DeMTS: Denoising Trajectories as Multivariate Time Series for Hallucination Detection in Diffusion Language Models

# Denoising Trajectories as Multivariate Time Series for Hallucination Detection in Diffusion Language Models
Source: [https://arxiv.org/html/2608.14632](https://arxiv.org/html/2608.14632)
Xin Zhang1\\equalcontrib, Yili Wang1\\equalcontrib, Yue Tan2, Xin He1, Yanyu Qian3, Yixin Liu2\\corresponding, Yi Chang1, Shirui Pan2, Xin Wang1\\corresponding

###### Abstract

Diffusion large language models \(D\-LLMs\) have emerged as a promising paradigm for text generation\. However, similar to autoregressive LLMs, D\-LLMs remain vulnerable to hallucinations, where fluent outputs may contain factually incorrect or unsupported content\. Although existing hallucination detection methods for D\-LLMs attempt to leverage uncertainty trajectories of the denoising process to better identify hallucination signals, they typically compress the trajectories along either the temporal or token dimension, overlooking the useful information encoded in the complete two\-dimensional token\-step structure\. Consequently, they may fail to capture hallucination\-relevant patterns, such as inconsistent convergence and cross\-token fault propagation, leading to suboptimal detection performance\. To bridge this gap, we propose a D\-LLM hallucination detection framework that formulates theDenoising trajectories asMultivariateTimeSeries over learnable latent variables \(DeMTSfor short\)\.DeMTSemploys a trajectory\-preserving token\-to\-variable assignment module to convert token signals into stable latent variables\. Based on these variables, we propose dynamic multivariate temporal modeling to progressively integrate inter\-variable dependency modeling with temporal encoding for hallucination prediction\. Extensive experiments on two D\-LLMs backbones and three benchmarks demonstrate thatDeMTSoutperforms existing hallucination detection methods while maintaining strong robustness, efficiency, and cross\-task transferability\.

## Introduction

Diffusion large language models \(D\-LLMs\) have recently gained increasing attention as an emerging paradigm for efficient and high\-quality text generation\(Nieet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib50); Bieet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib52); Yeet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib51)\)\. Rather than decoding tokens strictly from left to right\(Mehri and Sigal[2018](https://arxiv.org/html/2608.14632#bib.bib15); Guet al\.[2019](https://arxiv.org/html/2608.14632#bib.bib16); Ghazvininejadet al\.[2019](https://arxiv.org/html/2608.14632#bib.bib17)\), D\-LLMs generate a sequence through iterative denoising\. Despite their advantages over autoregressive LLMs, D\-LLMs remain vulnerable to hallucinations, where fluent outputs may contain factually incorrect or unsupported content\(Banget al\.[2025](https://arxiv.org/html/2608.14632#bib.bib56); Kalaiet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib57)\)\. Detecting such errors from output\-level evidence \(i\.e\., predictive uncertainty of output tokens\) is often insufficient\(Shobyet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib10); Badaveet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib9); Bhatnagaret al\.[2026](https://arxiv.org/html/2608.14632#bib.bib8)\), because hallucination\-related cues may appear during the denoising process in the form of uncertainty rebound or inconsistent refinement among token\(Luet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib54); Jiet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib55)\)\. This motivates D\-LLM\-specific hallucination detection methods that explicitly model denoising trajectories, rather than solely relying on static features of the generated output\(Chenet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib13); Zhanget al\.[2025](https://arxiv.org/html/2608.14632#bib.bib12)\)\.

Recent studies have begun to exploit predictive uncertainty throughout the denoising trajectory, typically quantified by token\-level entropy, for hallucination detection in D\-LLMs\(Guo and Tan[2026](https://arxiv.org/html/2608.14632#bib.bib61); Qiuet al\.[2025a](https://arxiv.org/html/2608.14632#bib.bib64)\)\. One line of work focuses onstep\-levelevidence, which compresses the denoising trajectory along the diffusion\-step dimension by selecting or reweighting the steps whose uncertainty patterns are most informative for hallucination prediction\(Changet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib59); Wenget al\.[2026](https://arxiv.org/html/2608.14632#bib.bib14)\)\. Another line modelstoken\-leveluncertainty dynamics, which compresses the trajectory along the token dimension by identifying hallucination\-indicative tokens and using their denoising dynamics as indicators of factual reliability\(Qianet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib63)\)\. Although these methods demonstrate the usefulness of uncertainty at intermediate denoising states, both lines of works compress the original trajectory along one dimension before detection\. As a result, the token\-step structure of D\-LLM denoising trajectories, which jointly captures how token\-level uncertainty evolves across diffusion steps, is not fully preserved, potentially leading to information loss for hallucination detection\.

![Refer to caption](https://arxiv.org/html/2608.14632v1/x1.png)Figure 1:Observed patterns in denoising trajectories\.To investigate whether compressing denoising trajectories may overlook critical cues for hallucination detection, we examine the evolution of token\-level uncertainty under different Q&A scenarios, and then identify two representative patterns that require explicit modeling of token\-step uncertainty evidence\.Pattern ❶: Inconsistent Convergence in Hallucination Answers\(Fig\.[1](https://arxiv.org/html/2608.14632#Sx1.F1)a\)\. In hallucinated responses, token uncertainties may converge to markedly different levels during the late denoising stage: some tokens become highly confident, while others remain relatively uncertain\. Such inconsistent convergence across tokens reveals abnormal inter\-token relations and provides hallucination evidence beyond any individual token\.Pattern ❷: Cross\-Token Fault Propagation in Hallucination Answers\(Fig\.[1](https://arxiv.org/html/2608.14632#Sx1.F1)b\)\. During the denoising process of some hallucination answers, an unstable or incorrect token state can affect the decoding of other contextually connected tokens, causing their entropy trajectories to deviate together from a coherent factual path\. Capturing this propagative pattern for hallucination detection requires explicit modeling of dynamic dependencies among token trajectories, further highlighting the necessity of preserving the full token\-step uncertainty structure\. Recognizing the importance of two\-dimensional evidence modeling, we pose a central research question:

How can we explicitly modeltoken\-step uncertainty trajectoriesto capture cross\-token dependencies and temporal dynamics for D\-LLM hallucination detection?

A promising solution to this question is to formulate the token\-step uncertainty trajectory as amultivariate time series, which naturally preserves its two\-dimensional structure: denoising steps define the temporal axis, while token\-associated uncertainty signals serve as multiple interacting variables\. Although this formulation is natural, directly applying existing approaches for multivariate time series to D\-LLM hallucination detection is non\-trivial, since they typically assume that each variable has a stable identity over time and among different variables\(Huet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib65); Zhang and Yan[2023](https://arxiv.org/html/2608.14632#bib.bib66)\)\. This assumption does not hold for D\-LLMs, where the semantic role of a potential variable \(i\.e\. token position\) may vary across samples and change during denoising process, thus giving rise to two key challenges\.*Challenge 1: Stable Variable Construction\.*Raw token positions cannot directly serve as variables, as they lack consistent semantic identities\. They must be reorganized into semantically stable variables while preserving fine\-grained denoising dynamics relevant to hallucination detection\.*Challenge 2: Relational Temporal Modeling\.*The temporal dynamics of the constructed variables are coupled with evolving dependencies among them\. In this case, the hallucination detector must jointly capture both aspects, as hallucination cues can manifest as inconsistent convergence or propagate across related trajectories\.

To address these challenges, we proposeDeMTS, a hallucination detection framework that formulates D\-LLMDenoising trajectories asMultivariateTimeSeries over learnable latent variables\. To handle*Challenge 1*,DeMTSintroduces atrajectory\-preserving Token\-to\-Variable assignment\(T2V\) module, which reorganizes token uncertainty trajectories into latent variables with more consistent identities, considering uncertainty states, contextual semantics, and positional structure\. A trajectory\-preservation constraint is further imposed to retain local denoising variations during token\-to\-variable assignment\. To address*Challenge 2*,DeMTSintroducesDynamic Multivariate Temporal Modeling\(DMTM\), which jointly captures relational and temporal patterns over the constructed latent variables\. It performs step\-adaptive inter\-variable interaction to suppress unreliable early\-step dependencies and strengthen interactions among stabilized variables, followed by variable\-wise temporal modeling to encode the evolution of each latent variable\. In this way,DeMTScaptures both hallucination\-related temporal dynamics and cross\-token dependencies\. In summary, our contributions are as follows:

- •We formulate D\-LLMs hallucination detection as a multivariate time series learning problem, and identify two key challenges in adapting existing time series approaches to denoising trajectories\.
- •We proposeDeMTS, a trajectory\-based detector which transforms semantically unstable token signals into stable latent variables and jointly captures dynamic inter\-variable dependencies and variable\-wise temporal evolution\.
- •Experiments on two D\-LLM backbones and three benchmarks show thatDeMTSconsistently outperforms baselines, while remaining robust, transferable, and efficient\.

## Related Works

Hallucination Detection in D\-LLMs\.The iterative denoising process of D\-LLMs\(Yanget al\.[2023](https://arxiv.org/html/2608.14632#bib.bib26)\)provides rich trajectory\-level uncertainty evidence for hallucination detection\(Guo and Tan[2026](https://arxiv.org/html/2608.14632#bib.bib61)\)\. One line of existing methods focuses onstep\-oriented evidence selection\(Changet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib59); Hemmatet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib58); Zhaoet al\.[2026a](https://arxiv.org/html/2608.14632#bib.bib60)\), which identifies or reweights informative denoising stages and uses the selected trajectory segments for hallucination prediction\. The second line focuses ondynamics\-oriented trajectory modeling\(Qianet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib63)\), which detects hallucinations by modeling abnormal token evolution during denoising\. They show that hallucination evidence can emerge from discriminative denoising steps or dynamic trajectory patterns\. Differently,DeMTSmodels D\-LLMs two\-dimensional denoising trajectories as multivariate time series for hallucination detection\.

Multivariate Time Series Learning\.Deep learning for multivariate time series has been widely studied to capture evolving patterns across multiple variables\(Linet al\.[2025b](https://arxiv.org/html/2608.14632#bib.bib49); Zhanget al\.[2026](https://arxiv.org/html/2608.14632#bib.bib47); Nguyenet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib48)\)\. Existing methods are commonly categorized by their channel modeling strategies\. Channel\-independent methods model each variable separately to preserve variable\-specific temporal patterns\(Zhang and Yan[2023](https://arxiv.org/html/2608.14632#bib.bib66); Liuet al\.[2024](https://arxiv.org/html/2608.14632#bib.bib46); Shenet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib1)\), while channel\-dependent methods jointly model all variables to capture global inter\-variable correlations\(Nieet al\.[2022](https://arxiv.org/html/2608.14632#bib.bib44); Zhaoet al\.[2026b](https://arxiv.org/html/2608.14632#bib.bib2); Zenget al\.[2023](https://arxiv.org/html/2608.14632#bib.bib43)\)\. More recent channel\-partial methods seek a balance between these two views by allowing each variable to interact only with relevant variables\(Qiuet al\.[2025b](https://arxiv.org/html/2608.14632#bib.bib41); Zhaoet al\.[2023](https://arxiv.org/html/2608.14632#bib.bib40)\)\. These strategies assume that variables have stable identities across samples and time\(Yanget al\.[2024](https://arxiv.org/html/2608.14632#bib.bib18); Linet al\.[2025a](https://arxiv.org/html/2608.14632#bib.bib19)\)\. However, in D\-LLM denoising trajectories, token positions may correspond to changing semantic roles during generation\. Therefore,DeMTSconverts changing token signals into stable latent variables for temporal and inter\-variable modeling\. Detailed literature review please refer to Appendix[A](https://arxiv.org/html/2608.14632#A1)\.

## Preliminary

Diffusion Large Language Models\.Given an input query𝐪\\mathbf\{q\}, the diffusion large language models \(D\-LLMs\) generates a fixed\-length response through iterative denoising\. Let𝐫\(t\)=\(r1\(t\),r2\(t\),…,rN\(t\)\)∈𝒱N\\mathbf\{r\}^\{\(t\)\}=\(r\_\{1\}^\{\(t\)\},r\_\{2\}^\{\(t\)\},\\ldots,r\_\{N\}^\{\(t\)\}\)\\in\\mathcal\{V\}^\{N\}denote the intermediate response at denoising stept∈\{T,…,0\}t\\in\\\{T,\\ldots,0\\\}, whereNNis the sequence length and𝒱\\mathcal\{V\}is the vocabulary\. Starting from a highly masked sequence𝐫\(T\)\\mathbf\{r\}^\{\(T\)\}, the model progressively reconstructs the response according to𝐫\(t−1\)∼pθ​\(𝐫\(t−1\)∣𝐫\(t\),𝐪\),\\mathbf\{r\}^\{\(t\-1\)\}\\sim p\_\{\\theta\}\\left\(\\mathbf\{r\}^\{\(t\-1\)\}\\mid\\mathbf\{r\}^\{\(t\)\},\\mathbf\{q\}\\right\),until obtaining the final response𝐫\(0\)\\mathbf\{r\}^\{\(0\)\}\.

At each denoising steptt, the D\-LLMs predicts a categorical distribution over the vocabulary for each token positionii:𝝅i,t=pθ​\(ri\(0\)∣𝐫\(t\),𝐪\)∈ℝ\|𝒱\|\.\\boldsymbol\{\\pi\}\_\{i,t\}=p\_\{\\theta\}\\left\(r\_\{i\}^\{\(0\)\}\\mid\\mathbf\{r\}^\{\(t\)\},\\mathbf\{q\}\\right\)\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}\|\}\.The corresponding predictive uncertainty is measured by token entropy:Hi,t=−∑v∈𝒱πi,t​\(v\)​log⁡πi,t​\(v\)\.H\_\{i,t\}=\-\\sum\_\{v\\in\\mathcal\{V\}\}\\pi\_\{i,t\}\(v\)\\log\\pi\_\{i,t\}\(v\)\.Collecting token uncertainties across all denoising steps and positions yields the uncertainty trajectory𝐇=\[Hi,t\]∈ℝN×\(T\+1\),\\mathbf\{H\}=\[H\_\{i,t\}\]\\in\\mathbb\{R\}^\{N\\times\(T\+1\)\},where the temporal axis corresponds to denoising steps and the token axis corresponds to sequence positions\. We further denote by𝐞i,t∈ℝdh\\mathbf\{e\}\_\{i,t\}\\in\\mathbb\{R\}^\{d\_\{h\}\}the contextual embedding of token positioniiat steptt, and collect them as𝐄∈ℝN×\(T\+1\)×dh\\mathbf\{E\}\\in\\mathbb\{R\}^\{N\\times\(T\+1\)\\times d\_\{h\}\}\. The uncertainty trajectory𝐇\\mathbf\{H\}provides the primary detection signal, while𝐄\\mathbf\{E\}supplies contextual semantics for constructing the latent\-variable trajectory\.

Hallucination Detection in D\-LLMs\.Given the uncertainty trajectory𝐇\\mathbf\{H\}, contextual trajectory𝐄\\mathbf\{E\}, and the generated response, hallucination detection aims to determine whether the response contains factually incorrect or unsupported content\. Let𝒟=\{\(𝐇n,𝐄n,yn\)\}n=1M\\mathcal\{D\}=\\left\\\{\(\\mathbf\{H\}\_\{n\},\\mathbf\{E\}\_\{n\},y\_\{n\}\)\\right\\\}\_\{n=1\}^\{M\}denote the training set, whereMMis the number of samples andyn∈\{0,1\}y\_\{n\}\\in\\\{0,1\\\}is the hallucination label, withyn=1y\_\{n\}=1indicating a hallucinated response andyn=0y\_\{n\}=0indicating a factual response\. The objective is to learn a detectorfϕf\_\{\\phi\}that maps the denoising trajectories to a hallucination probability:y^n=fϕ​\(𝐇n,𝐄n\)\.\\hat\{y\}\_\{n\}=f\_\{\\phi\}\\left\(\\mathbf\{H\}\_\{n\},\\mathbf\{E\}\_\{n\}\\right\)\.The task is formulated as trajectory\-based binary classification:

minϕ⁡1M​∑n=1Mℒcls​\(yn,fϕ​\(𝐇n,𝐄n\)\),\\min\_\{\\phi\}\\frac\{1\}\{M\}\\sum\_\{n=1\}^\{M\}\\mathcal\{L\}\_\{\\mathrm\{cls\}\}\\left\(y\_\{n\},f\_\{\\phi\}\(\\mathbf\{H\}\_\{n\},\\mathbf\{E\}\_\{n\}\)\\right\),\(1\)whereℒcls\\mathcal\{L\}\_\{\\mathrm\{cls\}\}denotes the binary cross\-entropy loss\.

## Methodology

![Refer to caption](https://arxiv.org/html/2608.14632v1/x2.png)Figure 2:Overview ofDeMTS, which transforms token\-level denoising signals into latent variables and jointly models their temporal evolution and inter\-variable interactions for hallucination detection\.In this section, we provide an overview ofDeMTS, a hallucination detection framework that models D\-LLMs denoising trajectories as multivariate time series\.DeMTStakes a token\-step uncertainty trajectory as input which jointly preserves temporal evolution and cross\-token relationships\. It first performsTrajectory\-Preserving Token\-to\-Variable \(T2V\) Assignmentto map schedule\-biased and semantically unstable token signals into latent variables with more consistent identities\. Based on the resulting latent variable trajectories,DeMTSthen appliesDynamic Multivariate Temporal Modelingto jointly capture variable\-wise temporal evolution and dynamically evolving interactions among latent variables throughout the denoising process\. The learned trajectory representation is finally used for hallucination detection\. The overall framework ofDeMTSis illustrated in Fig\.[2](https://arxiv.org/html/2608.14632#Sx4.F2)\.

### Trajectory\-Preserving T2V Assignment

The raw denoising trajectory of a D\-LLM is organized by token positions, where each trajectory corresponds to the uncertainty evolution of a fixed sequence position\. However, the same position may correspond to different semantic roles and evolve into different token contents during denoising, making it unsuitable as a variable for multivariate temporal modeling\. To construct a more reliable variable axis, we proposeTrajectory\-Preserving Token\-to\-Variable Assignment\(T2V\), which converts the position\-indexed token trajectory into a latent variable\-indexed trajectory, providing a more stable basis for multivariate time series learning\.

The goal of T2V is to learn how each token\-level signal should contribute toKKlatent variables with stable semantics while preserving its denoising dynamics\. To this end, a schedule\-aware uncertainty normalization scheme is introduced to reduce the global uncertainty bias induced by the denoising process\. Then, a latent variable axis is constructed by assigning token\-level signals to latent variables according to their uncertainty, semantic content, and positional structure\. Moreover, T2V imposes a trajectory\-preservation constraint on the assignment, preventing local hallucination cues from being smoothed out during token\-to\-variable aggregation\.

Schedule\-Aware Uncertainty Normalization\.Raw token uncertainty can be affected by the denoising schedule, where early steps are generally more uncertain and later steps become more stable\. To reduce this step\-wise bias, we normalize token uncertainty within each denoising step by

μt=1N​∑i=1NHi,t,σt=1N​∑i=1N\(Hi,t−μt\)2,H^i,t=Hi,t−μtσt\+ϵ,\\displaystyle\\begin\{gathered\}\\mu\_\{t\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}H\_\{i,t\},\\qquad\\sigma\_\{t\}=\\sqrt\{\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\left\(H\_\{i,t\}\-\\mu\_\{t\}\\right\)^\{2\}\},\\\\ \\widehat\{H\}\_\{i,t\}=\\frac\{H\_\{i,t\}\-\\mu\_\{t\}\}\{\\sigma\_\{t\}\+\\epsilon\},\\end\{gathered\}\(4\)whereHi,tH\_\{i,t\}denotes the raw uncertainty of token positioniiat denoising steptt,H^i,t\\widehat\{H\}\_\{i,t\}denotes the normalized uncertainty, andϵ\\epsilonis a small constant\. After normalization,H^i,t\\widehat\{H\}\_\{i,t\}can capture the relative uncertainty level of tokeniiwithin the uncertainty distribution at each denoising step, providing a schedule\-calibrated signal for token\-to\-variable assignment\.

Variable Axis Construction\.After normalization, the denoising trajectory remains indexed by raw token positions, preventing their direct use as stable variables for multivariate temporal modeling\. Since a token position lacks a stable semantic identity in most scenarios, we construct a latent variable axis by learning a soft assignment from each token\-level signal toKKlatent variables, grouping semantically related token signals into latent variables with more consistent identities\. In this way, the position\-indexed trajectory is reorganized into a stable variable\-indexed trajectory, providing a more reliable basis for subsequent procedures\.

To construct a latent\-variable axis with stable semantics, the assignment should depend on both uncertainty dynamics and token semantics\. We therefore combine the normalized uncertainty with the contextual token embedding extracted from the D\-LLMs via:

𝐮i,t=Linearh​\(H^i,t\)\+γ​Lineare​\(𝐞i,t\),\\displaystyle\\mathbf\{u\}\_\{i,t\}=\\mathrm\{Linear\}\_\{h\}\(\\widehat\{H\}\_\{i,t\}\)\+\\gamma\\,\\mathrm\{Linear\}\_\{e\}\(\\mathbf\{e\}\_\{i,t\}\),\(5\)where𝐮i,t∈ℝN×\(T\+1\)×du\\mathbf\{u\}\_\{i,t\}\\in\\mathbb\{R\}^\{N\\times\(T\+1\)\\times d\_\{u\}\}denotes the joint representation of token \(i\) at denoising step \(t\), encoding both its uncertainty state and contextual semantics for subsequent latent\-variable assignment\. Specifically,𝐞i,t\\mathbf\{e\}\_\{i,t\}denotes the contextual embedding of tokeniiat denoising steptt,Linearh​\(⋅\)\\mathrm\{Linear\}\_\{h\}\(\\cdot\)andLineare​\(⋅\)\\mathrm\{Linear\}\_\{e\}\(\\cdot\)project the uncertainty and semantic signals into the same space, respectively, andγ\\gammacontrols the contribution of contextual information\.

Based on𝐮i,t\\mathbf\{u\}\_\{i,t\}, we further compute the content\-conditioned assignment logits overKKlatent variables as:

𝐬i,tcon=ℱcon​\(𝐮i,t\),\\displaystyle\\mathbf\{s\}^\{\\mathrm\{con\}\}\_\{i,t\}=\\mathcal\{F\}\_\{\\mathrm\{con\}\}\(\\mathbf\{u\}\_\{i,t\}\),\(6\)whereℱcon​\(⋅\)\\mathcal\{F\}\_\{\\mathrm\{con\}\}\(\\cdot\)is a learnable mapping function, and𝐬i,tcon\\mathbf\{s\}^\{\\mathrm\{con\}\}\_\{i,t\}represents the assignment preference of token signal\(i,t\)\(i,t\)over theKKlatent variables\.

Although token positions cannot directly serve as stable variables due to their unstable semantic identities, positional information still provides useful structural cues for latent variable construction\. Specifically, relative positions preserve the sequential order and local neighborhood relationships among tokens, offering structural guidance for grouping semantically related token\-level signals into latent variables\. To capture structural cues at different spatial scales, we adopt a multi\-frequency Fourier representation to encode each position:

ϕi=\[sin⁡\(2​π​f​iN\),cos⁡\(2​π​f​iN\)\]f=1F,\\boldsymbol\{\\phi\}\_\{i\}=\\left\[\\sin\\left\(\\frac\{2\\pi fi\}\{N\}\\right\),\\cos\\left\(\\frac\{2\\pi fi\}\{N\}\\right\)\\right\]\_\{f=1\}^\{F\},\(7\)whereFFdenotes the number of Fourier frequencies\. Low\-frequency components capture coarse global ordering, whereas high\-frequency components preserve fine\-grained local neighborhood relations, providing complementary structural information for latent\-variable construction\. We then map the Fourier representation into structural assignment logits:

𝐬istr=ℱstr​\(ϕi\),\\mathbf\{s\}^\{\\mathrm\{str\}\}\_\{i\}=\\mathcal\{F\}\_\{\\mathrm\{str\}\}\(\\boldsymbol\{\\phi\}\_\{i\}\),\(8\)whereℱstr​\(⋅\)\\mathcal\{F\}\_\{\\mathrm\{str\}\}\(\\cdot\)is a learnable mapping function and𝐬istr\\mathbf\{s\}^\{\\mathrm\{str\}\}\_\{i\}represents the structural assignment preference of positioniiover theKKlatent variables\. Since positional structure should be adaptively injected according to the current token state, we introduce a content\-adaptive gate

𝐠i,t=σ​\(ℱgate​\(𝐮i,t\)\)∈ℝK,\\mathbf\{g\}\_\{i,t\}=\\sigma\\left\(\\mathcal\{F\}\_\{\\mathrm\{gate\}\}\(\\mathbf\{u\}\_\{i,t\}\)\\right\)\\in\\mathbb\{R\}^\{K\},\(9\)to regulate their contribution\. The final token\-to\-variable assignment weights are obtained by combining content\-conditioned and gated structural assignment logits:

𝐚i,t=Softmax⁡\(𝐬i,tcon\+𝐠i,t⊙𝐬istr\)∈ℝK,\\displaystyle\\mathbf\{a\}\_\{i,t\}=\\operatorname\{Softmax\}\\left\(\\mathbf\{s\}^\{\\mathrm\{con\}\}\_\{i,t\}\+\\mathbf\{g\}\_\{i,t\}\\odot\\mathbf\{s\}^\{\\mathrm\{str\}\}\_\{i\}\\right\)\\in\\mathbb\{R\}^\{K\},\(10\)where𝐚i,t\\mathbf\{a\}\_\{i,t\}denotes the soft assignment distribution of token signal\(i,t\)\(i,t\)over theKKlatent variables\. In this way, token semantics and positional structure are jointly considered for token\-to\-variable assignment\.

Preservation\-Constrained Variable Construction\.Given the token\-to\-variable assignment, we further construct the variable\-indexed trajectory through assignment\-weighted aggregation, transforming token\-level representations into latent\-variable representations\. We first define the aggregation operator for latent variablekkat denoising steptt:

𝒜k,t​\(𝐱t\)=∑i=1Nai,t,k​𝐱i,t∑i=1Nai,t,k\+ϵ,\\mathcal\{A\}\_\{k,t\}\(\\mathbf\{x\}\_\{t\}\)=\\frac\{\\sum\_\{i=1\}^\{N\}a\_\{i,t,k\}\\mathbf\{x\}\_\{i,t\}\}\{\\sum\_\{i=1\}^\{N\}a\_\{i,t,k\}\+\\epsilon\},\(11\)where𝐱t=\{𝐱i,t\}i=1N\\mathbf\{x\}\_\{t\}=\\\{\\mathbf\{x\}\_\{i,t\}\\\}\_\{i=1\}^\{N\}denotes a generic token\-indexed input to the aggregation operator instantiated as the joint token representation𝐮t\\mathbf\{u\}\_\{t\}\. Moreover,𝐚i,t=\[ai,t,1,ai,t,2,…,ai,t,K\]∈ℝK\\mathbf\{a\}\_\{i,t\}=\[a\_\{i,t,1\},a\_\{i,t,2\},\\ldots,a\_\{i,t,K\}\]\\in\\mathbb\{R\}^\{K\}is the soft assignment distribution of tokeniiover all latent variables at denoising steptt, andai,t,ka\_\{i,t,k\}denotes itskk\-th element\. It measures the contribution of token\-level input𝐱i,t\\mathbf\{x\}\_\{i,t\}to thekk\-th latent variable, whileϵ\\epsilonis a small constant for numerical stability\. Using this operator, the latent variablekkis written as𝐳k,t=𝒜k,t​\(𝐮t\)∈ℝdu,\\mathbf\{z\}\_\{k,t\}=\\mathcal\{A\}\_\{k,t\}\(\\mathbf\{u\}\_\{t\}\)\\in\\mathbb\{R\}^\{d\_\{u\}\},and the complete variable\-indexed trajectory is𝐙=\[𝐳k,t\]k=1,…,Kt=0,…,T\.\\mathbf\{Z\}=\[\\mathbf\{z\}\_\{k,t\}\]\_\{\\begin\{subarray\}\{c\}k=1,\\ldots,K\\\\ t=0,\\ldots,T\\end\{subarray\}\}\.

A potential concern with assignment\-weighted aggregation is that it may smooth out local denoising dynamics that are informative for hallucination detection\. We therefore introduce a trajectory\-preservation constraint to retain these dynamics\. Specifically, we project the uncertainty change of tokeniiinto the representation space:Δ​𝐇^i,t=Linearh​\(H^i,t−H^i,t−1\)\.\\Delta\\widehat\{\\mathbf\{H\}\}\_\{i,t\}=\\mathrm\{Linear\}\_\{h\}\(\\widehat\{H\}\_\{i,t\}\-\\widehat\{H\}\_\{i,t\-1\}\)\.Collecting all token\-level variations givesΔ​𝐇^t=\[Δ​𝐇^1,t,…,Δ​𝐇^N,t\]⊤\.\\Delta\\widehat\{\\mathbf\{H\}\}\_\{t\}=\[\\Delta\\widehat\{\\mathbf\{H\}\}\_\{1,t\},\\ldots,\\Delta\\widehat\{\\mathbf\{H\}\}\_\{N,t\}\]^\{\\top\}\.We then aggregate the token\-level variations into the latent\-variable space:Δ​𝐙k,t=𝒜k,t​\(Δ​𝐇^t\)\.\\Delta\\mathbf\{Z\}\_\{k,t\}=\\mathcal\{A\}\_\{k,t\}\(\\Delta\\widehat\{\\mathbf\{H\}\}\_\{t\}\)\.The latent\-variable variations are subsequently projected back to the token level:Δ​𝐇~i,t=∑k=1Kai,t,k​Δ​𝐙k,t\.\\Delta\\widetilde\{\\mathbf\{H\}\}\_\{i,t\}=\\sum\_\{k=1\}^\{K\}a\_\{i,t,k\}\\Delta\\mathbf\{Z\}\_\{k,t\}\.This aggregation and reconstruction process constrains the T2V assignment to preserve token\-level denoising dynamics during variable construction\.

To preserve both the magnitude and temporal direction of these dynamics, we decompose the preservation constraint into two complementary terms\. The first term is theAmplitude Preservation Loss, which addresses amplitude smoothing, where token\-to\-variable aggregation weakens the magnitude of local uncertainty changes:

ℒa​m​p=𝔼i,t​\[\(\|Δ​𝐇^i,t\|−\|Δ​𝐇~i,t\|\)2\]\.\\mathcal\{L\}\_\{amp\}=\\mathbb\{E\}\_\{i,t\}\\left\[\\left\(\|\\Delta\\mathbf\{\\widehat\{H\}\}\_\{i,t\}\|\-\|\\Delta\\mathbf\{\\widetilde\{H\}\}\_\{i,t\}\|\\right\)^\{2\}\\right\]\.\(12\)By matching the magnitude of the reconstructed variation to that of the original token\-level variation,ℒa​m​p\\mathcal\{L\}\_\{amp\}encourages variable construction to retain the strength of local denoising dynamics\. The second term is theDirection Consistency Loss, which addresses direction distortion, where the reconstructed variation may evolve in the opposite temporal direction:

ℒd​i​r=𝔼i,t​\[max⁡\(0,−Δ​𝐇^i,t​Δ​𝐇~i,t\)\]\.\\mathcal\{L\}\_\{dir\}=\\mathbb\{E\}\_\{i,t\}\\left\[\\max\\left\(0,\-\\Delta\\mathbf\{\\widehat\{H\}\}\_\{i,t\}\\Delta\\mathbf\{\\widetilde\{H\}\}\_\{i,t\}\\right\)\\right\]\.\(13\)This penalty is activated when the original and reconstructed variations have opposite signs, thereby discouraging variable construction from reversing uncertainty trends\. The final preservation constraint \(with balance hyperparametersλ\\lambda\) is defined as

ℒp​r​e​s=λa​m​p​ℒa​m​p\+λd​i​r​ℒd​i​r\.\\mathcal\{L\}\_\{pres\}=\\lambda\_\{amp\}\\mathcal\{L\}\_\{amp\}\+\\lambda\_\{dir\}\\mathcal\{L\}\_\{dir\}\.\(14\)

### Dynamic Multivariate Temporal Modeling

With the latent\-variable trajectory constructed by T2V, the remaining challenge is to model the temporal evolution with the evolving inter\-variable dependency for hallucination detection\. To this end, we proposeDynamic Multivariate Temporal Modeling\(DMTM\), which progressively integrates inter\-variable dependency modeling with variable\-wise temporal encoding for hallucination prediction\.

Stage\-Adaptive Inter\-Variable Interaction\.In D\-LLM denoising trajectories, the dependencies among latent variables evolve throughout denoising and provide important relational evidence for hallucination detection\. To model the dependencies, we aggregate cross\-variable relational evidence through attention\-based interaction\. Specifically, we construct contextualized variable representation via:

𝐂=Softmax⁡\(𝐐𝐊⊤d\)​𝐕,\\mathbf\{C\}=\\operatorname\{Softmax\}\\left\(\\frac\{\\mathbf\{Q\}\\mathbf\{K\}^\{\\top\}\}\{\\sqrt\{d\}\}\\right\)\\mathbf\{V\},\(15\)where𝐐=𝐙𝐖Q\\mathbf\{Q\}=\\mathbf\{Z\}\\mathbf\{W\}\_\{Q\},𝐊=𝐙𝐖K\\mathbf\{K\}=\\mathbf\{Z\}\\mathbf\{W\}\_\{K\}, and𝐕=𝐙𝐖V\\mathbf\{V\}=\\mathbf\{Z\}\\mathbf\{W\}\_\{V\}denote the query, key, and value projections, respectively\. The row\-wise softmax normalizes the dependency scores for each latent variable across all variables, producing relation weights that specify how much information it should receive from the others at steptt\. Using these weights, each latent variable aggregates information from its related variables, adaptively capturing step\-specific cross\-variable dependencies during denoising\.

While the above interaction captures step\-specific dependencies, their reliability varies with the noise level at each denoising step\. In the D\-LLMs denoising process, largerttcorresponds to earlier and noisier stages, where variable are less reliable and the estimated dependencies are more susceptible to spurious correlations\. Asttdecreases, the variables gradually stabilize, making their relational patterns increasingly representative of the underlying semantic dependencies\(Wenget al\.[2026](https://arxiv.org/html/2608.14632#bib.bib14)\)\. To calibrate cross\-variable information fusion according to this reliability transition, we introduce a trajectory\-reliability gate:

𝐔=𝐙\+ρ​𝐂,ρ=T−tT,\\mathbf\{U\}=\\mathbf\{Z\}\+\\rho\\mathbf\{C\},\\qquad\\rho=\\frac\{T\-t\}\{T\},\(16\)whereρt\\rho\_\{t\}increases as the trajectory approaches the final step, assigning limited fusion strength to noisy early\-step relations while progressively strengthening information exchange among stabilized variables\. By limiting interaction at noisy early stages and strengthening it as the variables stabilize, this design reduces the influence of spurious dependencies and emphasizes more reliable semantic relations\.

ModelMethodTriviaQAHotpotQACSQAAvg\.641286412864128LLaDA\-8B\-InstructOutput\-basedMethodsPerplexity47\.650\.451\.249\.365\.065\.654\.9LN\-Entropy53\.554\.654\.754\.864\.464\.657\.8Semantic Entropy67\.368\.953\.857\.643\.944\.155\.9Lexical Similarity59\.062\.557\.164\.260\.757\.360\.1Latent\-basedMethodsEigenScore66\.969\.259\.264\.760\.658\.563\.2CCS54\.257\.155\.857\.658\.550\.555\.6TSV61\.160\.259\.465\.055\.252\.959\.0Trajectory\-basedMethodsTraceDet74\.173\.963\.766\.177\.177\.272\.0DynHD86\.186\.785\.384\.281\.381\.684\.2DeMTS89\.889\.188\.188\.083\.485\.187\.3Dream\-7B\-InstructOutput\-basedMethodsSemantic Entropy72\.573\.767\.762\.748\.651\.462\.8Lexical Similarity64\.058\.362\.759\.776\.977\.366\.5Latent\-basedMethodsEigenScore69\.166\.067\.062\.577\.576\.969\.8CCS50\.356\.958\.251\.753\.254\.254\.1TSV74\.775\.663\.058\.756\.862\.365\.2Trajectory\-basedMethodsTraceDet86\.778\.176\.075\.184\.184\.780\.8DynHD84\.487\.385\.680\.184\.683\.584\.3DeMTS87\.788\.988\.284\.686\.987\.987\.4

Table 1:AUROC \(%\) comparison of hallucination detection methods on two D\-LLMs across three QA datasets\. The highest score isbolded, and the second\-highest score isunderlined\.Variable\-Wise Temporal Dynamics Modeling\.After incorporating stage\-adaptive relational evidence into each latent variable, we further model how its dependency\-enhanced state evolves throughout the denoising trajectory\. To capture temporal evolution patterns, we first incorporate trajectory\-order information into the dependency\-enhanced variable states:𝐔¯k=𝐔k\+𝐞time,\\bar\{\\mathbf\{U\}\}\_\{k\}=\\mathbf\{U\}\_\{k\}\+\\mathbf\{e\}^\{\\mathrm\{time\}\},where𝐞time\\mathbf\{e\}^\{\\mathrm\{time\}\}denotes the temporal embedding of trajectory steptt\. For latent variablekk, we then apply temporal self\-attention to its complete trajectory:

𝐑k=TemporalAttn​\(𝐔¯k\)\.\\mathbf\{R\}\_\{k\}=\\mathrm\{TemporalAttn\}\\left\(\\bar\{\\mathbf\{U\}\}\_\{k\}\\right\)\.\(17\)Applying it to all variables yields the multivariate temporal representation𝐑\\mathbf\{R\}for hallucination prediction\. Finally, the hallucination probability is predicted through a lightweight classifier:y^=MLP⁡\(𝐑\)\\hat\{y\}=\\operatorname\{MLP\}\(\\mathbf\{R\}\)\.

Training Objective\.The detector is trained with a classification objective together with the preservation constraint introduced in T2V\. Given the ground\-truth hallucination labely∈\{0,1\}y\\in\\\{0,1\\\}and the predicted probabilityy^\\hat\{y\}, we define the classification loss as the binary cross\-entropy:

ℒc​l​s=−𝔼​\[y​log⁡y^\+\(1−y\)​log⁡\(1−y^\)\]\.\\mathcal\{L\}\_\{cls\}=\-\\mathbb\{E\}\\left\[y\\log\\hat\{y\}\+\(1\-y\)\\log\(1\-\\hat\{y\}\)\\right\]\.\(18\)The overall training objective is:

ℒ=ℒc​l​s\+ℒp​r​e​s,\\mathcal\{L\}=\\mathcal\{L\}\_\{cls\}\+\\mathcal\{L\}\_\{pres\},\(19\)which optimizes the detector for hallucination prediction while encouraging the learned variable trajectory to retain fine\-grained token\-level denoising dynamics\. Algorithm[1](https://arxiv.org/html/2608.14632#alg1)summarizes the procedure ofDeMTSin Appendix[B](https://arxiv.org/html/2608.14632#A2), with complexity analysis in Appendix[C](https://arxiv.org/html/2608.14632#A3)\.

## Experiments

### Experimental Setup

Datasets\.Our evaluation covers three complementary question\-answering scenarios\.TriviaQA\(Joshiet al\.[2017](https://arxiv.org/html/2608.14632#bib.bib38)\)focuses on open\-domain knowledge recall,HotpotQA\(Yanget al\.[2018](https://arxiv.org/html/2608.14632#bib.bib37)\)requires the integration of multiple factual clues, andCommonsenseQA\(Talmoret al\.[2019](https://arxiv.org/html/2608.14632#bib.bib36)\)examines reasoning grounded in everyday knowledge\. We collect responses and their complete denoising trajectories from two representative D\-LLMs, LLaDA\-8B\-Instruct\(Nieet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib50)\)and Dream\-7B\-Instruct\(Yeet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib51)\), using step generation lengths of 64 and 128\.

Baselines and Evaluation\.We compareDeMTSwith three categories of hallucination detectors: ❶Output\-based methods, including Perplexity\(Renet al\.[2022](https://arxiv.org/html/2608.14632#bib.bib35)\), Length\-Normalized Entropy \(LN\-Entropy\)\(Malinin and Gales[2020](https://arxiv.org/html/2608.14632#bib.bib34)\), Semantic Entropy\(Kuhnet al\.[2023](https://arxiv.org/html/2608.14632#bib.bib33)\), and Lexical Similarity\(Linet al\.[2024](https://arxiv.org/html/2608.14632#bib.bib32)\); ❷Latent\-based methods, including EigenScore\(Chernet al\.[2023](https://arxiv.org/html/2608.14632#bib.bib31)\), Contrast\-Consistent Search \(CCS\)\(Burnset al\.[2022](https://arxiv.org/html/2608.14632#bib.bib30)\), and Truthfulness Separator Vector \(TSV\)\(Parket al\.[2025](https://arxiv.org/html/2608.14632#bib.bib29)\); ❸Trajectory\-based methods, including TraceDet\(Changet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib59)\)and DynHD\(Qianet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib63)\)\. We use AUROC as the evaluation metric\. Hallucination labels are determined by Qwen3\-8B\(Yanget al\.[2025](https://arxiv.org/html/2608.14632#bib.bib28); Guet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib27)\)based on the question, answer, and generated response\. Detailed descriptions of the baselines and implementation settings are provided in Appendix[D\.1](https://arxiv.org/html/2608.14632#A4.SS1)and Appendix[D\.2](https://arxiv.org/html/2608.14632#A4.SS2)\.

Table 2:Ablation study on LLaDA\-8B\-Instruct\.
### Experimental Results and Analysis

Performance Comparison\.Table[1](https://arxiv.org/html/2608.14632#Sx4.T1)presents the main comparison across two D\-LLM backbones, three QA benchmarks, and two generation\-length settings\. We further analyze the experimental results from the perspectives of these three model categories: ❶ To examine whether final\-output statistics are sufficient for D\-LLMs hallucination detection, we compareDeMTSwith output\-level methods\.DeMTSconsistently outperforms output\-based baselines across different settings\. These results show that output\-level evidence cannot adequately characterize hallucinations formed throughout iterative denoising, highlighting the necessity of exploiting denoising\-process information\. ❷ To evaluate the benefit of explicitly modeling the generation process over using static internal representations, we compareDeMTSwith latent\-based detectors\. Although latent\-based methods use richer model\-side signals than output\-level statistics, they still underperformDeMTSin most cases\. This suggests that static representation signals are insufficient to capture the evolving temporal dependencies involved in hallucination formation\. ❸ To verify whether preserving the multivariate structure of denoising trajectories provides additional detection evidence, we compareDeMTSwith TraceDet and DynHD\.DeMTSachieves the best overall performance across different settings\. The consistent improvement supports our central motivation: jointly modeling variable\-wise evolution and inter\-variable interactions better captures abnormal temporal relations for hallucination detection\.

![Refer to caption](https://arxiv.org/html/2608.14632v1/paper/image/visual_K.png)Figure 3:Latent\-variable analysis\.Left: performance sensitivity to the number of latent variables \(K\)\. Right: visualization of learned variable assignments across different samples\.Ablation Study\.As shown in Table[2](https://arxiv.org/html/2608.14632#Sx5.T2), removing any component consistently degrades performance on backbones, indicating the effectiveness of multivariate dynamics modeling\. ❶ Replacing schedule\-normalized signals with raw uncertainty trajectories leads to clear performance drops, suggesting that the schedule\-induced global uncertainty trend can mask sample\-specific abnormal dynamics\. ❷ Using token positions as variables causes the most pronounced degradation, confirming that semantically unstable token positions do not provide reliable variable identities for time\-series modeling\. ❸ Removing eitherℒa​m​p\\mathcal\{L\}\_\{amp\}orℒd​i​r\\mathcal\{L\}\_\{dir\}weakens detection performance, demonstrating the importance of preserving fine\-grained denoising dynamics during T2V assignment\. We provide a sensitivity analysis of the loss hyperparameters in the Appendix[D\.3](https://arxiv.org/html/2608.14632#A4.SS3)\. ❹ Removing either inter\-variable interaction or temporal evolution modeling also substantially weakens detection performance, showing that trajectory\-level hallucination evidence depends on both variable\-wise evolution and cross\-variable dependency modeling\. Comprehensive ablation studies are provided in Appendix[D\.4](https://arxiv.org/html/2608.14632#A4.SS4)\.

Table 3:Zero\-shot cross\-task generalization in AUROC \(%\)\.Robustness to the Number of Latent Variables\.As shown in Figure[3](https://arxiv.org/html/2608.14632#Sx5.F3)a, we vary the number of latent variablesKKto examine the effect of variable resolution\. A smallKKover\-compresses token\-level denoising signals, while a largeKKleads to fragmented and redundant variables\. The results show that a moderateKKbetter balances information preservation and variable stability\.

Visualization of Learned Variables\.To examine whether the Token\-to\-variable assignment \(T2V\) module constructs variables with stable identities, we visualize the assignment patterns across multiple samples in Figure[3](https://arxiv.org/html/2608.14632#Sx5.F3)b\. Although token contents and semantic roles vary substantially across samples, each latent variable consistently aggregates token\-level denoising signals with similar semantic or dynamic characteristics\. This cross\-sample consistency allows the learned latent variables to capture transferable trajectory patterns, thereby improving their generalization to unseen questions and generation contexts\.

Cross\-Task Generalization\.To assess whetherDeMTScaptures transferable hallucination dynamics rather than dataset\-specific semantic patterns, we train the detector on one benchmark and test it on the other benchmarks without further adaptation\. As shown in Table[3](https://arxiv.org/html/2608.14632#Sx5.T3),DeMTSachieves the highest average AUROC indicating that the stable latent variables constructed through T2V assignment, along with the modeling of temporal evolution and inter\-variable interactions, capture transferable hallucination patterns\.

![Refer to caption](https://arxiv.org/html/2608.14632v1/paper/image/time2.png)Figure 4:Average inference time and AUROC of different methods for comprehensive performance evaluation\.Efficiency Analysis\.We compare the inference time and AUROC of different methods to evaluate their practical efficiency as shown in Figure[4](https://arxiv.org/html/2608.14632#Sx5.F4)\.DeMTSachieves the highest AUROC with the shortest runtime\. These results show thatDeMTSprovides the best AUROC\-efficiency trade\-off by effectively exploiting multivariate denoising dynamics\.

## Conclusion

In this work, we investigated hallucination detection in D\-LLMs from a multivariate time\-series perspective\. Unlike existing approaches that compress denoising trajectories along the temporal or variable dimension, we preserve their two\-dimensional token\-step structure to capture how hallucination\-related uncertainty evolves and propagates throughout generation\. Extensive experiments across multiple D\-LLMs, benchmarks, and generation settings demonstrate thatDeMTSexhibits strong effectiveness, cross\-task generalization, robustness, and computational efficiency\.

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## Appendix AExtended Related Work

### A\.1Hallucination Detection in D\-LLMs

D\-LLMs generate text through iterative denoising, which exposes intermediate predictions, uncertainty estimates, and hidden states before the final response is obtained\(Yanget al\.[2023](https://arxiv.org/html/2608.14632#bib.bib26); Nieet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib50); Yeet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib51)\)\. This process provides trajectory\-level evidence for hallucination detection, since factual errors may emerge during refinement rather than only in the final output\(Guo and Tan[2026](https://arxiv.org/html/2608.14632#bib.bib61)\)\. Existing studies mainly explore this evidence from two perspectives\.

The first line focuses onstep\-oriented evidence selection\. TraceDet\(Changet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib59)\)formulates the denoising process as a decoding trace and selects informative sub\-traces for hallucination prediction, showing that different denoising stages contribute unequally to detection\. HIVE\(Zhaoet al\.[2026a](https://arxiv.org/html/2608.14632#bib.bib60)\)further exploits hidden evidence from intermediate denoising states and uses selected evidence for verification\. These methods demonstrate the value of intermediate states, but they mainly rely on selecting or compressing denoising stages\.

The second line focuses ondynamics\-oriented trajectory modeling\. DynHD\(Qianet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib63)\)models the evolution of uncertainty evidence and detects hallucinations by measuring deviations from expected denoising dynamics\. TDGNet\(Hemmatet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib58)\)constructs temporal dynamic graphs to capture evolving token\-level relations during denoising\. These methods reveal that hallucination cues can lie in dynamic uncertainty patterns and token interactions\. However, they still do not fully preserve the complete token\-by\-step trajectory structure\. In contrast,DeMTSmodels D\-LLM denoising trajectories as multivariate time series, aiming to jointly capture token\-wise temporal evolution and evolving inter\-token dependencies for hallucination detection\.

### A\.2Multivariate Time\-Series Modeling

Multivariate time\-series modeling aims to capture temporal patterns across multiple variables and their dependencies\(Linet al\.[2025b](https://arxiv.org/html/2608.14632#bib.bib49); Liet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib3); Nguyenet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib48)\)\. Existing methods are often discussed according to how they model variable channels\. Channel\-independent methods, such as PatchTST\(Nieet al\.[2022](https://arxiv.org/html/2608.14632#bib.bib44)\), model each variable separately to preserve variable\-specific temporal patterns and reduce interference from noisy correlations\. Channel\-dependent methods, such as Crossformer\(Zhang and Yan[2023](https://arxiv.org/html/2608.14632#bib.bib66)\)and iTransformer\(Liuet al\.[2024](https://arxiv.org/html/2608.14632#bib.bib46)\), jointly model variables to capture cross\-variable dependencies\. More recent channel\-partial methods, such as DUET\(Qiuet al\.[2025b](https://arxiv.org/html/2608.14632#bib.bib41)\), CrossGNN\(Huanget al\.[2023](https://arxiv.org/html/2608.14632#bib.bib39)\), and TimeFilter\(Huet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib65)\), allow each variable to interact with only relevant variables, balancing variable\-specific modeling and dependency modeling\.

These methods provide useful inspiration for modeling D\-LLMs denoising trajectories\(Chenet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib4); Tanet al\.[2024](https://arxiv.org/html/2608.14632#bib.bib5); Panet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib6); Heet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib7)\), where denoising steps form the temporal axis and token signals form the raw variable axis\. However, conventional multivariate time\-series methods usually assume that each variable has a stable identity and meaning across samples and time\. This assumption does not hold for D\-LLMs, because token positions may correspond to different semantic roles across samples and may change during denoising\. Therefore, directly treating token positions as variables can lead to unstable temporal patterns and unreliable dependencies\. To address this issue,DeMTSfirst converts changing token signals into stable latent variables, and then models both their temporal evolution and stage\-adaptive inter\-variable interactions\.

## Appendix BAlgorithm

For completeness, we summarize the overall workflow and optimization procedure of the proposed framework in Algorithm[1](https://arxiv.org/html/2608.14632#alg1), providing a concise overview of its implementation\.

Algorithm 1Overall Procedure ofDeMTS1:Input:Token uncertainty trajectory

𝐇∈ℝT×N\\mathbf\{H\}\\in\\mathbb\{R\}^\{T\\times N\}, contextual token embeddings

\{𝐞i,t\}\\\{\\mathbf\{e\}\_\{i,t\}\\\}, label

yy
2:Output:Hallucination probability

y^\\hat\{y\}and objective

ℒ\\mathcal\{L\}
3://Module 1: Trajectory\-Preserving Token\-to\-Variable Assignment \(T2V\)

4:Normalize token uncertainty

𝐇^\\widehat\{\\mathbf\{H\}\}via Eq\. \([4](https://arxiv.org/html/2608.14632#Sx4.E4)\)\.

5:foreach denoising step

ttand token position

iido

6:Compute uncertainty\-semantic representation

𝐮i,t\\mathbf\{u\}\_\{i,t\}via Eq\. \([5](https://arxiv.org/html/2608.14632#Sx4.E5)\)\.

7:Compute content assignment logits

𝐬i,tcon\\mathbf\{s\}^\{\\mathrm\{con\}\}\_\{i,t\}via Eq\. \([6](https://arxiv.org/html/2608.14632#Sx4.E6)\)\.

8:Compute positional representation

ϕi\\boldsymbol\{\\phi\}\_\{i\}and structural logits

𝐬istr\\mathbf\{s\}^\{\\mathrm\{str\}\}\_\{i\}via Eqs\. \([7](https://arxiv.org/html/2608.14632#Sx4.E7)\)–\([8](https://arxiv.org/html/2608.14632#Sx4.E8)\)\.

9:Compute gate

𝐠i,t\\mathbf\{g\}\_\{i,t\}and assignment weights

𝐚i,t\\mathbf\{a\}\_\{i,t\}via Eqs\. \([9](https://arxiv.org/html/2608.14632#Sx4.E9)\)–\([10](https://arxiv.org/html/2608.14632#Sx4.E10)\)\.

10:endfor

11:foreach denoising step

ttand latent variable

kkdo

12:Construct latent\-variable state

𝐳k,t=𝒜k,t​\(𝐮t\)\\mathbf\{z\}\_\{k,t\}=\\mathcal\{A\}\_\{k,t\}\(\\mathbf\{u\}\_\{t\}\)via Eq\. \([11](https://arxiv.org/html/2608.14632#Sx4.E11)\)\.

13:endfor

14:Form latent\-variable trajectory

𝐙=\{𝐳k,t\}k=1,t=1K,T\\mathbf\{Z\}=\\\{\\mathbf\{z\}\_\{k,t\}\\\}\_\{k=1,t=1\}^\{K,T\}\.

15:Compute preservation loss

ℒp​r​e​s\\mathcal\{L\}\_\{pres\}via Eqs\. \([12](https://arxiv.org/html/2608.14632#Sx4.E12)\)–\([14](https://arxiv.org/html/2608.14632#Sx4.E14)\)\.

16://Module 2: Dynamic Multivariate Temporal Modeling \(DMTM\)

17:foreach denoising step

ttdo

18:Compute inter\-variable interaction

𝐂t\\mathbf\{C\}\_\{t\}via Eq\. \([15](https://arxiv.org/html/2608.14632#Sx4.E15)\)\.

19:Compute reliability\-gated variable states

𝐔t\\mathbf\{U\}\_\{t\}via Eq\. \([16](https://arxiv.org/html/2608.14632#Sx4.E16)\)\.

20:endfor

21:foreach latent variable

kkdo

22:Encode variable\-wise temporal dynamics

𝐑k\\mathbf\{R\}\_\{k\}via Eq\. \([17](https://arxiv.org/html/2608.14632#Sx4.E17)\)\.

23:endfor

24:Aggregate

\{𝐑k\}k=1K\\\{\\mathbf\{R\}\_\{k\}\\\}\_\{k=1\}^\{K\}into

𝐑\\mathbf\{R\}and predict

y^=MLP⁡\(𝐑\)\\hat\{y\}=\\operatorname\{MLP\}\(\\mathbf\{R\}\)\.

25:Compute

ℒc​l​s\\mathcal\{L\}\_\{cls\}and the overall objective

ℒ\\mathcal\{L\}via Eqs\. \([18](https://arxiv.org/html/2608.14632#Sx4.E18)\)–\([19](https://arxiv.org/html/2608.14632#Sx4.E19)\)\.

26:return

y^\\hat\{y\}and

ℒ\\mathcal\{L\}\.

## Appendix CComputational Complexity

LetTT,NN, andKKdenote the numbers of denoising steps, token positions, and latent variables, respectively, and letd=dud=d\_\{u\}denote the dimension of the token and latent\-variable representations\. Following the standard convention for attention complexity, we report the dominant assignment, aggregation, and pairwise interaction costs introduced byDeMTS, excluding the original D\-LLM denoising process, token\-wise transformations, and the lightweight classifier\.

In T2V, SAUN normalizes the uncertainty distribution over all token positions at each denoising step, resulting in𝒪​\(T​N\)\\mathcal\{O\}\(TN\)complexity\. The subsequent token\-to\-variable assignment computes the assignment weightsai,t,ka\_\{i,t,k\}, while assignment\-weighted aggregation constructs the latent\-variable states𝐳k,t\\mathbf\{z\}\_\{k,t\}from the token representations\. These operations require𝒪​\(T​N​K​d\)\\mathcal\{O\}\(TNKd\)computation\. During training, the trajectory\-preservation constraint aggregates thedd\-dimensional token\-level variationsΔ​𝐇^i,t\\Delta\\widehat\{\\mathbf\{H\}\}\_\{i,t\}intoΔ​𝐙k,t\\Delta\\mathbf\{Z\}\_\{k,t\}and reconstructsΔ​𝐇~i,t\\Delta\\widetilde\{\\mathbf\{H\}\}\_\{i,t\}, introducing an additional𝒪​\(T​N​K​d\)\\mathcal\{O\}\(TNKd\)cost\. This constraint incurs no additional cost at inference\.

In DMTM, SAIVI models interactions amongKKlatent variables independently at each denoising step, resulting in a pairwise attention cost of𝒪​\(T​K2​d\)\\mathcal\{O\}\(TK^\{2\}d\)\. VTDM performs temporal self\-attention overTTdenoising steps for each latent variable, requiring𝒪​\(K​T2​d\)\\mathcal\{O\}\(KT^\{2\}d\)computation\. Therefore, the overall inference\-time complexity is

𝒪​\(T​N​K​d\+T​K2​d\+K​T2​d\)\.\\mathcal\{O\}\\left\(TNKd\+TK^\{2\}d\+KT^\{2\}d\\right\)\.\(20\)
During training, the trajectory\-preservation constraint introduces an additional𝒪​\(T​N​K​d\)\\mathcal\{O\}\(TNKd\)term, which has the same asymptotic order as the token\-to\-variable assignment and aggregation\.

The main efficiency advantage ofDeMTSarises from conducting relational and temporal modeling in the compact latent\-variable space\. Applying analogous axial attention directly to the original token\-indexed trajectory would model interactions amongNNtoken positions at each denoising step and temporal dependencies overTTsteps for each token position, requiring𝒪​\(T​N2​d\+N​T2​d\)\\mathcal\{O\}\(TN^\{2\}d\+NT^\{2\}d\)attention computation\. In contrast,DeMTSfirst transforms the token\-indexed trajectory intoKKlatent\-variable trajectories with an assignment and aggregation cost of𝒪​\(T​N​K​d\)\\mathcal\{O\}\(TNKd\), and reduces the subsequent attention cost to𝒪​\(T​K2​d\+K​T2​d\)\\mathcal\{O\}\(TK^\{2\}d\+KT^\{2\}d\)\. SinceK<NK<Nin practice, this design reduces the quadratic relational cost fromN2N^\{2\}toK2K^\{2\}and decreases the number of temporally modeled trajectories fromNNtoKK\. Moreover,DeMTSdirectly reuses the uncertainty and contextual signals collected during denoising, requiring neither repeated generation nor additional forward passes through the D\-LLM backbone\.

Table 4:Ablation study in AUROC \(%\) on two D\-LLMs backbones\.## Appendix DExperiments

### D\.1Baselines

We compareDeMTSwith three categories of hallucination detectors, covering output\-level signals, internal representations, and denoising\-trajectory evidence\.

❶Output\-based methods\.These methods assess hallucination risk using information derived from the generated responses, including model likelihood, token\-level uncertainty, and consistency among sampled outputs\. They serve as widely used baselines for evaluating whether final\-output signals are sufficient for hallucination detection\.

⊳\\rhdPerplexity\(Renet al\.[2022](https://arxiv.org/html/2608.14632#bib.bib35)\)uses the likelihood of the generated response as a confidence signal, where responses with lower model likelihood are considered less reliable\.

⊳\\rhdLength\-Normalized Entropy \(LN\-Entropy\)\(Malinin and Gales[2020](https://arxiv.org/html/2608.14632#bib.bib34)\)estimates predictive uncertainty from token\-level entropy and normalizes it by sequence length, reducing the bias caused by different output lengths\.

⊳\\rhdSemantic Entropy\(Kuhnet al\.[2023](https://arxiv.org/html/2608.14632#bib.bib33)\)samples multiple responses, groups them according to semantic equivalence, and computes the entropy over semantic clusters to measure meaning\-level uncertainty\.

⊳\\rhdLexical Similarity\(Linet al\.[2024](https://arxiv.org/html/2608.14632#bib.bib32)\)measures the surface\-level agreement among multiple sampled responses, where lower similarity indicates higher generation instability\.

❷Latent\-based methods\.These methods exploit internal representations of LLMs to identify factuality\-related signals beyond surface outputs\. They are included to examine whether static hidden\-state features are sufficient for detecting hallucinations in D\-LLMs without explicitly modeling the denoising trajectory\.

⊳\\rhdEigenScore\(Chernet al\.[2023](https://arxiv.org/html/2608.14632#bib.bib31)\)detects hallucinations by measuring the semantic consistency of internal representations, using the spectral structure of hidden\-state covariance matrices\.

⊳\\rhdContrast\-Consistent Search \(CCS\)\(Burnset al\.[2022](https://arxiv.org/html/2608.14632#bib.bib30)\)discovers truth\-related directions in the activation space without direct supervision by enforcing consistency between contrastive statements\.

⊳\\rhdTruthfulness Separator Vector \(TSV\)\(Parket al\.[2025](https://arxiv.org/html/2608.14632#bib.bib29)\)learns a lightweight steering vector in the latent space to improve the separation between truthful and hallucinated responses\.

❸Trajectory\-based methods\.These methods are specifically designed for D\-LLMs and leverage intermediate information exposed during iterative denoising\. They provide the most direct comparison toDeMTS, since they also use process\-level evidence rather than relying only on final outputs or static representations\.

⊳\\rhdTraceDet\(Changet al\.[2025](https://arxiv.org/html/2608.14632#bib.bib59)\)is designed for D\-LLMs and formulates the denoising process as a decoding trace\. It selects hallucination\-relevant sub\-traces from intermediate denoising steps for final prediction\.

⊳\\rhdDynHD\(Qianet al\.[2026](https://arxiv.org/html/2608.14632#bib.bib63)\)constructs semantic\-aware uncertainty evidence and learns reference denoising dynamics for factual responses\. Hallucinations are detected by measuring the deviation between observed evidence trajectories and the learned reference trajectories\.

HyperparameterValueNumber of latent variablesKK32, 48Hidden dimensiondd64Fourier frequenciesFF16Cross\-variable attention heads2Temporal attention heads8Dropout rate0\.05OptimizerAdamWLearning rate2×10−42\\times 10^\{\-4\}Weight decay0\.1Batch size8Maximum epochs100Early stopping patience20Trajectory preservation weight0\.1Random seed42Table 5:Hyperparameter settings of DeMTS\.Table 6:Examples of factual and hallucinated responses from HotpotQA, TriviaQA, and CommonsenseQA\. Correct responses are highlighted in blue, while hallucinated responses are highlighted in red\.
### D\.2Implementation Details

We conduct experiments on TriviaQA, HotpotQA, and CommonsenseQA \(show in Table[6](https://arxiv.org/html/2608.14632#A4.T6)\) using LLaDA\-8B\-Instruct and Dream\-7B\-Instruct as D\-LLM backbones\. For each dataset, we sample 1,600 training examples and split an additional 400 examples into 200 validation and 200 test instances\. The random seed is fixed to 42\. The number of denoising steps is set to 64 or 128\. At each denoising step, we record token\-wise predictive entropy and token IDs\.

Our detector uses a hidden dimension of 64 and constructs \(K=32,48\) latent variables from the denoising trajectory\. It is trained with Cross Entropy and the trajectory\-preservation loss\. We use AdamW with a learning rate of2×10−42\\times 10^\{\-4\}, weight decay of 0\.1, and batch size of 8\. Training lasts for at most 100 epochs and early stopping with a patience of 20\. The checkpoint with the highest validation AUROC is selected for testing\. AUROC is used as the primary evaluation metric\. All experiments are implemented in PyTorch 2\.6\.0 and conducted on 4 NVIDIA L40 GPUs\. All experimental hyperparameter settings are shown in Table[5](https://arxiv.org/html/2608.14632#A4.T5)\.

### D\.3Sensitivity Analysis of Trajectory Preservation Losses

We further analyze the sensitivity of the two trajectory\-preservation losses, i\.e\., the amplitude consistency lossℒa​m​p\\mathcal\{L\}\_\{amp\}and the direction consistency lossℒd​i​r\\mathcal\{L\}\_\{dir\}\. As shown in Fig[5](https://arxiv.org/html/2608.14632#A4.F5), the detector maintains stable performance across a broad range of loss weights, indicating that the proposed trajectory\-preservation objective is not overly sensitive to specific hyperparameter choices\. When either loss is removed or assigned a very small weight, the performance decreases, suggesting that preserving only the classification objective is insufficient to retain fine\-grained denoising dynamics during token\-to\-variable assignment\. Meanwhile, excessively large weights also lead to performance degradation, since overly strong trajectory constraints may suppress discriminative feature learning for hallucination detection\. Overall, the heatmaps show thatℒa​m​p\\mathcal\{L\}\_\{amp\}andℒd​i​r\\mathcal\{L\}\_\{dir\}play complementary roles:ℒa​m​p\\mathcal\{L\}\_\{amp\}helps preserve the magnitude of local uncertainty variations, whileℒd​i​r\\mathcal\{L\}\_\{dir\}maintains the temporal direction of denoising evolution\. Their joint use provides a stable and effective regularization scheme for constructing reliable variable\-level trajectories\.

![Refer to caption](https://arxiv.org/html/2608.14632v1/paper/image/loss_sensitivity.png)Figure 5:Sensitivity analysis of the amplitude consistency lossℒa​m​p\\mathcal\{L\}\_\{amp\}and direction consistency lossℒd​i​r\\mathcal\{L\}\_\{dir\}\.
### D\.4Complete Ablation Results

To provide a more comprehensive analysis of each component inDeMTS, we report the complete ablation results on both D\-LLMs backbones in Table[C](https://arxiv.org/html/2608.14632#A3)\. The results exhibit consistent trends across different models, datasets, and generation\-length settings, further verifying the robustness and generality of our design\.

The complete ablation results further confirm the effectiveness of each component inDeMTS\. Replacing schedule\-normalized trajectories with raw uncertainty signals consistently reduces performance, showing the importance of suppressing schedule\-induced global trends\. Directly using token positions as variables also leads to clear degradation, verifying the necessity of constructing semantically stable latent variables before time\-series modeling\. Removing eitherℒa​m​p\\mathcal\{L\}\_\{amp\}orℒd​i​r\\mathcal\{L\}\_\{dir\}also weakens performance, indicating that both amplitude consistency and direction consistency contribute to preserving fine\-grained denoising dynamics during token\-to\-variable assignment\. Moreover, removing inter\-variable interaction or temporal evolution modeling consistently degrades performance, demonstrating thatDeMTSbenefits from jointly preserving local denoising dynamics and modeling multivariate temporal dependencies\.

Overall, the complete ablation results are consistent with the observations in the main paper\. They demonstrate that the performance gain ofDeMTSdoes not come from a single isolated component, but from the integration of schedule normalization, stable latent\-variable construction, trajectory\-preserving regularization, and dynamic multivariate temporal modeling\.

## Appendix EPrompt Templates

### E\.1D\-LLM Response Generation

We design task\-specific generation prompts according to the answer format and evidence setting of each benchmark\. TriviaQA requires concise factual answers, HotpotQA requires responses grounded in the provided context, and CommonsenseQA requires selecting a single option without additional explanations\. Across all datasets, the final answer is enclosed within<answer\></answer\>tags to enable consistent answer extraction and subsequent hallucination annotation\. The complete prompt templates are provided below\.

`TriviaQA Generation Prompt HotpotQA Generation Prompt CommonsenseQA Generation Prompt`

`E\.2 Automatic Hallucination Annotation For TriviaQA and HotpotQA, we use Qwen3\-8B to automatically determine whether the generated response is hallucinated\. For TriviaQA and HotpotQA, the judgment yes is mapped to label 11, indicating a hallucinated response, whereas no is mapped to label 0, indicating a non\-hallucinated response\. CommonsenseQA does not use the Qwen3\-8B judge; its labels are obtained through rule\-based matching of the predicted option or answer text\. Hallucination Evaluation Instructions`