIREMENTS for Merging Claims triggers: 1. Include at least 2 pairs of related but distinct claims in adjacent sentences. 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. Make the claims similar enough to invite merging but with distinct specifics. OQ – Omission of Qualifiers. STRUCTURAL REQUIREMENTS for Omission of Qualifiers triggers: 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. At least 4–5 key claims should be heavily qualified. 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. Make every claim narrowly scoped: specify region, time period, client segment, product line, or division. 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. Include at least 3–4 such narrowly scoped claims. CTRL – Control (easy). STRUCTURAL REQUIREMENTS for Control (easy) texts: 1. Write a clear, well-structured finance text with unambiguous facts. 2. Use straightforward sentence structure with no buried negations or tricky phrasing. 3. Include a reasonable number of facts (4–6) that are easy to verify in a summary. 4. Avoid any adversarial structural patterns. This text should be easy to summarise correctly. C.6 LLM 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.6.1 Blind 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. 24 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. Suspiciously specific details (e.g. invented names, dates, statistics) 2. Contradiction of widely known facts 3. False confidence about uncertain or obscure topics 4. Internal inconsistency within the answer 5. Confabulation: plausible-sounding but fabricated information Return: "hallucination_probability": <0.0–1.0>, "confidence": <0.0–1.0>, "reasoning": "..." C.6.2 Binary 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] and Farquhar et al. [2024], 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.6.3 Scalar 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": "..." 25 C.6.4 Summarisation 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. D Extended 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. 26 D.1 Complementary failure modes Figures 6 and 7 examine the principal blind spot of semantic entropy. Among hallucinated queries, the proportion whoseNresponses 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. AmbigQA A Omni Finance Cheque Generation HotpotQA Long Text QA SQuAD Summarisation Finance 0 50 100 150 200 Hallucinated samples 39\% 60\% 76\% 85\% 93\% 95\% 99\% |S| = 1 (SE blind)|S| > 1 (SE has signal) Figure 6: Single-cluster collapse among hallucinated queries. Hallucinated queries are divided according to whether theirNsampled responses form one semantic cluster (K = 1) or multiple clusters (K ≥ 2). Standard SE is necessarily zero in the single-cluster group. Percentages above the bars report the K = 1 share for each dataset. SQuAD Summarisation Finance Cheque Generation AmbigQA Long Text QA HotpotQA A Omni Finance 0.0 0.1 0.2 0.3 Median U topk within | S | = 1 Non-hallucinated, |S| = 1 (median)Hallucinated, |S| = 1 (median) Figure 7: TopK uncertainty within the semantic-entropy blind spot. The analysis is restricted toK = 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. 27 D.2 Token-feature discriminability Figure 8 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. 0.00.20.40.60.81.0 Best token feature |d| 0.5 0.6 0.7 0.8 0.9 1.0 Stacked classifier AUROC SQuAD Long Text QA A Omni Finance HotpotQA AmbigQA Cheque Generation Summarisation Finance Figure 8: Token-feature discriminability and Stacked performance. Stacked AUROC plotted against the largest absolute Cohen’sdamong 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.3 ROC curves across datasets Figure 9 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. 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR A Omni Finance 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR AmbigQA 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR Summarisation Finance 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR HotpotQA 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR SQuAD 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR Long Text QA 0.00.51.0 FPR 0.00 0.25 0.50 0.75 1.00 TPR Cheque Generation Standard SE SE UEigV SE Hybrid SE Von Neumann TopK Spectral Epistemic (P3) Gated (Hybrid) Gated (Spectral) Stacked (Hybrid) Stacked (Spectral) Stacked (Von Neumann) CoCoA SP CoCoA PPL 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. 28 D.4 Cheque Generation robustness Figure 10 compares method categories across response counts for the three vision-capable models. IncreasingNgenerally 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. 357101520 Sampled responses per prompt, N 0.5 0.6 0.7 0.8 AUROC GPT-4.1-mini 357101520 Sampled responses per prompt, N GPT-5.1 357101520 Sampled responses per prompt, N GPT-5.4 Best SETopKAvg StackedAvg Gated 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. 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. E Ablation studies This section examines sensitivity to model family, response countN, sampling temperature, the number of returned token candidatesk, and classifier calibration. We report each factor separately to distinguish changes in the available uncertainty evidence from changes in the fitted method. E.1 Model-family ablation Table 7 reports the primary GPT-4.1-mini fold-mean results discussed in Section 4.1. Tables 8–11 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 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-specificτandCvalues are held fixed across models rather than retuned, and the single-class Llama Long-Text QA setting is marked as N/A. 29 Table 7: GPT-4.1-mini AUROC by method and dataset on the held-out folds of 5-fold cross-validation. Generation usesN =10, temperature 1.0, and five returned log-probability candidates per token (k=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. MethodAA Fin. AmbigQA Summ. Fin. HotpotQA SQuAD Long QA Cheque Standard SE0.5080.6310.5040.5040.4540.4750.481 SE UEigV0.4690.6180.5040.5000.4540.4750.498 SE Hybrid0.4730.6180.5040.5000.4550.4750.501 SE Von Neumann0.5180.6520.4680.6380.5580.5890.581 TopK0.7470.6370.6110.6640.4510.6810.600 Spectral Epistemic0.5170.6490.5040.5030.4540.4750.482 CoCoA SP0.6870.6670.5130.6810.5750.6830.618 CoCoA PPL0.7040.6540.5160.6960.5460.6610.609 Gated Hybrid0.6190.6250.5750.6770.7290.7310.637 Gated Spectral0.6310.6170.5730.6510.7530.7110.607 Stacked Hybrid0.7630.5320.5710.6210.7580.6830.760 Stacked Spectral0.7560.5600.5690.6280.7620.6830.740 Stacked Von Neumann0.7510.5520.5860.6210.7600.6830.738 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 reports pooled estimates and 95% bootstrap intervals for every cell. E.2 Response-count ablation Figure 11 examines sensitivity to the number of sampled responses. IncreasingNcan 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. 30 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. MethodAA FinanceAmbigQAHotpotQASQuADLong TextSumm. Finance Standard SE0.5220.6340.5400.4420.5620.500 SE UEigV0.5090.6170.5350.4410.5620.500 SE Hybrid0.5150.6070.5330.4420.5620.500 SE Von Neumann0.5610.6230.6760.5240.7710.442 TopK0.6210.5890.6870.4570.7290.309 Spectral Epistemic (P3)0.5340.6550.5370.4440.5620.500 CoCoA SP0.6210.6610.6710.5620.7710.327 CoCoA PPL0.6330.6830.6840.5380.7220.329 Gated (Hybrid)0.5070.5810.6360.7350.5210.240 Gated (Spectral)0.4940.5750.5930.7530.4930.240 Stacked (Hybrid)0.6340.4480.6530.7490.6040.524 Stacked (Spectral)0.6400.4740.6650.7630.6110.506 Stacked (Von Neumann)0.6360.4510.6740.7670.5970.520 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. MethodAA FinanceAmbigQAHotpotQASQuADLong TextSumm. Finance Standard SE0.5150.5030.5550.523N/A0.487 SE UEigV0.5140.5020.5550.523N/A0.487 SE Hybrid0.5160.5060.5570.523N/A0.487 SE Von Neumann0.5010.5380.5690.479N/A0.306 TopK0.5840.5930.6500.541N/A0.416 Spectral Epistemic (P3)0.5130.5070.5560.523N/A0.487 CoCoA SP0.5330.6540.6420.598N/A0.325 CoCoA PPL0.5420.6320.6560.547N/A0.305 Gated (Hybrid)0.6090.5930.5790.621N/A0.389 Gated (Spectral)0.5760.6140.5320.621N/A0.390 Stacked (Hybrid)0.5670.6410.6650.746N/A0.495 Stacked (Spectral)0.5580.6350.6610.744N/A0.515 Stacked (Von Neumann)0.5790.6450.6510.747N/A0.523 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. MethodAA FinanceAmbigQAHotpotQASQuADLong TextSumm. Finance Standard SE0.5230.7060.5790.5620.2400.500 SE UEigV0.5240.7050.5790.5590.1760.501 SE Hybrid0.5220.7060.5780.5600.2160.500 SE Von Neumann0.5330.7250.6610.6350.3280.496 TopK0.7670.6820.7060.5830.7120.440 Spectral Epistemic (P3)0.5300.7130.5790.5570.2560.500 CoCoA SP0.6080.7400.7060.6810.7840.521 CoCoA PPL0.6150.7140.6860.6610.7520.518 Gated (Hybrid)0.6530.6960.6920.6810.2160.481 Gated (Spectral)0.6590.7070.6850.7060.5840.486 Stacked (Hybrid)0.7990.6650.6850.7350.6640.483 Stacked (Spectral)0.8010.6820.6890.7330.6240.472 Stacked (Von Neumann)0.8040.6770.6880.7330.6160.491 31 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC A Omni Finance Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC AmbigQA Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC HotpotQA Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC Long Text QA Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC SQuAD Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK 5101520 Number of samples (N) 0.00 0.25 0.50 0.75 1.00 AUROC Summarisation Finance Stacked (Hybrid) Gated (Hybrid) SE Hybrid TopK AUROC vs Number of samples (N) Figure 11: AUROC versus response count. Performance is shown overN ∈3, 5, 7, 10, 15, 20 for the strongest semantic method, TopK, and the averaged Gated and Stacked variants. The effect of increasingNvaries across datasets and methods. CoCoA is omitted because the required main- answer embeddings are unavailable for this sweep. 32 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. MethodAA FinanceAmbigQAHotpotQASQuADLong TextSumm. Finance Standard SE0.5030.6280.6240.4810.6190.492 SE UEigV0.5060.6360.6240.4810.5950.492 SE Hybrid0.5090.6410.6280.4810.5950.492 SE Von Neumann0.5290.6400.6660.5560.6600.722 TopK0.7250.6330.7030.7360.5850.562 Spectral Epistemic (P3)0.5010.6210.6290.4810.6190.492 CoCoA SP0.5840.6760.6800.6230.7070.679 CoCoA PPL0.5930.6550.6980.6130.6260.641 Gated (Hybrid)0.6110.6510.6840.6050.4690.388 Gated (Spectral)0.5870.6410.6410.6060.4900.388 Stacked (Hybrid)0.6980.6010.7020.5880.7140.496 Stacked (Spectral)0.6980.6270.7130.5730.6940.579 Stacked (Von Neumann)0.6940.6100.6950.5740.7010.522 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. EvidenceCoverageStatus Primary evaluationGPT-4.1-mini, all detector methodspooled OOF N-sweepGPT-4.1-mini, N = 3–20external evaluator Model-family sweeptext-only agentsN/A by modality 33 E.3 Sampling-temperature ablation Sampling temperature changes both response diversity and token confidence. Figure 12 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. 0.40.60.81.01.2 Sampling temperature 0.00 0.25 0.50 0.75 1.00 AUROC (Stacked Hybrid) (a) AUROC vs. temperature A Omni Finance AmbigQA HotpotQA Long Text QA SQuAD Summarisation Finance 0.40.60.81.01.2 Sampling temperature 0.00 0.25 0.50 0.75 1.00 Single-cluster rate ( | S | = 1 ) (b) Single-cluster rate vs. temperature A Omni Finance AmbigQA HotpotQA Long Text QA SQuAD Summarisation Finance 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. Higher response diversity does not consistently improve hallucination separation. CoCoA is omitted because the required main-answer embeddings are unavailable for this sweep. E.4 Top-k candidate ablation Table 13 evaluates the number of token candidates retained at each generation step. Most improve- ments occur when moving fromk = 1tok = 2ork = 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 acrossk: 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 = 5cannot be reconstructed without new generations. E.5 Calibration sensitivity We sweep five clustering thresholdsτ ∈ 0.80, 0.85, 0.90, 0.95, 0.98and three regularisation strengthsC ∈ 0.1, 1, 10, giving 15 calibration settings. Figure 13 shows the method-level minimum–maximum AUROC ranges, with colour indicating interval width; Table 14 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. F Statistical 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.1 Significance tests Table 15 reports Mann–WhitneyUtests on the pooled out-of-fold method scores. All three Stacked variants significantly separate the labels on A Omni Finance, Cheque Generation, and SQuAD, but not on AmbigQA, HotpotQA, Long-Text QA, or Financial Summaries atp < 0.05. Gated Hybrid is significant on five datasets, with Long-Text QA and Financial Summaries as the exceptions. 34 Table 13: Top-klog-probability sensitivity, by detector category. AUROC vs. the number of top-klog-probability alternatives requested per token, recomputed from existing saved responses (no new generation;k > 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 acrossk because both depend only on the generated answer’s own token log-probabilities, never the truncated alternatives. AUROC at k = DatasetCategory12345 A Omni FinanceBest SE0.5430.5430.5430.5430.543 A Omni FinanceTopK0.6210.7040.7210.7280.735 A Omni FinanceAvg Stacked0.6340.6700.7410.7400.742 A Omni FinanceAvg Gated0.5420.5660.5870.6010.611 A Omni FinanceCoCoA SP0.6870.6870.6870.6870.687 A Omni FinanceCoCoA PPL0.7040.7040.7040.7040.704 AmbigQABest SE0.6520.6520.6520.6520.652 AmbigQATopK0.6140.6360.6380.6380.639 AmbigQAAvg Stacked0.5540.5260.5480.5460.543 AmbigQAAvg Gated0.5480.5890.5860.5810.581 AmbigQACoCoA SP0.6670.6670.6670.6670.667 AmbigQACoCoA PPL0.6540.6540.6540.6540.654 Summarisation FinanceBest SE0.5080.5080.5080.5080.508 Summarisation FinanceTopK0.5300.6090.6150.6170.619 Summarisation FinanceAvg Stacked0.5970.5380.5180.5270.556 Summarisation FinanceAvg Gated0.4980.5380.5880.5250.540 Summarisation FinanceCoCoA SP0.5130.5130.5130.5130.513 Summarisation FinanceCoCoA PPL0.5160.5160.5160.5160.516 HotpotQABest SE0.6270.6270.6270.6270.627 HotpotQATopK0.5960.6550.6540.6560.656 HotpotQAAvg Stacked0.6210.5970.6140.5940.599 HotpotQAAvg Gated0.4930.6390.6390.6380.640 HotpotQACoCoA SP0.6810.6810.6810.6810.681 HotpotQACoCoA PPL0.6960.6960.6960.6960.696 SQuADBest SE0.5370.5370.5370.5370.537 SQuADTopK0.4630.4520.4590.4610.460 SQuADAvg Stacked0.7670.7680.7680.7600.759 SQuADAvg Gated0.5350.7380.7380.7390.740 SQuADCoCoA SP0.5750.5750.5750.5750.575 SQuADCoCoA PPL0.5460.5460.5460.5460.546 Long Text QABest SE0.6470.6470.6470.6470.647 Long Text QATopK0.6960.7050.7100.7050.705 Long Text QAAvg Stacked0.6410.6080.5860.5210.611 Long Text QAAvg Gated0.5250.7370.7190.7010.681 Long Text QACoCoA SP0.6830.6830.6830.6830.683 Long Text QACoCoA PPL0.6610.6610.6610.6610.661 Cheque GenerationBest SE0.5630.5630.5630.5630.563 Cheque GenerationTopK0.5680.6020.5980.5940.593 Cheque GenerationAvg Stacked0.7250.7420.7410.7390.731 Cheque GenerationAvg Gated0.4510.6260.6240.6220.617 Cheque GenerationCoCoA SP0.6180.6180.6180.6180.618 Cheque GenerationCoCoA PPL0.6090.6090.6090.6090.609 AmbigQA shows the clearest significance for semantic-only methods. CoCoA SP and CoCoA PPL are significant on A 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 < 0.03) despite sub- chance AUROC (≈0.48) — indicating a significant but inverted relationship, consistent with the positional-entropy inversion for SQuAD discussed in Section 4.2, rather than an absence of signal. 35 A Omni Finance AmbigQA Summarisation Finance HotpotQASQuADLong Text QA Cheque Generation Standard SE SE UEigV SE Hybrid SE Von Neumann Spectral Epistemic (P3) TopK Gated (Hybrid) Gated (Spectral) Stacked (Hybrid) Stacked (Spectral) Stacked (Von Neumann) CoCoA SP CoCoA PPL 0.506--0.5430.540--0.6360.461--0.5080.503--0.6160.454--0.5000.469--0.5710.454--0.640 0.471--0.5160.530--0.6160.459--0.5080.499--0.6170.454--0.4990.469--0.5670.470--0.644 0.478--0.5190.531--0.6140.453--0.5080.501--0.6160.455--0.5040.469--0.5670.470--0.625 0.543--0.5430.652--0.6520.470--0.4700.627--0.6270.537--0.5370.647--0.6470.563--0.563 0.516--0.5430.540--0.6540.453--0.5080.502--0.6080.455--0.4950.469--0.5620.457--0.562 0.735--0.7350.639--0.6390.619--0.6190.656--0.6560.460--0.4600.705--0.7050.593--0.593 0.581--0.6560.537--0.5940.478--0.5980.522--0.6540.678--0.7420.616--0.7100.615--0.702 0.599--0.6470.542--0.6310.522--0.5770.423--0.6250.661--0.7530.500--0.6830.553--0.672 0.730--0.7520.500--0.5620.550--0.5640.553--0.5970.753--0.7810.482--0.6270.689--0.749 0.730--0.7470.513--0.5860.543--0.5590.570--0.6020.753--0.7730.487--0.6230.694--0.726 0.732--0.7470.509--0.5800.548--0.5600.563--0.6020.759--0.7760.487--0.6230.696--0.724 0.687--0.6870.667--0.6670.513--0.5130.681--0.6810.575--0.5750.683--0.6830.618--0.618 0.704--0.7040.654--0.6540.516--0.5160.696--0.6960.546--0.5460.661--0.6610.609--0.609 Min--max AUROC over 15 grid points (colour = max − min), with CoCoA 0.00 0.05 0.10 0.15 0.20 AUROC width Figure 13: Calibration sensitivity, with CoCoA. Cells report the minimum–maximum AUROC over 15(τ,C)settings; colour encodes the width of that range. CoCoA SP/PPL are included as zero-width reference rows. 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.2 Bootstrap confidence intervals and AUROC aggregation Table 7 reports the mean of five held-out fold AUROCs, whereas Table 16 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 approximately0.001. 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 A Omni Finance with a 95% interval of [0.67, 0.82], overlapping TopK at 0.735[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. 36 Table 14: Calibration-threshold sensitivity, by detector category. AUROC sensitivity to clustering thresholdτand regularisationC, swept overτ ∈ 0.80,..., 0.98,C ∈ 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. DatasetCategoryAUROC range over 15 settings A Omni FinanceBest SE0.543–0.543 A Omni FinanceTopK0.735–0.735 A Omni FinanceAvg Stacked0.731–0.748 A Omni FinanceAvg Gated0.590–0.648 A Omni FinanceCoCoA SP0.687–0.687 A Omni FinanceCoCoA PPL0.704–0.704 AmbigQABest SE0.652–0.652 AmbigQATopK0.639–0.639 AmbigQAAvg Stacked0.509–0.576 AmbigQAAvg Gated0.540–0.610 AmbigQACoCoA SP0.667–0.667 AmbigQACoCoA PPL0.654–0.654 Summarisation FinanceBest SE0.470–0.508 Summarisation FinanceTopK0.619–0.619 Summarisation FinanceAvg Stacked0.549–0.561 Summarisation FinanceAvg Gated0.520–0.588 Summarisation FinanceCoCoA SP0.513–0.513 Summarisation FinanceCoCoA PPL0.516–0.516 HotpotQABest SE0.627–0.627 HotpotQATopK0.656–0.656 HotpotQAAvg Stacked0.566–0.599 HotpotQAAvg Gated0.472–0.640 HotpotQACoCoA SP0.681–0.681 HotpotQACoCoA PPL0.696–0.696 SQuADBest SE0.537–0.537 SQuADTopK0.460–0.460 SQuADAvg Stacked0.755–0.777 SQuADAvg Gated0.675–0.747 SQuADCoCoA SP0.575–0.575 SQuADCoCoA PPL0.546–0.546 Long Text QABest SE0.647–0.647 Long Text QATopK0.705–0.705 Long Text QAAvg Stacked0.485–0.624 Long Text QAAvg Gated0.574–0.683 Long Text QACoCoA SP0.683–0.683 Long Text QACoCoA PPL0.661–0.661 Cheque GenerationBest SE0.563–0.644 Cheque GenerationTopK0.593–0.593 Cheque GenerationAvg Stacked0.693–0.731 Cheque GenerationAvg Gated0.590–0.683 Cheque GenerationCoCoA SP0.618–0.618 Cheque GenerationCoCoA PPL0.609–0.609 37 Table 15: Mann–Whitney U testp-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 < 0.01, and ∗ p < 0.001. MethodAA Fin.AmbigQACheque HotpotQA Long QASQuAD Summ. Fin. Standard SE0.9000.002**0.6910.9340.6040.028*0.547 SE UEigV0.4870.009**0.8330.9890.6040.027*0.547 SE Hybrid0.6080.011*0.9880.9850.6040.029*0.547 SE Von Neumann0.6070.001**0.2540.021*0.3290.2330.588 TopK< 10 −4 ***0.004**0.0910.004**0.0590.3540.027* Spectral Epistemic0.705 < 10 −3 ***0.6250.9520.6610.030*0.547 CoCoA SP0.0002***0.0003***0.032* 0.0009***0.036*0.1150.821 CoCoA PPL< 10 −4 ***0.0003***0.045* 0.0005***0.0590.3470.692 Gated Hybrid0.038*0.029*0.021*0.004**0.084 < 10 −4 ***0.125 Gated SpectralN/AN/AN/AN/AN/AN/AN/A Stacked Hybrid< 10 −4 ***0.487 < 10 −4 ***0.0710.318 < 10 −4 ***0.435 Stacked Spectral< 10 −4 ***0.215 < 10 −4 ***0.0570.280 < 10 −4 ***0.484 Stacked Von Neumann < 10 −4 ***0.230 < 10 −3 ***0.0760.299 < 10 −4 ***0.335 Table 16: Pooled out-of-fold AUROC with 95% bootstrap confidence intervals (2000 resamples; percentile method) for every (dataset, method) cell in Table 7. Point estimates here are the pooled- OOF AUROC, which can differ slightly from Table 7’s mean-of-fold AUROC (see that table’s footnote and §F.2 for the two datasets where this changes the outright winner). MethodAA Omni FinanceAmbigQASummarisation FinanceHotpotQASQuADLong Text QACheque Generation Standard SE0.506 [0.42, 0.59]0.632 [0.54, 0.72]0.504 [0.50, 0.51]0.503 [0.43, 0.57]0.454 [0.41, 0.50]0.469 [0.36, 0.58]0.483 [0.40, 0.56] SE UEigV0.471 [0.39, 0.55]0.612 [0.52, 0.69]0.504 [0.50, 0.51]0.499 [0.43, 0.56]0.454 [0.41, 0.50]0.469 [0.36, 0.58]0.492 [0.42, 0.56] SE Hybrid0.478 [0.39, 0.57]0.610 [0.52, 0.69]0.504 [0.50, 0.51]0.501 [0.43, 0.56]0.455 [0.41, 0.50]0.469 [0.36, 0.58]0.499 [0.42, 0.57] SE Von Neumann0.525 [0.43, 0.62]0.653 [0.56, 0.74]0.473 [0.38, 0.57]0.625 [0.51, 0.73]0.551 [0.46, 0.64]0.607 [0.39, 0.84]0.562 [0.46, 0.66] TopK0.735 [0.66, 0.81]0.636 [0.54, 0.73]0.610 [0.51, 0.70]0.656 [0.55, 0.76]0.460 [0.37, 0.55]0.705 [0.50, 0.90]0.593 [0.49, 0.69] Spectral Epistemic (P3)0.516 [0.43, 0.60]0.652 [0.56, 0.74]0.504 [0.50, 0.51]0.502 [0.43, 0.57]0.455 [0.41, 0.50]0.473 [0.36, 0.58]0.479 [0.40, 0.56] CoCoA SP0.685 [0.60, 0.77]0.672 [0.57, 0.76]0.511 [0.42, 0.60]0.679 [0.57, 0.78]0.567 [0.48, 0.65]0.728 [0.53, 0.92]0.617 [0.51, 0.71] CoCoA PPL0.697 [0.62, 0.78]0.673 [0.57, 0.76]0.520 [0.43, 0.61]0.689 [0.59, 0.78]0.540 [0.45, 0.63]0.705 [0.50, 0.90]0.609 [0.50, 0.71] Gated (Hybrid)0.602 [0.51, 0.69]0.604 [0.51, 0.70]0.576 [0.48, 0.67]0.654 [0.56, 0.75]0.729 [0.65, 0.80]0.688 [0.47, 0.88]0.626 [0.54, 0.71] Gated (Spectral)0.620 [0.53, 0.70]0.605 [0.51, 0.70]0.575 [0.48, 0.67]0.625 [0.53, 0.73]0.752 [0.68, 0.82]0.674 [0.44, 0.88]0.608 [0.51, 0.69] Stacked (Hybrid)0.746 [0.67, 0.82]0.533 [0.44, 0.63]0.539 [0.44, 0.63]0.597 [0.48, 0.71]0.759 [0.68, 0.83]0.609 [0.38, 0.82]0.749 [0.65, 0.84] Stacked (Spectral)0.742 [0.66, 0.81]0.559 [0.46, 0.65]0.535 [0.44, 0.63]0.603 [0.49, 0.72]0.759 [0.68, 0.83]0.618 [0.39, 0.83]0.725 [0.62, 0.81] Stacked (Von Neumann)0.739 [0.66, 0.81]0.557 [0.46, 0.65]0.548 [0.45, 0.64]0.596 [0.48, 0.71]0.760 [0.68, 0.83]0.614 [0.38, 0.82]0.720 [0.62, 0.81] 38