基于低秩多线性分解的大语言模型幻觉检测

# LLM Hallucination Detection via Low-rank Multilinear Decomposition
Source: [https://arxiv.org/html/2610.36191](https://arxiv.org/html/2610.36191)
Faizul Rakib SayemShahana IbrahimAffiliation:Department of Electrical & Computer Engineering, University of Central FloridaAffiliation:\{devansh\.sharma, faizulrakib\.sayem, shahana\.ibrahim\}@ucf\.edu

###### Abstract

Large language models \(LLMs\) frequently generate fluent yet factually incorrect responses, raising concerns about their reliability\. Recent hallucination detectors increasingly exploit internal representations, but extracting informative signals from high\-dimensional hidden states remains challenging\. Existing approaches often rely on selected activations or pooling\-based compression, potentially overlooking discriminative information distributed across Transformer layers, feature modes, and tokens\. We proposeLOTUS\(Low\-rankOrder\-statisticTUcker\-basedSubspace learning\), a multilinear low\-rank decomposition framework for hallucination detection that extracts compact representations from LLM hidden states while reducing dependence on labeled data\. Our approach first summarizes variable\-length token representations using complementary statistical descriptors that capture peak activations, activation distributions, and cross\-layer changes\. We then leverage low\-rank multilinear decomposition to extract compact representations that capture the underlying structure of LLM hidden states, which are subsequently used to train a lightweight hallucination classifier\. Experiments across multiple LLMs and question\-answering benchmarks demonstrate improved hallucination detection performance over recent state\-of\-the\-art baselines\.

## 1Introduction

Large language models \(LLMs\) have demonstrated remarkable capabilities in language understanding, generation, and reasoning\([Huang & Chang, 2023](https://arxiv.org/html/2610.36191#bib.bib17)\), enabling widespread use in question answering, summarization, and dialogue systems\([Brown et al\., 2020](https://arxiv.org/html/2610.36191#bib.bib5);[Min et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib36)\)\. They nonetheless remain susceptible to*hallucination*, generating fluent and plausible responses that are factually incorrect or unsupported by evidence\([Huang et al\., 2025a](https://arxiv.org/html/2610.36191#bib.bib18);[Bang et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib3)\)\. Detection is difficult because hallucinated responses are often indistinguishable from correct ones at the surface level, remaining coherent and linguistically well formed even when their content is incorrect\([Wang et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib47);[Singh & Singh, 2025](https://arxiv.org/html/2610.36191#bib.bib41)\)\. Large reasoning models compound the problem by producing logically structured reasoning traces that appear sound yet reach an incorrect conclusion\([Mei et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib34);[Yao et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib48)\)\.

Existing hallucination detection methods are broadly either output\-based\([Farquhar et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib14);[Manakul et al\., 2023](https://arxiv.org/html/2610.36191#bib.bib33);[Zhang et al\., 2023](https://arxiv.org/html/2610.36191#bib.bib51);[Shelmanov et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib40)\)or internal\-representation\-based\([Duan et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib12);[Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16);[Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4);[Zeng et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib49)\)\. The former often exploit uncertainty or consistency across multiple generations, using signals such as lexical agreement\([Lin et al\., 2022b](https://arxiv.org/html/2610.36191#bib.bib30);[Manakul et al\., 2023](https://arxiv.org/html/2610.36191#bib.bib33)\)or semantic entropy\([Farquhar et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib14)\), and therefore do not use the internal representations that give rise to those outputs\. The latter treat hidden states as a richer source of hallucination\-related information, e\.g\.,[Hu et al\. \(2026\)](https://arxiv.org/html/2610.36191#bib.bib16)projects hidden representations onto subspaces derived from the model’s unembedding matrix, and[Bar\-Shalom et al\. \(2026\)](https://arxiv.org/html/2610.36191#bib.bib4)models representations across layers and tokens using a Vision Transformer \(ViT\)\-inspired architecture\. Both train supervised detectors on hallucination labels\.

Despite their promise, hidden\-state representations present several challenges for hallucination detection\. First, they are extremely high\-dimensional, spanning multiple Transformer layers, response tokens, and hidden features\. Existing methods therefore reduce these representations before classification, e\.g\., by selecting particular hidden states\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16);[Zeng et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib49)\)or by applying fixed\-resolution pooling\([Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)\. Such reductions make detection tractable but may discard complementary information distributed across layers, tokens, and features\. Directly learning from the full hidden\-state tensor avoids these reductions but results in a substantially higher\-dimensional learning problem, increasing computational cost and the risk of overfitting under limited supervision\. Second, response lengths vary, so the number of hidden states along the token mode differs across generations, making a common fixed\-size representation nontrivial\. Third, hidden\-state representations are model\-specific\. LLMs differ in both the number of layers and hidden dimensions, making detectors defined directly in the original hidden space architecture\-dependent and difficult to transfer across LLMs\.

Our contributions\.We address these challenges with a framework based on a low\-rank multilinear formulation of LLM hidden representations\. Our key contributions are:

- ❏Structured low\-rank hidden\-state representation\.We introduceLOTUS\(Low\-rankOrder\-statisticTUcker\-basedSubspace learning\) for LLM hallucination detection, which converts variable\-length token representations into complementary layer\-by\-feature summaries and applies an unsupervised Tucker\-2 projection\([Tucker, 1966](https://arxiv.org/html/2610.36191#bib.bib45);[Kolda & Bader, 2009](https://arxiv.org/html/2610.36191#bib.bib25)\)over the layer and feature modes, preserving the multilinear structure of the hidden states\.
- ❏Compact and efficient detection pipeline\.LOTUSlearns its multilinear representation from unlabeled hidden states, with hallucination labels used only by a lightweight classifier\. The low\-rank decomposition reduces representation size and memory cost by mapping model\-specific hidden states into fixed\-dimensional cores, which also facilitates detector reuse across LLMs with different internal dimensions\.
- ❏Strong empirical performance with comprehensive evaluation\.We evaluateLOTUSacross two LLMs and four question\-answering benchmarks, where it delivers strong and consistent detection performance, and report ablations on the statistical summaries, Tucker ranks, projection design, performance under limited labeled data, and cross\-model transfer\.

## 2Problem Statement

Letℳ\\mathcal\{M\}be a decoder\-only language model withLLtransformer layers, each with hidden dimensionDD\. Consider prompts\{𝒙i\}i=1N\\\{\\bm\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\}, each associated with a reference answer𝒂i∗\\bm\{a\}\_\{i\}^\{\*\}, and for each prompt sampleKKresponses𝒜i=\{𝒂i,1,…,𝒂i,K\}\{\\cal A\}\_\{i\}=\\\{\\bm\{a\}\_\{i,1\},\\dots,\\bm\{a\}\_\{i,K\}\\\}fromℳ\\mathcal\{M\}\. Lets⁡\(𝒂i,k,𝒂i∗\)s\(\\bm\{a\}\_\{i,k\},\\bm\{a\}\_\{i\}^\{\*\}\)denote the similarity between a generated answer and the reference answer\. Given a similarity thresholdλ\\lambda, the hallucination label of thekk\-th response is

zi,k=𝕀\[s\(𝒂i,k,𝒂i∗\)<λ\],z\_\{i,k\}=\\mathbb\{I\}\\\!\\left\[s\(\\bm\{a\}\_\{i,k\},\\bm\{a\}\_\{i\}^\{\*\}\)<\\lambda\\right\],where𝕀⁡\[⋅\]\\mathbb\{I\}\[\\cdot\]is the indicator function, so thatzi,k=1z\_\{i,k\}=1denotes a hallucinated response andzi,k=0z\_\{i,k\}=0a correct one\. The goal of hallucination detection is to learn a scoring functionc⁡\(𝒂i,k\)c\(\\bm\{a\}\_\{i,k\}\)that assigns larger values to hallucinated responses, from which a binary prediction follows as

gτ\(𝒂i,k\)=𝕀\[c\(𝒂i,k\)≥τ\],g\_\{\\tau\}\(\\bm\{a\}\_\{i,k\}\)=\\mathbb\{I\}\\\!\\left\[c\(\\bm\{a\}\_\{i,k\}\)\\geq\\tau\\right\],whereτ\\tauis a decision threshold, withgτ​\(𝒂i,k\)=1g\_\{\\tau\}\(\\bm\{a\}\_\{i,k\}\)=1denoting a hallucinated response andgτ​\(𝒂i,k\)=0g\_\{\\tau\}\(\\bm\{a\}\_\{i,k\}\)=0a non\-hallucinated one\.

Hidden\-state tensor\.Recent studies provide growing evidence that LLM internal representations encode cues relevant to hallucination detection across Transformer layers and token positions\([Azaria & Mitchell, 2023](https://arxiv.org/html/2610.36191#bib.bib2);[Chen et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib6);[Orgad et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib37);[Zhang et al\., 2025a](https://arxiv.org/html/2610.36191#bib.bib50);[Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)\. Since the location and strength of these cues vary across responses and datasets, detectors that rely on a few selected activations can miss useful information, which motivates modeling hidden representations jointly across layers, tokens, and feature dimensions\.

Accordingly, we consider the internal representations generated byℳ\\mathcal\{M\}for each response\. For thekk\-th response to prompt𝒙i\\bm\{x\}\_\{i\}, let𝒉i,k,t\(ℓ\)∈ℝD\\bm\{h\}\_\{i,k,t\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{D\}denote the hidden representation of itstt\-th response token after layerℓ\\ell, withℓ∈\{1,…,L\}\\ell\\in\\\{1,\\ldots,L\\\}andt∈\{1,…,Ti,k\}t\\in\\\{1,\\ldots,T\_\{i,k\}\\\}, whereTi,kT\_\{i,k\}is the token length of that response\. Sinceℳ\\mathcal\{M\}is autoregressive,𝒉i,k,t\(ℓ\)\\bm\{h\}\_\{i,k,t\}^\{\(\\ell\)\}is computed conditioned on the prompt𝒙i\\bm\{x\}\_\{i\}and the preceding generated tokens𝒂i,k,<t\\bm\{a\}\_\{i,k,<t\}\. Collecting these representations across layers and answer tokens yields thehidden\-state tensorgiven by

𝑯¯i,k∈ℝL×Ti,k×D,\{\\underline\{\\bm\{H\}\}\}\_\{i,k\}\\in\\mathbb\{R\}^\{L\\times T\_\{i,k\}\\times D\},with𝑯¯i,k\[ℓ,t,:\]=𝒉i,k,t\(ℓ\)\{\\underline\{\\bm\{H\}\}\}\_\{i,k\}\[\\ell,t,:\]=\\bm\{h\}\_\{i,k,t\}^\{\(\\ell\)\}, whose entries preserve three complementary modes of variation: Transformer layer, token position, and hidden feature dimension\. Using these representations for hallucination detection poses three practical challenges:\(i\)the hidden\-state tensor is extremely high\-dimensional, so direct processing is memory\- and computation\-intensive and requires substantially more labeled data to avoid overfitting;\(ii\)the token dimensionTi,kT\_\{i,k\}varies across responses, requiring a fixed\-size representation without token alignment; and\(iii\)LLandDDvary across LLMs, so detectors defined directly in the original hidden space do not transfer readily across models\. This raises a central research question:

How can we extract compact, informative hidden\-state representations that enable computationally efficient and label\-efficient hallucination detection while facilitating transfer across LLMs?

## 3Proposed approach

Our approach is motivated by the hypothesis that hallucination\-discriminative information can be captured by compact, low\-rank multilinear representations of LLM hidden states\. Rather than learning a detector directly from the high\-dimensional hidden\-state tensor, we propose a two\-stage framework that separates representation learning from hallucination classification\. The first stage constructs fixed\-size statistical summaries of hidden states and learns label\-free Tucker\-2 projections over the layer and feature modes to obtain compact response representations\. The second stage concatenates the resulting low\-dimensional representations and trains a lightweight hallucination classifier\. Fig\.[1](https://arxiv.org/html/2610.36191#S3.F1)gives the complete overview ofLOTUS, and we describe each component below\.

![Refer to caption](https://arxiv.org/html/2610.36191v1/final_method.png)Figure 1:Overview of our proposed frameworkLOTUS\. Hidden states are summarized across tokens using peak activations, activation quantiles, and cross\-layer differences, projected with summary\-specific Tucker\-2 bases learned from unlabeled activations, and concatenated for lightweight hallucination classification\.Order statistics over the token mode\.The hidden\-state tensor𝑯¯i,k∈ℝL×Ti,k×D\{\\underline\{\\bm\{H\}\}\}\_\{i,k\}\\in\\mathbb\{R\}^\{L\\times T\_\{i,k\}\\times D\}varies in size along the token mode, and hallucination\-related activations may appear at different token positions\. Hence, representations taken from predefined positions may overlook informative signals\. We therefore summarize the token dimension using complementary statistical descriptors that produce fixed\-size representations while retaining information about activation magnitude, distribution, and cross\-layer changes\. For each response𝒂i,k\\bm\{a\}\_\{i,k\}and each selected layerℓ\\ell, we construct three descriptor matrices:

𝑪i,k\[ℓ,:\]\\displaystyle\\bm\{C\}\_\{i,k\}\[\\ell,:\]=\(maxt⁡\{𝒉i,k,t\(ℓ\)\}\)⊤,\(Peak activation matrix\)\\displaystyle=\\big\(\\max\_\{t\}~\\\{\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}\\\}\\big\)^\{\\top\},~~~\\text\{\(\{\\bf Peak activation matrix\}\)\}𝑺i,k\[ℓ,:\]\\displaystyle\\bm\{S\}\_\{i,k\}\[\\ell,:\]=\(Q0\.95​\(\{𝒉i,k,t\(ℓ\)\}t\)⊤,Q0\.05​\(\{𝒉i,k,t\(ℓ\)\}t\)⊤\),\(Activation quantile matrix\)\\displaystyle=\\big\(\\mathrm\{Q\}\_\{0\.95\}\(\\\{\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}\\\}\_\{t\}\)^\{\\top\},\\;\\mathrm\{Q\}\_\{0\.05\}\(\\\{\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}\\\}\_\{t\}\)^\{\\top\}\\big\),~~~\\text\{\(\{\\bf Activation quantile matrix\}\)\}𝑽i,k\[ℓ,:\]\\displaystyle\\bm\{V\}\_\{i,k\}\[\\ell,:\]=\(Q0\.95​\(\{Δ​𝒉i,k,t\(ℓ\)\}t\)⊤,Q0\.05​\(\{Δ​𝒉i,k,t\(ℓ\)\}t\)⊤\),\(Cross\-layer difference matrix\)\\displaystyle=\\big\(\\mathrm\{Q\}\_\{0\.95\}\(\\\{\\Delta\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}\\\}\_\{t\}\)^\{\\top\},\\;\\mathrm\{Q\}\_\{0\.05\}\(\\\{\\Delta\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}\\\}\_\{t\}\)^\{\\top\}\\big\),~~~\\text\{\(\{\\bf Cross\-layer difference matrix\}\)\}wheremaxt⁡\{𝒉t\}∈ℝD\\max\_\{t\}~\\\{\\bm\{h\}\_\{t\}\\\}\\in\\mathbb\{R\}^\{D\}andQα​\(\{𝒉t\}t\)∈ℝDQ\_\{\\alpha\}\(\\\{\\bm\{h\}\_\{t\}\\\}\_\{t\}\)\\in\\mathbb\{R\}^\{D\}denote the element\-wise maximum and the element\-wise empiricalα\\alpha\-quantile of the hidden\-state vectors over the token dimension, and the cross\-layer differenceΔ​𝒉i,k,t\(ℓ\)=𝒉i,k,t\(ℓ\+1\)−𝒉i,k,t\(ℓ\)\\Delta\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}=\\bm\{h\}^\{\(\\ell\+1\)\}\_\{i,k,t\}\-\\bm\{h\}^\{\(\\ell\)\}\_\{i,k,t\}is the change in the hidden states between consecutive layers\.

The three summaries capture complementary properties of hidden representations\. The peak activation matrix captures localized activation peaks irrespective of their positions, since hallucination\-related patterns may concentrate in a few tokens whose positions vary across responses\([Orgad et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib37);[Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)\. Because a maximum depends on a single token, the activation quantile matrix characterizes how feature activations are distributed across the response, distinguishing isolated peaks from sustained patterns\. The cross\-layer difference matrix summarizes how representations evolve as successive Transformer layers process them\([Zhang et al\., 2025b](https://arxiv.org/html/2610.36191#bib.bib52)\)\. Together, these position\-independent summaries turn variable\-length token representations into fixed\-size matrices without token alignment\.

Unsupervised Tucker\-2 projection over layer and feature modes\.The token summaries are of fixed\-size but remain high\-dimensional, particularly along the feature mode\. To extract compact representations that capture dependencies across layers and hidden features, we employ aTucker\-2 decomposition\([Tucker, 1966](https://arxiv.org/html/2610.36191#bib.bib45);[Kolda & Bader, 2009](https://arxiv.org/html/2610.36191#bib.bib25)\), which learns a compact, separable projection comprising a layer basis and a feature basis, shared across all training responses and learned without any hallucination labels\. Projecting each summary matrix onto these subspaces yields a low\-dimensional core matrix that retains interactions between the layer and feature dimensions\.

Let𝑿i,k∈ℝP×F\\bm\{X\}\_\{i,k\}\\in\\mathbb\{R\}^\{P\\times F\}denote any one of the summary matrices𝑪i,k\\bm\{C\}\_\{i,k\},𝑺i,k\\bm\{S\}\_\{i,k\}, or𝑽i,k\\bm\{V\}\_\{i,k\}, wherePPis the layer\-mode dimension andF∈\{D,2​D\}F\\in\\\{D,2D\\\}the feature\-mode dimension\. The layer dimensionPPneed not equal the total number of layersLL—it can be set smaller to reduce computational cost, and the choice of which layers to retain can incorporate prior knowledge \(see more details about the layer selection in Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1)\)\. Before decomposition, we robustly scale each summary𝑿i,k\{\\bm\{X\}\}\_\{i,k\}with scaling parameters estimated from the training responses only, which prevents a small number of large\-magnitude activations from dominating the projection, and denote the scaled summary by𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}\(more details about this scaling are provided in Appendix[D](https://arxiv.org/html/2610.36191#A4)\)\. From these scaled hidden representations𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}, we seekshared orthonormal bases𝑼L∈ℝP×rL\\bm\{U\}\_\{L\}\\in\\mathbb\{R\}^\{P\\times r\_\{L\}\}and𝑼F∈ℝF×rF\\bm\{U\}\_\{F\}\\in\\mathbb\{R\}^\{F\\times r\_\{F\}\}, withrL<Pr\_\{L\}<PandrF<Fr\_\{F\}<F, that maximize the energy retained after projection onto the corresponding low\-dimensional subspaces:

max𝑼L,𝑼F\\displaystyle\\max\_\{\\bm\{U\}\_\{L\},\\bm\{U\}\_\{F\}\}∑i=1N∑k=1K‖𝑼L⊤​𝑿~i,k​𝑼F‖F2\\displaystyle\\sum\_\{i=1\}^\{N\}\\sum\_\{k=1\}^\{K\}\\left\\\|\\bm\{U\}\_\{L\}^\{\\top\}\\tilde\{\\bm\{X\}\}\_\{i,k\}\\bm\{U\}\_\{F\}\\right\\\|\_\{F\}^\{2\}\(1\)s\.t\.\\displaystyle\\text\{s\.t\.\}𝑼L⊤𝑼L=𝑰,𝑼F⊤𝑼F=𝑰\.\\displaystyle\\bm\{U\}\_\{L\}^\{\\top\}\\bm\{U\}\_\{L\}=\\bm\{I\},\\qquad\\bm\{U\}\_\{F\}^\{\\top\}\\bm\{U\}\_\{F\}=\\bm\{I\}\.where𝑰\\bm\{I\}is an identity matrix of appropriate size\. This is the Tucker\-2 decomposition of the third\-order tensor formed by stacking the training summaries along a response mode\([Tucker, 1966](https://arxiv.org/html/2610.36191#bib.bib45);[Kolda & Bader, 2009](https://arxiv.org/html/2610.36191#bib.bib25)\)\. It compresses only the layer and feature modes, preserving a separate low\-dimensional representation for each response\.

We handle the optimization problem in equation[1](https://arxiv.org/html/2610.36191#S3.E1)with truncated higher\-order singular value decomposition \(HOSVD\)\([De Lathauwer et al\., 2000](https://arxiv.org/html/2610.36191#bib.bib10)\)\. Specifically, the layer basis𝑼L\\bm\{U\}\_\{L\}consists of therLr\_\{L\}leading eigenvectors of∑i=1N∑k=1K𝑿~i,k​𝑿~i,k⊤\\sum\_\{i=1\}^\{N\}\\sum\_\{k=1\}^\{K\}\\tilde\{\\bm\{X\}\}\_\{i,k\}\\tilde\{\\bm\{X\}\}\_\{i,k\}^\{\\top\}, while the feature basis𝑼F\\bm\{U\}\_\{F\}is obtained from therFr\_\{F\}leading eigenvectors of∑i=1N∑k=1K𝑿~i,k⊤​𝑿~i,k\\sum\_\{i=1\}^\{N\}\\sum\_\{k=1\}^\{K\}\\tilde\{\\bm\{X\}\}\_\{i,k\}^\{\\top\}\\tilde\{\\bm\{X\}\}\_\{i,k\}, We compute the feature eigenspace with randomized SVD on the vertically stacked rows of all training summaries\([Halko et al\., 2011](https://arxiv.org/html/2610.36191#bib.bib15)\)rather than forming theF×FF\\times FGram matrix, which reachesF=8192F=8192for LLaMA\-3\.1\-8B\. Once the bases are estimated, each response is mapped to a low\-dimensional core matrix defined as

𝑮i,k=𝑼L⊤​𝑿~i,k​𝑼F∈ℝrL×rF,\\bm\{G\}\_\{i,k\}=\\bm\{U\}\_\{L\}^\{\\top\}\\tilde\{\\bm\{X\}\}\_\{i,k\}\\bm\{U\}\_\{F\}\\in\\mathbb\{R\}^\{r\_\{L\}\\times r\_\{F\}\},\(2\)where𝑮i,k\\bm\{G\}\_\{i,k\}containsrL​rFr\_\{L\}r\_\{F\}coefficients that represent interactions between the layer and feature directions\. In this way, we learn a separate core representation for each summary type𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}\.

Hallucination detector from low\-dimensional representations\.To learn the hallucination detector, we vectorize and concatenate the low\-dimensional core representations𝑮i,kC\\bm\{G\}^\{C\}\_\{i,k\},𝑮i,kS\\bm\{G\}^\{S\}\_\{i,k\}, and𝑮i,kV\\bm\{G\}^\{V\}\_\{i,k\}of𝒂i,k\\bm\{a\}\_\{i,k\}into theresponse feature vector:

ϕi,k=\[vec⁡\(𝑮i,kC\),vec⁡\(𝑮i,kS\),vec⁡\(𝑮i,kV\)\]⊤∈ℝdϕ,\\bm\{\\phi\}\_\{i,k\}=\\begin\{bmatrix\}\\mathrm\{vec\}\(\\bm\{G\}^\{C\}\_\{i,k\}\),\\mathrm\{vec\}\(\\bm\{G\}^\{S\}\_\{i,k\}\),\\mathrm\{vec\}\(\\bm\{G\}^\{V\}\_\{i,k\}\)\\end\{bmatrix\}^\{\\top\}\\in\\mathbb\{R\}^\{d\_\{\\phi\}\},\(3\)wheredϕ=rLC​rFC\+rLS​rFS\+rLV​rFVd\_\{\\phi\}=r\_\{L\}^\{C\}r\_\{F\}^\{C\}\+r\_\{L\}^\{S\}r\_\{F\}^\{S\}\+r\_\{L\}^\{V\}r\_\{F\}^\{V\}is the dimension of the resulting feature vector\. We then train a lightweight classifierf𝜽f\_\{\\bm\{\\theta\}\}, with parameters𝜽\\bm\{\\theta\}, on the feature vectors of the training responses and their hallucination labelszi,kz\_\{i,k\}\. At inference, the statistical summaries of an unseen response𝒂test\\bm\{a\}\_\{\\mathrm\{test\}\}are computed from its hidden\-state tensor, transformed with the learned projection bases using equation[2](https://arxiv.org/html/2610.36191#S3.E2), and assembled as in equation[3](https://arxiv.org/html/2610.36191#S3.E3)intoϕtest\\bm\{\\phi\}\_\{\\mathrm\{test\}\}\. The classifier returns the hallucination scorec⁡\(𝒂test\)=f𝜽​\(ϕtest\)c\(\\bm\{a\}\_\{\\mathrm\{test\}\}\)=f\_\{\\bm\{\\theta\}\}\(\\bm\{\\phi\}\_\{\\mathrm\{test\}\}\), which can be thresholded atτ\\tauto obtain the binary decisiongτg\_\{\\tau\}\.

## 4Experiments

### 4\.1Evaluation Setup

Models and datasets\.We evaluate on two open\-weight models of comparable scale: Qwen2\.5\-7B\-Instruct, which is instruction\-tuned, and the base LLaMA\-3\.1\-8B\. We use four question answering datasets: TruthfulQA\([Lin et al\., 2022a](https://arxiv.org/html/2610.36191#bib.bib29)\), TriviaQA\([Joshi et al\., 2017](https://arxiv.org/html/2610.36191#bib.bib22)\)without evidence documents, NQ\-Open\([Kwiatkowski et al\., 2019](https://arxiv.org/html/2610.36191#bib.bib27)\), and the English questions of TyDiQA\-GP\([Clark et al\., 2020](https://arxiv.org/html/2610.36191#bib.bib9)\), in which each question comes with a passage containing its answer\. For each question we generateK=10K=10responses by beam sampling with1010beams, returning1010sequences each capped at6464new tokens\. The datasets contain817817,9,9609\{,\}960,3,6103\{,\}610, and440440questions, respectively\. A question is*model\-known*if at least one of itsKKresponses is labeled correct, and*model\-unknown*otherwise\. Only model\-known questions are eligible for detector training; their number varies substantially across models and datasets, and Qwen2\.5\-7B\-Instruct has notably fewer on NQ\-Open, which contributes to the larger variance in that setting\. Appendix[D](https://arxiv.org/html/2610.36191#A4)provides the exact counts, prompts, generation parameters, and reproducibility details\.

Hallucination labels\.We label responses following the procedure in\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16)\)\. A response𝒂i,k\\bm\{a\}\_\{i,k\}is compared with the reference answers of questioniiand labeled correct \(zi,k=0z\_\{i,k\}=0\) if its ROUGE\-L score\([Lin, 2004](https://arxiv.org/html/2610.36191#bib.bib28)\), averaged over the references, is at least0\.70\.7, or if its BLEURT\-20 score\([Sellam et al\., 2020](https://arxiv.org/html/2610.36191#bib.bib38)\)exceeds0\.50\.5for at least one reference; otherwise it is labeled hallucinated \(zi,k=1z\_\{i,k\}=1\)\. All methods are evaluated on the same generated responses and labels\. Appendix[P](https://arxiv.org/html/2610.36191#A16)compares these labels with those of the judge released by[Hu et al\. \(2026\)](https://arxiv.org/html/2610.36191#bib.bib16)\.

Train–test splitting protocol\.Since each question is associated with multiple generated responses, the set of question\-response pairs can be divided by question or by response\. We use the*question\-level*train\-test split throughout, where we randomly assign75%75\\%of the model\-known questions to training and the remaining25%25\\%to testing, keeping all responses to each question in the same split\. The test set additionally includes all responses to model\-unknown questions\. Consequently, no question appears in both the training and test sets\. The*answer\-level*alternative pools responses instead, which puts responses to the same question on both train and test splits and evaluates a detector on questions it was trained on \(e\.g\., HARP[Hu et al\. \(2026\)](https://arxiv.org/html/2610.36191#bib.bib16)follows this protocol in their released implementation\)\. Appendix[E](https://arxiv.org/html/2610.36191#A5)reports both splits and demonstrates how question overlap can inflate reported hallucination detection performance\. The choice affects only detectors that score individual responses, since one producing a single score per question is unaffected\.

Evaluation metrics\.We use pooled AUROC as our primary metric, following standard practice in hallucination detection\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16);[Du et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib11);[Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)\. Pooled AUROC compares responses across different questions, so it can reward a detector for identifying questions that are generally more prone to hallucination rather than for separating correct and hallucinated responses to the same question\. We therefore also report*within\-question AUROC*, which restricts the pairwise comparison to responses of the same question and so measures response\-level discrimination while controlling for question\-specific effects\. A detector that assigns an identical score to every response of a question obtains50%50\\%, however high its pooled AUROC\. It is closely related to the within\-question discrimination of[Taubenfeld et al\. \(2025\)](https://arxiv.org/html/2610.36191#bib.bib44)\. Question\- and dataset\-level shortcuts in hallucination detection have likewise been reported\([Dubanowska et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib13);[Seo et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib39)\)\. Within\-question AUROC is defined only for questions containing at least one correct and one hallucinated response—more details in Appendix[N](https://arxiv.org/html/2610.36191#A14)\.

LOTUS implementation details\.LOTUS uses a fixed window of nine consecutive Transformer blocks,1515–2323, for both models and all datasets, motivated by prior evidence that hallucination\- and factuality\-related information is prominent in intermediate Transformer representations\([Azaria & Mitchell, 2023](https://arxiv.org/html/2610.36191#bib.bib2);[Orgad et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib37);[Zhang et al\., 2025b](https://arxiv.org/html/2610.36191#bib.bib52)\)\. We use ranks\(rL,rF\)=\(5,64\)\(r\_\{L\},r\_\{F\}\)=\(5,64\)for𝑪\\bm\{C\}and𝑺\\bm\{S\}, and\(4,64\)\(4,64\)for𝑽\\bm\{V\}, producing cores of dimensions5×645\\times 64,5×645\\times 64, and4×644\\times 64\. After vectorization and concatenation, each response is represented by an896896\-dimensional feature vector\. We then train a random\-forest classifier with200200trees and balanced class weights on these features and use its predicted hallucination probability for AUROC evaluation\. Layer window, Tucker ranks, and classifier settings are fixed across models and datasets rather than tuned per benchmark\. Appendix[C](https://arxiv.org/html/2610.36191#A3)evaluates alternative windows of the same width and Appendix[I](https://arxiv.org/html/2610.36191#A9)the rank choices\.

Baselines\.We compareLOTUSagainst eight different hallucination detection baselines\.

Label\-free baselines\.We consider six methods that require no hallucination\-labeled training responses: answer\-level*Perplexity*; LN\-Entropy\([Malinin & Gales, 2021](https://arxiv.org/html/2610.36191#bib.bib32)\), which aggregates predictive uncertainty across generated responses; Semantic Entropy\([Kuhn et al\., 2023](https://arxiv.org/html/2610.36191#bib.bib26)\), over semantically equivalent answer clusters; Lexical Similarity\([Lin et al\., 2022b](https://arxiv.org/html/2610.36191#bib.bib30)\), based on pairwise agreement among sampled responses; EigenScore\([Chen et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib6)\), based on the eigenspectrum of hidden\-state representations; and HaloScope\([Du et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib11)\), which estimates a hallucination\-related subspace from unlabeled generations\.

Supervised baselines\.We additionally compare with two recent hidden\-representation\-based detectors trained on hallucination labels\. HARP\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16)\)projects final\-layer token representations onto a low\-dimensional subspace derived from the model’s unembedding matrix and applies a lightweight MLP to obtain token\-level hallucination scores\. ACT\-ViT\([Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)operates on layer–token activation tensors, reducing them to a fixed resolution and training a ViT\-based classifier for response\-level detection\. Both are run with their released implementations and training settings; further implementation details are given in Appendix[D](https://arxiv.org/html/2610.36191#A4)\.

### 4\.2Experiment Results

Comparison with baselines\.Table[1](https://arxiv.org/html/2610.36191#S4.T1)reports pooled AUROC under the question\-level split across two LLMs and four QA datasets\. LOTUS outperforms HARP in all eight model–dataset combinations, by2\.152\.15to11\.5611\.56AUROC points and by7\.107\.10points on average, with the largest gains on NQ\-Open and TruthfulQA\. Compared with ACT\-ViT, LOTUS achieves the higher AUROC in six of the eight settings: by0\.180\.18,2\.432\.43, and5\.045\.04points on NQ\-Open, TruthfulQA, and TyDiQA\-GP with Qwen2\.5\-7B\-Instruct, and by2\.382\.38and3\.063\.06points on NQ\-Open and TyDiQA\-GP with LLaMA\-3\.1\-8B, where the two are effectively tied on TruthfulQA \(89\.1989\.19vs\.89\.0789\.07\)\. ACT\-ViT is ahead on TriviaQA, by1\.081\.08and3\.113\.11points\. TriviaQA is the largest of the four datasets, at99,60099\{,\}600responses per model, and ACT\-ViT retains a layer–token activation tensor per response: on TriviaQA it stores106106–122122GB against28\.628\.6–32\.732\.7GB for LOTUS at the token resolution we report, and about nineteen times the LOTUS figure at its released resolution \(Appendix[K](https://arxiv.org/html/2610.36191#A11)\)\. The remaining margin on that dataset is therefore obtained at a considerably higher representation cost\. LOTUS also outperforms every label\-free baseline in all eight settings\. Overall, the proposed low\-rank multilinear representation gives strong and consistent detection performance across models and datasets at a small fraction of the representation size of the strongest supervised baseline\.

Table[2](https://arxiv.org/html/2610.36191#S4.T2)reports within\-question AUROC under the question\-level split\. Label\-free baselines that assign a single score per question attain50%50\\%by construction and are omitted\. LOTUS outperforms HARP in all four settings by5\.405\.40–14\.9514\.95points and exceeds ACT\-ViT in three of four, trailing by0\.770\.77points on TruthfulQA with LLaMA\-3\.1\-8B\. The larger gap between pooled and within\-question AUROC for HARP on TruthfulQA indicates that more of its pooled performance is attributable to differences between questions, whereas LOTUS retains stronger response\-level discrimination\. Standard deviations are larger than for pooled AUROC because the metric uses only questions containing both correct and hallucinated responses; Appendix[N](https://arxiv.org/html/2610.36191#A14)reports the number of eligible pairs\.

Table 1:Pooled AUROC \(%\) under the question\-level split across two LLMs and four QA benchmarks\. Results are averaged over five random trials; mean±\\pmstandard deviation is reported for supervised detectors\. Best results are in bold\.Table 2:Within\-question AUROC \(%\) under the question\-level split\. Results are reported as mean±\\pmstandard deviation over five random trials\.Different configurations of statistical summaries\.Table[3](https://arxiv.org/html/2610.36191#S4.T3)evaluates how each token\-mode summary defined in Section[3](https://arxiv.org/html/2610.36191#S3)contributes to LOTUS, which compresses and concatenates the peak activation \(𝑪\\bm\{C\}\), activation quantile \(𝑺\\bm\{S\}\), and cross\-layer difference \(𝑽\\bm\{V\}\) summaries independently \(see equation[3](https://arxiv.org/html/2610.36191#S3.E3)\)\. No single configuration is optimal in every model–dataset setting, but LOTUS is the most consistent\. It is the best configuration in two of the eight settings and stays within0\.740\.74AUROC points of the best alternative in the others\. Removing individual summaries generally reduces performance, so activation magnitude, distribution across tokens, and cross\-layer variation carry complementary signals\. Projecting𝑺\\bm\{S\}and𝑽\\bm\{V\}with one shared pair of bases underperforms the default in seven of eight settings, supporting separate bases per summary type\.

Cross\-model transfer\.LOTUS produces representations of the same dimensionality across LLMs, but in model\-specific coordinate systems, since the Tucker bases are learned independently from each model’s activations\. Therefore, a detector trained on one model cannot directly interpret another model’s LOTUS features even when their dimensions match\. To enable transfer, we learn a lightweight ridge mapping from the target model’s representation to the source model’s on paired, unlabeled responses, and apply the source detector to the aligned target representations without any target hallucination labels \(see more details of the implementation in Appendix[J](https://arxiv.org/html/2610.36191#A10)\)\. Table[4](https://arxiv.org/html/2610.36191#S4.T4)shows that the aligned detector transfers in both directions\. Across the four model–dataset settings, its pooled AUROC stays within1\.781\.78–6\.036\.03points of a detector trained directly on the target model, whereas removing the alignment causes substantial degradation, particularly on TyDiQA\-GP\.

Table 3:The full LOTUS configuration independently projects and concatenates the peak activation \(𝑪\\bm\{C\}\), activation quantile \(𝑺\\bm\{S\}\), and cross\-layer difference \(𝑽\\bm\{V\}\) summaries\. The LOTUS row reports absolute pooled AUROC \(%\), while all ablated variants report the change in AUROC relative to LOTUS \(Δ\\Delta\)\. Positive values indicate improvement over LOTUS and negative values indicate degradation\. Results use the question\-level split and are averaged over five random draws\.Table 4:AUROC \(%\) on the target model under the question\-level split\.*Target\-trained*uses target labels, while transferred detectors reuse the source classifier with or without unlabeled ridge alignment\. Pooled AUROC is mean±\\pmstandard deviation over five draws; within\-question AUROC is also reported\.Spectral evidence for low\-rank hallucination structure\.LOTUS relies on a low\-dimensional feature basis, raising a natural question:*how much of the hidden\-state variation is concentrated in a small number of directions?*We analyze the peak\-activation summary𝑪i,k\\bm\{C\}\_\{i,k\}one layer at a time\. At layerℓ\\ell, each response is represented by𝑪i,k\[ℓ,:\]∈ℝD\\bm\{C\}\_\{i,k\}\[\\ell,:\]\\in\\mathbb\{R\}^\{D\}, and separate covariance matrices are formed for correct and hallucinated responses\. To control for class imbalance,*size\-matched*sampling uses an equal number of responses from each class;*question\-paired*sampling additionally controls for question identity by selecting one correct and one hallucinated response from each question containing both outcomes\. Each scheme is repeated over2020draws\. We characterize each covariance spectrum by the variance captured by its leading6464eigen\-directions and by the effective rankreff=exp\(−∑jpjlogpj\)r\_\{\\mathrm\{eff\}\}=\\exp\(\-\\sum\_\{j\}p\_\{j\}\\log p\_\{j\}\), wherepj=λj/∑mλmp\_\{j\}=\\lambda\_\{j\}/\\sum\_\{m\}\\lambda\_\{m\}\.

Across the nine LOTUS layers, the leading6464ofD=3584D=3584directions explain64\.764\.7–66\.4%66\.4\\%of the variance of correct responses and52\.352\.3–55\.5%55\.5\\%for hallucinated responses, indicating substantial low\-dimensional concentration\. Hallucinated responses nevertheless exhibit broader spectra: their effective rank is2\.012\.01–2\.22×2\.22\\timesthat of correct responses under size\-matched sampling and1\.521\.52–1\.60×1\.60\\timesunder question\-paired sampling \(Fig\.[2](https://arxiv.org/html/2610.36191#S4.F2)a,b\)\. This ordering persists across all layers, suggesting that hallucinated responses distribute their variance over a broader set of feature directions even after controlling for question identity\. Spectral separation alone, however, does not determine the most discriminative layers\. The embedding representation has the largest rank ratio but only76\.5%76\.5\\%single\-layer AUROC, whereas blocks1515–2323average86\.8%86\.8\\%\(Fig\.[2](https://arxiv.org/html/2610.36191#S4.F2)c\)\. We therefore use this analysis to motivate low\-rank compression while selecting the layer window separately\.

Figure 2:Covariance spectra of the peak activation matrix for Qwen2\.5\-7B\-Instruct on TyDiQA\-GP, at every layer\. \(a\) effective rank of correct and hallucinated responses, \(b\) hallucinated\-to\-correct effective\-rank ratio under size\-matched and question\-paired sampling, and \(c\) single\-layer AUROC\. Shaded bands indicate95%95\\%intervals; the gray region marks the LOTUS layer window\.Performance with limited labeled data\.We further examine whether the compact Tucker\-2 representation helps when fewer labeled training questions are available\. Table[11](https://arxiv.org/html/2610.36191#A7.T11)in Appendix[G](https://arxiv.org/html/2610.36191#A7)reports the AUROC difference between Tucker\-2 and an equal\-dimensional flat\-PCA representation as the training set is reduced to5%5\\%–50%50\\%, with the test set held fixed\. With the full training set, the two representations perform similarly, within0\.800\.80AUROC points in all settings\. As the training data decrease, Tucker\-2 gains a clear advantage in several settings: on LLaMA\-3\.1\-8B with TruthfulQA it improves AUROC by2\.662\.66–3\.983\.98points in every reduced\-data regime, and on Qwen2\.5\-7B\-Instruct with TyDiQA\-GP the gain reaches5\.065\.06points\. The remaining settings are mixed, so the benefit is not universal, but the more compact, structured Tucker\-2 parameterization is advantageous when labeled training data are scarce\.

Tucker\-3 token\-mode extension\.LOTUS aggregates the token mode before decomposition, trading explicit token\-order information for a compact layer–feature representation\. To examine this trade\-off, we construct a Tucker\-3 variant that partitions each response intoBBcontiguous token bins and additionally decomposes the resulting token mode with rankrTr\_\{T\}\. Table[10](https://arxiv.org/html/2610.36191#A6.T10)in Appendix[F](https://arxiv.org/html/2610.36191#A6)shows that retaining this coarse temporal structure provides only limited benefit\. With the proposed order\-statistic summaries, no Tucker\-3 configuration improves pooled AUROC over the Tucker\-2 baseline by more than0\.370\.37points across the four settings, while increasing the representation from896896to as many as7,1687\{,\}168features, and increasing the token rank gives no consistent gain\. The same trend holds with average pooling, supporting the choice to aggregate the token mode before decomposition, i\.e\., the Tucker\-2 representation is substantially more compact at comparable detection performance\.

## 5Conclusion

In this work, we introducedLOTUS, a low\-rank multilinear framework for hallucination detection from LLM hidden representations\.LOTUSconverts variable\-length token representations into complementary layer–feature summaries and learns compact Tucker\-2 subspaces from unlabeled activations, requiring hallucination labels only for a lightweight downstream classifier\. Across two LLMs and four question\-answering benchmarks, our method achieves strong and consistent detection performance while substantially reducing representation size and supporting transfer across models with different internal dimensions\. Extensive ablations further validate the contributions of the proposed summaries, low\-rank projection, rank choices, and transfer design\. Overall, our results show that structured low\-dimensional representations provide an effective and efficient basis for detecting hallucinations from LLM internal states\.

#### AI use statement

We used a generative AI assistant in two ways\. First, for code: most of the pipeline, job and analysis scripts in the released repository were written or revised with AI assistance under the authors’ direction\. Every number we report is produced by that code, the scripts carry self\-tests that check them against independent implementations where one exists \(the ACT\-ViT preprocessing, for instance, is asserted bit\-identical to the released implementation of its authors\), and the authors reviewed both the code and its outputs\. Second, for writing: the assistant was used to draft and revise text throughout the paper, and to locate candidate related work, which the authors then verified against the original papers\. The assistant was not used to generate experimental results, data or labels: all generations, activations, metrics and tables come from running the released code\. We have reviewed all AI\-assisted work and take responsibility for the final content of this paper, including its text, claims and artifacts\.

#### Ethics statement

This work uses four public question answering benchmarks and two openly available language models, each under its own license\. It involves no human subjects, no new data collection, and no personally identifying information beyond what those public datasets already contain\. Hallucination labels are assigned by automatic metrics rather than by human annotation \(Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1)\), so they inherit the biases of those metrics; we therefore report their agreement with an independently released judge in Appendix[P](https://arxiv.org/html/2610.36191#A16)rather than treating them as ground truth\. A detector of this kind reduces, but does not remove, the risk of acting on an incorrect model output, and accuracy on these benchmarks does not imply reliability in deployment, in other languages, or on other domains; we make no such claim\. We are not aware of dual\-use concerns raised by this work beyond those already present in the publicly released models it analyses\.

#### Reproducibility statement

The detector is specified in Section[3](https://arxiv.org/html/2610.36191#S3): the token summaries, the entry\-wise scaling, the two Tucker\-2 bases and the classifier\. The configuration behind every reported number is given in Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1), which states the layer window, the ranks, the classifier, the split and the two metrics, and Appendix[D](https://arxiv.org/html/2610.36191#A4)adds the generation settings, the prompts, the per\-question seeding and how the two trained baselines were run\. A single harness owns the splits, the seeds and the metrics so that no method computes its own, and Appendix[E](https://arxiv.org/html/2610.36191#A5)defines both split procedures\. Code for generation, extraction, evaluation, the ablations and the analysis is available at[https://github\.com/devvansh1997/LOTUS](https://github.com/devvansh1997/LOTUS), one script per stage, each with a self\-test\. The generated responses and activations are not released because of their size; each question is seeded from a global seed and its identifier, so they can be regenerated exactly from the released configuration\.

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Supplementary Material of “LLM Hallucination Detection via Low\-rank Multilinear Decomposition”

## Appendix ARelated Work

##### Hallucination detection in LLMs\.

Hallucination refers to unfaithful, unsupported, or fabricated generations produced by large language models\([Ji et al\., 2023](https://arxiv.org/html/2610.36191#bib.bib20);[Huang et al\., 2025a](https://arxiv.org/html/2610.36191#bib.bib18)\)\. Since hallucinations reduce the reliability of LLMs in downstream applications, hallucination detection has become an important research direction\. Existing methods can be broadly grouped into uncertainty\-based approaches\([Chen et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib7);[Liu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib31)\)and internal\-representation\-based approaches\([Zhang et al\., 2025b](https://arxiv.org/html/2610.36191#bib.bib52)\)\. Uncertainty\-based methods assume that hallucinated outputs are associated with lower model confidence or higher variability across generations\. Early work measures token\-level uncertainty using predictive logits, entropy, or likelihood\-based scores such as perplexity\([Huang et al\., 2025b](https://arxiv.org/html/2610.36191#bib.bib19)\)\. Other studies extend this idea to sentence\-level or response\-level uncertainty\([Malinin & Gales, 2021](https://arxiv.org/html/2610.36191#bib.bib32)\)by aggregating token probabilities or comparing multiple sampled generations\. For example, semantic entropy\([Farquhar et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib14)\)estimates the consistency of meanings across multiple outputs, while lexical similarity\([Lin et al\., 2022b](https://arxiv.org/html/2610.36191#bib.bib30)\)measures whether sampled answers agree with each other at the surface text level\.

##### Internal representation and probing methods\.

To overcome the limitations of output\-only detection, recent work investigates whether internal representations of LLMs encode truthfulness\-related information\([Ji et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib21);[Orgad et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib37)\)\. Probing\-based approaches\([Alain & Bengio, 2016](https://arxiv.org/html/2610.36191#bib.bib1);[Ju et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib23);[Orgad et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib37)\)train classifiers on hidden states, logits, or activation vectors to predict whether a generated response is correct or hallucinated\. These methods are motivated by the observation that LLMs may internally encode more knowledge than they explicitly express in the final output\([Duan et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib12)\)\. Several works\([Kim et al\., 2025](https://arxiv.org/html/2610.36191#bib.bib24);[Bar\-Shalom et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib4)\)analyze layer\-wise or token\-wise representations and show that hallucination signals can emerge in intermediate hidden states\. HaloScope\([Du et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib11)\), for instance, leverages unlabeled embeddings and applies SVD to identify important subspace directions before probing them for hallucination detection\. EigenScore\([Chen et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib6)\)measures semantic agreement through covariance eigenvalues\.

##### Mechanistic interpretability of LLMs\.

Mechanistic interpretability provides another perspective for understanding hallucinations by analyzing model parameters, hidden states, and layer\-wise computation\([Merullo et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib35);[Cheng et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib8)\)\. Prior work has studied weight matrices\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16)\), attention heads\([Sriramanan et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib42)\), and feed\-forward modules\([Zhang et al\., 2025b](https://arxiv.org/html/2610.36191#bib.bib52)\)to uncover structural properties of LLMs\. In reasoning models, hallucinations may arise from shallow pattern matching, unstable reasoning trajectories, incorrect backtracking, or overthinking\([Wang et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib46)\)\. These findings suggest that hallucination is not only a surface\-level text generation problem, but also a consequence of abnormal internal computation\.

## Appendix BLOTUS workflow

Algorithms[1](https://arxiv.org/html/2610.36191#alg1)and[2](https://arxiv.org/html/2610.36191#alg2)summarize the training and inference procedures of LOTUS\. During training, the scaling statistics and the Tucker\-2 bases\(𝑼L,𝑼F\)\(\\bm\{U\}\_\{L\},\\bm\{U\}\_\{F\}\)are estimated separately for each summary type from the training responses without hallucination labels, which are used only to train the final classifier\. At inference, a test response is scaled with the training statistics and projected onto the same fixed bases, so scoring requires two matrix multiplications per summary followed by a single classifier evaluation\.

Algorithm 1Training the proposed detector0:Training responses

\{𝒂i,k\}i∈Ntrain,k=1K\\\{\\bm\{a\}\_\{i,k\}\\\}\_\{i\\in N\_\{\\text\{train\}\},k=1\}^\{K\}, labels

zi,kz\_\{i,k\}, layer window

𝒲\\mathcal\{W\}
1:foreach response

𝒂i,k\\bm\{a\}\_\{i,k\}do

2:Compute summaries

𝑪i,k,𝑺i,k,𝑽i,k\\bm\{C\}\_\{i,k\},\\bm\{S\}\_\{i,k\},\\bm\{V\}\_\{i,k\}
3:endfor

4:foreach summary

𝑿~i,k∈\{𝑪i,k,𝑺i,k,𝑽i,k\}\\tilde\{\\bm\{X\}\}\_\{i,k\}\\in\\\{\\bm\{C\}\_\{i,k\},\\bm\{S\}\_\{i,k\},\\bm\{V\}\_\{i,k\}\\\}do

5:Estimate robust scaling statistics from

𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}, without labels

6:Fit layer basis

𝑼L\\bm\{U\}\_\{L\}and feature basis

𝑼F\\bm\{U\}\_\{F\}with truncated HOSVD via \([1](https://arxiv.org/html/2610.36191#S3.E1)\)

7:endfor

8:foreach response

𝒂i,k\\bm\{a\}\_\{i,k\}do

9:Compute core

𝑮i,k=𝑼L⊤​𝑿~i,k​𝑼F\\bm\{G\}\_\{i,k\}=\\bm\{U\}\_\{L\}^\{\\top\}\\tilde\{\\bm\{X\}\}\_\{i,k\}\\bm\{U\}\_\{F\}for each summary

𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}with \([2](https://arxiv.org/html/2610.36191#S3.E2)\)

10:Concatenate cores into feature vector with \([3](https://arxiv.org/html/2610.36191#S3.E3)\)

ϕi,k=\(vec​\(𝑮i,kC\)⊤,vec​\(𝑮i,kS\)⊤,vec​\(𝑮i,kV\)⊤\)⊤\\bm\{\\phi\}\_\{i,k\}=\\left\(\\mathrm\{vec\}\(\\bm\{G\}^\{C\}\_\{i,k\}\)^\{\\top\},\\mathrm\{vec\}\(\\bm\{G\}^\{S\}\_\{i,k\}\)^\{\\top\},\\mathrm\{vec\}\(\\bm\{G\}^\{V\}\_\{i,k\}\)^\{\\top\}\\right\)^\{\\top\}
11:endfor

12:Train classifier on

\{\(ϕi,k,zi,k\)\}\\\{\(\\bm\{\\phi\}\_\{i,k\},z\_\{i,k\}\)\\\}

Algorithm 2Scoring a test response0:Test response

\{𝒂i,k\}i∈Ntest,k=1K\\\{\\bm\{a\}\_\{i,k\}\\\}\_\{i\\in N\_\{\\text\{test\}\},k=1\}^\{K\}, fitted scaling statistics, bases

\(𝑼L,𝑼F\)\(\\bm\{U\}\_\{L\},\\bm\{U\}\_\{F\}\), trained classifier

1:Compute summaries

𝑪i,k,𝑺i,k,𝑽i,k\\bm\{C\}\_\{i,k\},\\bm\{S\}\_\{i,k\},\\bm\{V\}\_\{i,k\}for each response

𝒂i,k\\bm\{a\}\_\{i,k\}
2:Scale each summary with the training statistics

3:Project each scaled summary onto

\(𝑼L,𝑼F\)\(\\bm\{U\}\_\{L\},\\bm\{U\}\_\{F\}\)to obtain its core

𝑮i,k\\bm\{G\}\_\{i,k\}via \([2](https://arxiv.org/html/2610.36191#S3.E2)\)

4:Concatenate cores into feature vector

ϕi,k\\bm\{\\phi\}\_\{i,k\}using \([3](https://arxiv.org/html/2610.36191#S3.E3)\)

5:return

c⁡\(𝒂i,k\)←c\(\\bm\{a\}\_\{i,k\}\)\\leftarrowclassifier’s predicted probability with

ϕi,k\\bm\{\\phi\}\_\{i,k\}

## Appendix CLayer window analysis

LOTUS uses residual\-stream representations from blocks 15–23 for both models\. To examine the sensitivity of different layer windows, we evaluate the hallucination signal available at each individual layer using Qwen2\.5\-7B\-Instruct on TyDiQA\-GP and TruthfulQA shown in Table[5](https://arxiv.org/html/2610.36191#A3.T5)\. Across both datasets, performance is lower for the earliest windows, the intermediate\-layer windows yield consistently strong AUROC with only small variations across neighboring windows, and performance generally declines as the window moves toward the final layers\. The selected nine\-block window obtains the best intermediate\-layer AUROC performance, indicating a strong hallucination\-discriminative signal\.

Table[6](https://arxiv.org/html/2610.36191#A3.T6)reports the AUROC of a detector trained on a single layer for Qwen2\.5\-7B\-Instruct\. On both datasets, performance is lowest near the embedding layer, rises through the early and middle blocks, and peaks within our window \(blocks 15–23\), reaching89\.0%89\.0\\%on TyDiQA\-GP and87\.9%87\.9\\%on TruthfulQA at blocks 16–18\. Beyond the window, AUROC declines toward the final block\. The range summary matches or exceeds the peak summary in almost every block, indicating that activation distributions across tokens carry additional signal beyond the maximum\.

Table 5:Pooled AUROC \(%\) under the question\-level split, averaged over five different runs\. Results are reported across different nine\-layer windows while keeping all other settings fixed\.Table 6:Single\-layer AUROC \(%\) for Qwen2\.5\-7B\-Instruct: the peak or range summary of one layer, projected to6464features and classified by logistic regression, with question\-grouped five\-fold cross\-validation\. Blocks are counted from zero; the rules enclose our window, blocks1515to2323\.
## Appendix DOther implementation details

We conducted all experiments for all methods on a single NVIDIA H100 80GB GPU\.

##### Generation\.

For each question, we generateK=10K=10answers using beam search with sampling:1010beams,1010returned sequences, temperature0\.50\.5, top\-k=5k=5, top\-p=0\.99p=0\.99, and at most6464new tokens\. Prompts use plain text without a chat template; TyDiQA\-GP additionally includes the passage and an instruction to answer from it\. Each question uses a fixed seed for reproducibility, duplicate TriviaQA questions are merged, and the same generated responses are used across all methods\.

##### HARP\.

We run the released implementation on our answers and labels\. HARP projects final\-layer token hidden states onto the256256smallest\-singular\-value directions of the unembedding matrix, applies a two\-layer MLP, and uses the maximum token score as the answer score\. We report the performance under both question\-level and answer\-level splits in Table[17](https://arxiv.org/html/2610.36191#A13.T17)\(Appendix[M](https://arxiv.org/html/2610.36191#A13)\)\.

##### ACT\-ViT\.

We follow the released ACT\-ViT implementation and same training setup\. We train for1515epochs with a batch size of128128, learning rate and weight decay of10−310^\{\-3\}, using a three\-layer transformer with four attention heads, hidden size128128, and dropout0\.30\.3\. Activations are max\-pooled using the released preprocessing to88layers andN=20N=20token positions\. We useN=20N=20instead of the defaultN=100N=100because our generated responses are short and larger values would mostly introduce padding\. However, we also report results withN=100N=100in Table[15](https://arxiv.org/html/2610.36191#A12.T15)\(Appendix[L](https://arxiv.org/html/2610.36191#A12)\)\.

##### Evaluation\.

For each seed in\{42,0,1,2,3\}\\\{42,0,1,2,3\\\}, all methods use the same train/test questions and compute the metrics from the scores the method returns\. This ensures that all methods use identical data partitions and evaluation procedures\.

##### Summary statistics scaling in LOTUS\.

Before doing Tucker\-2 decomposition in LOTUS, each summary is scaled entry by entry using statistics computed from the training responses alone: every entry is clipped to its0\.50\.5th and99\.599\.5th training percentiles, centered by its median, divided by its interquartile range rescaled to estimate a standard deviation, and clipped to\[−6,6\]\[\-6,6\]\. We use the median and the interquartile range rather than the mean and the standard deviation because a few features of the residual stream take values far larger than the rest\([Sun et al\., 2024](https://arxiv.org/html/2610.36191#bib.bib43)\)and would otherwise dominate the objective in equation[1](https://arxiv.org/html/2610.36191#S3.E1)\. We write𝑿~i,k\\tilde\{\\bm\{X\}\}\_\{i,k\}for the scaled summary\.

## Appendix EEffect of the data split

As discussed in Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1), the answer\-level split allows responses from the same question to appear in both training and test sets\. This appendix quantifies the resulting AUROC inflation across trained detectors and explains why this overlap is not apparent from aggregate results\. Figure[3](https://arxiv.org/html/2610.36191#A5.F3)illustrates the two splitting strategies\.

Figure 3:Comparison of question\-level and answer\-level splitting with ten responses per question\. Question\-level splitting keeps all responses from a known question in one partition, whereas answer\-level splitting can place responses from the same question in both training and test sets, affecting92\.6%92\.6\\%–95\.7%95\.7\\%of known questions in our data\.##### Overlap between training and test\.

A known question stays out of the test set only if all ten of its answers land in training, and out of training only if all ten land in test\. Under the answer\-level split, a question appears in both training and test sets with probability1−0\.7510−0\.2510≈0\.9441\-0\.75^\{10\}\-0\.25^\{10\}\\approx 0\.944\. Empirically, this overlap ranges from92\.6%92\.6\\%to95\.7%95\.7\\%across the two models, TruthfulQA and TyDiQA\-GP, over five runs\. The test answers are new, but their questions were seen in training, and answers to the same question share a prompt and a topic\. A detector may achieve high performance by recognizing question\-specific information rather than evaluating the factuality of the response itself\.

##### Impact of the splitting protocol\.

The question\- and answer\-level splits produce nearly identical train/test sizes, making the split choice difficult to infer from aggregate results alone\. The released HARP implementation splits answers, and ACT\-ViT splits responses in a similar way\. When multiple responses belong to the same question, this can place the same question in both the training and test sets\. To isolate the effect of the split, we evaluate each trained detector under both settings while keeping the answers, labels, features, and training procedure fixed as shown in Table[7](https://arxiv.org/html/2610.36191#A5.T7)\. Across HARP, LOTUS, and both ACT\-ViT variants, the answer\-level split consistently yields higher AUROC, with gains ranging from less than one point to over1313points\. Under this split, LOTUS achieves the highest AUROC in hallucination detection performance\.

Table 7:Pooled AUROC \(%\) of HARP and LOTUS under the answer\-level split \(Ans\.\), and its gain over the question\-level split \(Δ\\Delta= Ans\.−\-question\-level AUROC in Table[1](https://arxiv.org/html/2610.36191#S4.T1)\)\. ACT\-ViT under both splits is in Table[17](https://arxiv.org/html/2610.36191#A13.T17)\. Standard deviations are in Table[16](https://arxiv.org/html/2610.36191#A13.T16)\.NQ\-OpenTruthfulQATriviaQATyDiQA\-GPMethodAns\.Δ\\DeltaAns\.Δ\\DeltaAns\.Δ\\DeltaAns\.Δ\\DeltaQwen2\.5\-7B\-InstructHARP81\.03\+13\.30\+13\.3085\.16\+7\.62\+7\.6292\.50\+3\.75\+3\.7591\.75\+11\.85\+11\.85LOTUS91\.14\+11\.85\+11\.8594\.03\+6\.20\+6\.2094\.05\+2\.36\+2\.3694\.72\+6\.86\+6\.86LLaMA\-3\.1\-8BHARP86\.56\+6\.00\+6\.0084\.84\+5\.26\+5\.2688\.08\+5\.54\+5\.5485\.65\+10\.66\+10\.66LOTUS91\.96\+6\.60\+6\.6094\.41\+5\.22\+5\.2289\.48\+4\.79\+4\.7990\.07\+7\.58\+7\.58

## Appendix FComparison with Alternative Reductions

##### Reductions of equal width\.

Table[8](https://arxiv.org/html/2610.36191#A6.T8)compares Tucker\-2 with Tucker\-3, flat PCA, and random projection, all using the same random\-forest classifier \(Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1)\)\. Tucker\-2 matches flat PCA and Tucker\-3 in accuracy while using far fewer parameters and multiply\-adds, and random projection performs clearly worse\. The same pattern holds under within\-question AUROC \(defined in Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1)\)\.

Table 8:Performance comparison of reduction methods using the same random\-forest classifier under the question\-level split, averaged over five runs on TruthfulQA \(TQA\) and TyDiQA\-GP \(TyDi\)\. We report pooled AUROC \(%\), feature dimension, fitted basis parameters, and projection multiply\-adds for Qwen2\.5\-7B\-Instruct / LLaMA\-3\.1\-8B\.
##### Single\-layer reductions\.

Table[9](https://arxiv.org/html/2610.36191#A6.T9)compares Tucker\-2 with PCA applied either to the best single layer or to the average over the selected layer window, using the same896896features\. Their AUROC remains close to Tucker\-2, ranging from1\.551\.55points lower to0\.650\.65points higher\. However, both alternatives require about4\.6×4\.6\\timesmore fitted parameters, showing that Tucker\-2 achieves comparable performance with a substantially more compact representation\.

Table 9:Single\-layer reductions to896896features with the random forest, under the question\-level split: pooled AUROC \(%\), averaged over five draws\. Parameters are the entries of the fitted bases, for Qwen2\.5\-7B\-Instruct / LLaMA\-3\.1\-8B\.
##### Tucker\-3\.

We extend Tucker\-2 by dividing each response intoBBtoken bins and applying truncated HOSVD over the layer, token, and feature modes\. Within each bin, we use either the three order\-statistic summaries from §[3](https://arxiv.org/html/2610.36191#S3)or average pooling over hidden states and layer updates\. We evaluateB∈\{1,2,4,8\}B\\in\\\{1,2,4,8\\\}with corresponding token ranksrT∈\{B/2,B\}r\_\{T\}\\in\\\{B/2,B\\\}, while keeping the layer and feature ranks fixed\. All projection bases are fitted on training responses without labels, and the resulting cores are passed to the same random\-forest classifier\. The one\-bin order\-statistic setting corresponds to the Tucker\-2 baseline\. As shown in Table[10](https://arxiv.org/html/2610.36191#A6.T10), retaining an explicit token mode yields limited improvement\. No order\-statistic setting improves over one bin by more than0\.37%0\.37\\%in AUROC, while full token rank is usually worse\. Average pooling shows a similar pattern, suggesting that preserving token structure explicitly does not consistently improve detection performance\.

Table 10:Tucker\-3 withBBtoken bins and token rankrTr\_\{T\}, with the random forest, under the question\-level split: pooled AUROC \(%\), averaged over five draws, and in parentheses the difference from one bin of the same pooling over the same draws\. Features are passed to the classifier\.

## Appendix GPerformance with Limited Training Data

We further compare the Tucker\-2 core with flat PCA under reduced training data, using only5%5\\%–50%50\\%of the training questions while keeping the test sets fixed \(Table[11](https://arxiv.org/html/2610.36191#A7.T11)\)\. The Tucker\-2 core shows a clear advantage when training data are limited\. In the other settings, its relative performance varies with the amount of available training data\. Tucker\-2 improves AUROC by up to5\.06%5\.06\\%on Qwen2\.5\-7B\-Instruct with TyDiQA\-GP and by up to3\.98%3\.98\\%on LLaMA\-3\.1\-8B with TruthfulQA using only5%5\\%–10%10\\%of the training data\. At very small training sizes, flat PCA may not produce the full896896features because the number of principal components is limited by the number of training samples\. These results suggest that the more compact Tucker\-2 parameterization shows stronger robustness when training data is limited\.

Table 11:Reducing the number of labeled training questions, with the test sets held fixed and both reductions refit on each run\. For each setting the first row is the pooled AUROC \(%\) of the Tucker\-2 core, averaged over five runs, and the second is the difference of flat PCA of the same width from it, in AUROC points; negative values are below the core\. The last row averages the four differences\.†At these sizes flat PCA cannot supply all896896features, since each summary yields at most one component per training response\.
## Appendix HLogistic regression classifier

In Table[12](https://arxiv.org/html/2610.36191#A8.T12), we replace the random forest with logistic regression on standardized features for all six configurations\. Logistic regression is particularly competitive on TriviaQA, suggesting that classifier choice affects the relative performance of the different configurations\.

Table 12:Pooled AUROC \(%\) for LOTUS and its ablated variants using logistic regression under the question\-level split\. Results are averaged over five random draws\.ConfigurationSummariesFeaturesNQ\-OpenTruthfulQATriviaQATyDiQA\-GPQwen2\.5\-7B\-InstructLOTUS \(full\)C,S,VC,S,V89689669\.2469\.2480\.1680\.1692\.4892\.4884\.09\\mathbf\{84\.09\}Ablated variantsw/oSSC,VC,V57657668\.0168\.0181\.1781\.1792\.5692\.5681\.9281\.92Peak onlyCC32032071\.85\\mathbf\{71\.85\}81\.8081\.8092\.4492\.4480\.9980\.99Quantile onlySS32032070\.4370\.4382\.0782\.0792\.69\\mathbf\{92\.69\}82\.8282\.82Difference onlyVV25625671\.4571\.4582\.52\\mathbf\{82\.52\}91\.6991\.6982\.7082\.70Shared basesS,VS,V51251268\.9868\.9879\.3979\.3992\.5492\.5482\.8282\.82LLaMA\-3\.1\-8BLOTUS \(full\)C,S,VC,S,V89689679\.0279\.0279\.3879\.3885\.16\\mathbf\{85\.16\}76\.9776\.97Ablated variantsw/oSSC,VC,V57657680\.2080\.2080\.7880\.7884\.7184\.7174\.7074\.70Peak onlyCC32032081\.57\\mathbf\{81\.57\}83\.0483\.0484\.5584\.5577\.0977\.09Quantile onlySS32032081\.0881\.0882\.4882\.4885\.0585\.0578\.67\\mathbf\{78\.67\}Difference onlyVV25625679\.5679\.5683\.14\\mathbf\{83\.14\}82\.2282\.2274\.0874\.08Shared basesS,VS,V51251278\.5578\.5580\.9380\.9384\.5784\.5773\.9573\.95

## Appendix IRank sensitivity

Table[13](https://arxiv.org/html/2610.36191#A9.T13)reports the rank sensitivity across all thirteen configurations\. The peak and range summaries userL=kr\_\{L\}=k, while the update summary usesrL=max⁡\(1,k−1\)r\_\{L\}=\\max\(1,k\-1\)to account for its eight\-layer structure\. It is evident that performance remains relatively stable as the layer rankkkvaries from11to99\. The reported setting,k=5k=5andrF=64r\_\{F\}=64, remains competitive across different settings\.

Table 13:Rank sensitivity under the question\-level split, averaged over five runs\. We vary one rank at a time, and width denotes the number of features per response\.
## Appendix JTransfer across models

To enable cross\-model transfer, we align the target model’s LOTUS representation with the feature space expected by the source detector\. Letϕ\(t\)\\bm\{\\phi\}^\{\(t\)\}andϕ\(s\)\\bm\{\\phi\}^\{\(s\)\}denote the LOTUS representations produced by the target and source models, respectively\. For each response in the target model’s training set, we first computeϕ\(t\)\\bm\{\\phi\}^\{\(t\)\}using the target model’s own hidden states, scaling parameters, and Tucker bases\. We then present the*same response text*, together with its corresponding prompt, to the source model and compute its LOTUS representationϕ\(s\)\\bm\{\\phi\}^\{\(s\)\}using the source model’s bases\. This gives an unlabeled paired set\{\(ϕj\(t\),ϕj\(s\)\)\}j\\\{\(\\bm\{\\phi\}^\{\(t\)\}\_\{j\},\\bm\{\\phi\}^\{\(s\)\}\_\{j\}\)\\\}\_\{j\}, where the two vectors describe the same prompt–response pair in the coordinate systems of the two models\.

Using these pairs, we fit a ridge regressionϕ\(s\)≈𝑾​ϕ\(t\)\+𝒃\\bm\{\\phi\}^\{\(s\)\}\\approx\\bm\{W\}\\bm\{\\phi\}^\{\(t\)\}\+\\bm\{b\}that maps target\-model LOTUS features into the source model’s feature space\. Importantly, this alignment uses only paired representations and does not use whether a response is hallucinated or correct\. At test time, a target\-model response is represented only in the target LOTUS space, mapped through the learned ridge transformation, and then passed directly to the source model’s pretrained hallucination detector\. Thus, the source detector remains fixed and no target\-model hallucination labels are required\.

We train the detector separately for each model in Section[3](https://arxiv.org/html/2610.36191#S3), since the scaling statistics and Tucker\-2 bases are specific to that model’s hidden\-state space\. Its robust scaling and its Tucker\-2 bases live in that model’s hidden space, and the bases are defined only up to sign, order and rotation, so the coordinates the random forest reads have no counterpart in another model\. We therefore evaluate whether a detector trained on one source model can score responses from another target model without using any labels\.

##### Protocol\.

For each question\-level split, the source detector is trained as in Section[3](https://arxiv.org/html/2610.36191#S3), using only questions that appear in the training splits of both models\. Target responses are represented using the target model’s own scaling and Tucker\-2 bases, fitted without target labels\. The target’s answers are projected with the target model’s own scaling and bases, which are fitted without labels on the target’s training answers, as for the target’s own detector\. To carry these coordinates into the source’s, the source model reads the target’s training answers: each answer is appended to the source model’s own prompt for the same question, and the window layer features of Section[3](https://arxiv.org/html/2610.36191#S3)are pooled over the answer’s tokens and projected with the source’s bases\. A ridge regression from target to source coordinates is fitted on these pairs, with the penalty chosen by question\-level five\-fold cross\-validation\. No target labels are used during alignment or training\. Table[4](https://arxiv.org/html/2610.36191#S4.T4)compares the target\-specific detector, cross\-model transfer with and without alignment, and the source detector applied to the source model’s hidden states while it reads the target’s answers\.

## Appendix KComputational Cost

In Table[14](https://arxiv.org/html/2610.36191#A11.T14), we compare the storage required per response for the three trained detectors using uncompressed float16 representations across both model sizes\. It is evident that LOTUS is substantially more storage\-efficient than ACT\-ViT across all datasets\. AtN=20N=20, ACT\-ViT requires roughly33–4×4\\timesmore storage, and its defaultN=100N=100increases this gap to about19×19\\times\. The difference is largest on TriviaQA, where LOTUS uses28\.628\.6–32\.732\.7GB compared with106106–122122GB for ACT\-ViT atN=20N=20\. HARP avoids stored activations entirely by recomputing the final\-layer states during evaluation\. Thus, LOTUS provides a much smaller memory footprint while retaining a fixed\-size representation for each response\.

Table 14:Stored input \(GB\) per model and dataset, float16 and uncompressed, as a range over Qwen2\.5\-7B\-Instruct and LLaMA\-3\.1\-8B\.NNdenotes ACT\-ViT’s number of tokens, where we reportN=20N=20and the released default isN=100N=100\.
## Appendix LToken Budget Analysis of ACT\-ViT

ACT\-ViT max\-pools its activation tensor to a fixed number of tokensNN, withN=100N=100as the released default\. Since our generated responses are short, we useN=20N=20in the main experiments to reduce unnecessary padding and storage\. In Table[15](https://arxiv.org/html/2610.36191#A12.T15), we compare the performance ofN=20N=20andN=100N=100for Qwen2\.5\-7B\-Instruct using both pooled and within\-question AUROC\. The results show that increasingNN=100100yields only modest improvements on TruthfulQA and TyDiQA\-GP, while increasing activation storage by approximately5×5\\timesmore\. LOTUS remains ahead of ACT\-ViT under the question\-level split at both token settings\.

Table 15:ACT\-ViT performance across different numbers of tokensNN\. We report pooled AUROC \(%\) under question\-level \(Q\) and answer\-level \(A\) splits, and within\-question AUROC under the question\-level split, averaged over five runs with standard deviations\.
## Appendix MPerformance Across Splitting Protocols

We report the performance of HARP, ACT\-ViT, and LOTUS under both question\-level and answer\-level splits, including the standard deviation over five runs in Table[16](https://arxiv.org/html/2610.36191#A13.T16)\. Across most settings, the variability is relatively small, while the answer\-level split generally shows lower standard deviations than the question\-level split\. However, for Qwen2\.5\-7B\-Instruct on NQ\-Open dataset under the question\-level split, the standard deviation increases to5\.24%5\.24\\%because the test set contains only about280280correct responses\.

Table 16:HARP, ACT\-ViT and LOTUS under the question\-level \(Q\) and answer\-level \(A\) splits: pooled AUROC \(%\), mean±\\pmstandard deviation over five runs\.Table 17:HARP, both ACT\-ViT variants and LOTUS on TruthfulQA and TyDiQA\-GP under the question\-level \(Q\) and answer\-level \(A\) splits, with standard deviations: pooled AUROC \(%\), mean±\\pmstandard deviation over five draws\.
## Appendix NWithin\-question AUROC

Definition\.For each questionii, let

ℋi=\{k:zi,k=1\},𝒞i=\{k:zi,k=0\}\\mathcal\{H\}\_\{i\}=\\\{k:z\_\{i,k\}=1\\\},\\qquad\\mathcal\{C\}\_\{i\}=\\\{k:z\_\{i,k\}=0\\\}denote the sets of hallucinated and correct sampled responses, respectively\. We consider only questions containing at least one response of each class,

𝒬mix=\{i:\|ℋi\|\>0,\|𝒞i\|\>0\}\.\\mathcal\{Q\}\_\{\\mathrm\{mix\}\}=\\\{i:\|\\mathcal\{H\}\_\{i\}\|\>0,\\;\|\\mathcal\{C\}\_\{i\}\|\>0\\\}\.Given a hallucination scoresi,ks\_\{i,k\}, where larger values indicate stronger evidence of hallucination, we define the within\-question AUROC as

AUROCWQ=1Npair​∑i∈𝒬mix∑k∈ℋi∑k′∈𝒞i\[𝕀⁡\(si,k\>si,k′\)\+12​𝕀​\(si,k=si,k′\)\],\\mathrm\{AUROC\}\_\{\\mathrm\{WQ\}\}=\\frac\{1\}\{N\_\{\\mathrm\{pair\}\}\}\\sum\_\{i\\in\\mathcal\{Q\}\_\{\\mathrm\{mix\}\}\}\\sum\_\{k\\in\\mathcal\{H\}\_\{i\}\}\\sum\_\{k^\{\\prime\}\\in\\mathcal\{C\}\_\{i\}\}\\left\[\\mathbb\{I\}\(s\_\{i,k\}\>s\_\{i,k^\{\\prime\}\}\)\+\\frac\{1\}\{2\}\\mathbb\{I\}\(s\_\{i,k\}=s\_\{i,k^\{\\prime\}\}\)\\right\],\(4\)where

Npair=∑i∈𝒬mix\|ℋi\|​\|𝒞i\|\.N\_\{\\mathrm\{pair\}\}=\\sum\_\{i\\in\\mathcal\{Q\}\_\{\\mathrm\{mix\}\}\}\|\\mathcal\{H\}\_\{i\}\|\\,\|\\mathcal\{C\}\_\{i\}\|\.
We report within\-question AUROC under the question\-level split for LOTUS, HARP, and ACT\-ViT in Table[18](https://arxiv.org/html/2610.36191#A14.T18)\. The remaining baselines assign one score per question, so their within\-question AUROC is always50%50\\%and does not distinguish their responses\. LOTUS outperforms HARP in all four settings and ACT\-ViT in three of the four\. On TruthfulQA, HARP shows a noticeably larger drop from pooled to within\-question AUROC, suggesting that more of its pooled performance is driven by differences between questions\. LOTUS and ACT\-ViT show smaller gaps, indicating better separation among responses to the same question\. On TyDiQA\-GP, pooled and within\-question AUROC performance are much closer for all three detectors\.

Table 18:Within\-question AUROC \(%\) of the three detectors that score each answer, under the question\-level split: mean±\\pmstandard deviation over five draws\. LOTUS and ACT\-ViT share the same test sets; HARP uses the draws of its own code\. ACT\-ViT usesN=20N=20\.
## Appendix OReproduction of HARP

We ran the released implementation of HARP\([Hu et al\., 2026](https://arxiv.org/html/2610.36191#bib.bib16)\)on our generations and labels \(Section[4\.1](https://arxiv.org/html/2610.36191#S4.SS1)\)\. Our answers replace its cache of generations, so its own generation and labeling code does not run, and the only modification is to two device strings\. Table[19](https://arxiv.org/html/2610.36191#A15.T19)compares the published results, obtained on the authors’ generations with the answer\-level split of the released code, with our runs\. For Qwen2\.5\-7B\-Instruct, a single run of the released script differs from the published results by−1\.83\-1\.83to\+3\.47\+3\.47points,\+0\.21\+0\.21on average\. For LLaMA\-3\.1\-8B, five draws of the released split fall below the published results on every dataset, by0\.950\.95to4\.824\.82points and3\.073\.07on average; we have not identified the cause\. Under the question\-level split used throughout the paper, the same detector is3\.753\.75to13\.3013\.30points lower than under the released split \(Section[E](https://arxiv.org/html/2610.36191#A5)\)\.

Table 19:HARP with256256directions: published AUROC \(%\) on the authors’ generations, and our runs of the released code on our generations and labels\. The released split is the answer\-level split; five draws are reported as mean±\\pmstandard deviation\.
## Appendix PAgreement with the labels of HARP

Our labels follow the rubric of[Hu et al\. \(2026\)](https://arxiv.org/html/2610.36191#bib.bib16): an answer is correct if its ROUGE\-L value with the reference answers is at least0\.70\.7or its BLEURT\-20 score against a reference exceeds0\.50\.5\. We average the ROUGE\-L scores across all reference answers and compare the average with0\.70\.7, while the released judge uses the highest ROUGE\-L score and compares it with0\.70\.7\. We use the same BLEURT\-20 criterion in both implementations, where the highest score across the reference answers is compared with a threshold value of0\.50\.5\. Table[20](https://arxiv.org/html/2610.36191#A16.T20)compares the labels from the released judge with ours on all148,270148\{,\}270Qwen2\.5\-7B\-Instruct responses\. The two labeling procedures agree on99\.27%99\.27\\%–99\.94%99\.94\\%of responses across datasets, and only4444of the14,82714\{,\}827questions change between known and unknown status\.

Table 20:Our labels against those of the released judge of HARP, on the answers of Qwen2\.5\-7B\-Instruct\.*Changed*counts questions whose known or unknown status differs between the two sets of labels; the last two columns give the agreement \(%\) over answers and over the known or unknown status of questions\.
## Appendix QCovariance spectra

Table[21](https://arxiv.org/html/2610.36191#A17.T21)reports the values summarized in Section[4\.2](https://arxiv.org/html/2610.36191#S4.SS2)for the embedding output, our selected layer window, and the final block, together with the variance captured by the leading6464directions\. Hallucinated responses consistently have higher effective rank than correct responses, while both classes retain a large share of their variance in a low\-dimensional subspace\. The selected window also achieves substantially higher single\-layer AUROC than the embedding and final block\.

Table 21:Covariance spectra of peak summaries for correct \(C\) and hallucinated \(H\) responses from Qwen2\.5\-7B\-Instruct on TyDiQA\-GP, averaged over2020runs\. We report size\-matched and question\-paired comparisons for the embedding, blocks1515–2323, and the final block\. The last column reports single\-layer AUROC \(%\) using logistic regression with question\-grouped five\-fold cross\-validation \(Appendix[C](https://arxiv.org/html/2610.36191#A3)\)\.