\section{Experimental Details}
\label{app:HD_SM}

\paragraph{QA Task}We evaluate our methods on five question answering datasets: CoQA, SQuAD, HotpotQA, TriviaQA, and PsiLoQA.  
For each dataset, we construct a training set of 9{,}000 examples and a validation set of 1{,}000 examples, which are used for probe training and hyperparameter selection.  
The test sets are kept fixed and are used exclusively for evaluation. Specifically, the test set sizes are 7{,}983 for CoQA, 10{,}000 for SQuAD, 7{,}405 for HotpotQA, 10{,}000 for TriviaQA, and 8{,}103 for PsiLoQA. 
\paragraph{Summarization Task}
We evaluate on two summarization datasets: CNN/DailyMail and HaluEval. For CNN/DailyMail, we construct a training set of 9{,}000 examples and a validation set of 1{,}000 examples, with a test set of 10{,}000 examples. For HaluEval, we use 7{,}200 training examples, 800 validation examples, and 2{,}000 test examples.
\paragraph{Prompt Template}
To generate answers for hallucination detection, we use two prompt settings depending on whether a supporting passage is available. \cref{tab:prompt_templates} summarizes the templates used in the question-only setting (HotpotQA, TriviaQA) and the context-aware setting (CoQA, SQuAD, PsiLoQA, HaluEval, CNN/Daily ), where \texttt{[Q]} and \texttt{[C]} denote the question and context, respectively.
\paragraph{Implementation of First-Sentence Truncation}
\label{app:implementation_FST}
First-sentence truncation is implemented via a rule-based scanner. Specifically, the scanner processes the generated text from left to right and identifies the first period that does not fall under common exception cases as the end of the first sentence. These exceptions include ellipses (\texttt{…}), decimal numbers (e.g., \texttt{3.14}), multi-dot abbreviations (e.g., \texttt{U.S.}, \texttt{i.e.}), word-level abbreviations (e.g., \texttt{Dr.}, \texttt{etc.}, \texttt{No.\ 3}), and name initials (e.g., \texttt{G. Smith}). The scanner is simply implemented with regular expression operations.
\begin{table}[b]
\centering
\small
\caption{Prompt templates used for answer generation.
Here, [Q] denotes the question text and [C] denotes the provided context passage.}
\label{tab:prompt_templates}
\begin{tabular}{lcl}
\toprule
\textbf{}             & Dataset                                                               & Prompt Template                                                                                                                                                 \\ \midrule
Question-Only Setting & \begin{tabular}[c]{@{}c@{}}HotpotQA \\ TriviaQA\end{tabular}          & \begin{tabular}[c]{@{}l@{}}Answer the question as briefly as possible, using plain text only:\\ Question: {[}Q{]}\\ Answer:\end{tabular}                        \\ \midrule
Context-Aware Setting & \begin{tabular}[c]{@{}c@{}}CoQA \\ SQuAD\\ PsiLoQA\end{tabular}       & \begin{tabular}[c]{@{}l@{}}Answer the question as briefly as possible, based only on the context:\\ Context: {[}C{]}\\ Question: {[}Q{]}\\ Answer:\end{tabular} \\ \midrule
Summarization Tasks   & \begin{tabular}[c]{@{}c@{}}HaluEval\\ CNN/Daily Mail\end{tabular} & \begin{tabular}[c]{@{}l@{}}Summarize the following document in one or two concise sentences.\\ Document: {[}C{]}\\Summary:\end{tabular}                           \\ \bottomrule
\end{tabular}
\end{table}

\begin{table}[]
\centering
\caption{Results on the \textbf{LlaMA-3.1-8B base} model with $w=7$. The representations are extracted without FST. FEPoID maintains strong performance on a non-instruction-tuned 
model, demonstrating that our method \emph{generalizes} across both base and 
instruction-tuned models. }
\label{tab:model_8B_base_without_FST}
\begin{tabular}{ccccccc}
\toprule
          & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                \\ \midrule
RankME    & 0.6972                             & 0.7132                             & 0.6458                             & 0.6472                             & 0.8032                             & 0.7013                             \\
Curvature & \cellcolor{rankthree}0.7187        & \cellcolor{rankthree}0.7432        & \textbf{\cellcolor{rankone}0.7370} & \cellcolor{rankthree}0.7846        & \cellcolor{ranktwo}0.8639          & \cellcolor{rankthree}0.7695        \\
Val Loss  & \textbf{\cellcolor{rankone}0.7552} & \cellcolor{ranktwo}0.7811          & \cellcolor{ranktwo}0.7326          & \textbf{\cellcolor{rankone}0.8141} & \cellcolor{rankthree}0.8434        & \cellcolor{ranktwo}0.7853          \\
RGN       & 0.6859                             & 0.7313                             & \cellcolor{rankthree}0.6955        & 0.7257                             & 0.7816                             & 0.7240                             \\
SNR       & 0.5118                             & 0.6387                             & 0.6115                             & 0.6304                             & 0.7677                             & 0.6320                             \\
ID        & \cellcolor{ranktwo}0.7468          & \textbf{\cellcolor{rankone}0.7892} & 0.6672                             & 0.6179                             & \textbf{\cellcolor{rankone}0.8655} & 0.7373                             \\
FEPoID    & \cellcolor{ranktwo}0.7468          & \textbf{\cellcolor{rankone}0.7892} & \cellcolor{ranktwo}0.7326          & \cellcolor{ranktwo}0.8136          & \textbf{\cellcolor{rankone}0.8655} & \textbf{\cellcolor{rankone}0.7895} \\ \bottomrule
\end{tabular}
\end{table}

\begin{table}[]
\centering
\caption{Results on different model scales, where representations are extracted without FST. Forward horizon is set to $w=7$ for \textbf{LlaMA-3.2-3B} and $w = 3$ for \textbf{LlaMA-3.2-1B}. Top-3 results are highlighted, with darker color indicating better performance. FEPoID achieves consistently strong performance across both model scales, demonstrating its \emph{generalizability} to models of varying scales. 
}
\label{tab:model_scales_without_FST}
\footnotesize
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccccccc}
\toprule
          & \multicolumn{6}{c}{LlaMA-3.2-3B}                                                                                                                                                                                            & \multicolumn{6}{c}{LlaMA-3.2-1B}                                                                                                                                                                                            \\
          & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                \\ \midrule
RankME    & 0.5981                             & 0.6476                             & 0.6914                             & \cellcolor{rankthree}0.6639        & 0.7469                             & 0.6696                             & 0.5621                             & 0.5902                             & 0.6583                             & 0.5963                             & 0.6441                             & 0.6102                             \\
Curvature & \cellcolor{ranktwo}0.7123          & \cellcolor{rankthree}0.6616        & \cellcolor{rankthree}0.7268        & \cellcolor{ranktwo}0.6734          & \cellcolor{rankthree}0.8280        & 0.7204                             & \cellcolor{rankthree}0.6699        & \cellcolor{ranktwo}0.6442          & \cellcolor{ranktwo}0.7587          & \cellcolor{rankone}0.7028          & \cellcolor{rankthree}0.7355        & \cellcolor{rankthree}0.7022        \\
Val Loss  & \textbf{\cellcolor{rankone}0.7391} & \cellcolor{ranktwo}0.7153          & \textbf{\cellcolor{rankone}0.7439} & \textbf{\cellcolor{rankone}0.7198} & \textbf{\cellcolor{rankone}0.8498} & \cellcolor{ranktwo}0.7536          & \textbf{\cellcolor{rankone}0.6860} & \cellcolor{ranktwo}0.6442          & \textbf{\cellcolor{rankone}0.7641} & \cellcolor{ranktwo}0.6985          & \textbf{\cellcolor{rankone}0.7597} & \textbf{\cellcolor{rankone}0.7105} \\
RGN       & \cellcolor{rankthree}0.6369        & 0.6567                             & 0.6914                             & \textbf{\cellcolor{rankone}0.7198} & 0.7990                             & 0.7008                             & 0.5777                             & 0.5902                             & \cellcolor{rankthree}0.7370        & \cellcolor{rankthree}0.6050        & 0.6597                             & 0.6339                             \\
SNR       & 0.5116                             & 0.5544                             & \textbf{\cellcolor{rankone}0.7439} & 0.5748                             & 0.6878                             & 0.6145                             & 0.6226                             & \cellcolor{rankthree}0.6225        & 0.7265                             & 0.5963                             & 0.6126                             & 0.6361                             \\
ID        & \textbf{\cellcolor{rankone}0.7391} & \textbf{\cellcolor{rankone}0.7275} & 0.6982                             & 0.6554                             & 0.7990                             & \cellcolor{rankthree}0.7238        & \cellcolor{ranktwo}0.6709          & 0.5872                             & 0.7151                             & \cellcolor{rankthree}0.6050        & 0.6597                             & 0.6476                             \\
FEPoID    & \textbf{\cellcolor{rankone}0.7391} & \textbf{\cellcolor{rankone}0.7275} & \cellcolor{ranktwo}0.7364          & \textbf{\cellcolor{rankone}0.7198} & \cellcolor{ranktwo}0.8483          & \textbf{\cellcolor{rankone}0.7542} & \cellcolor{ranktwo}0.6709          & \textbf{\cellcolor{rankone}0.6514} & \textbf{\cellcolor{rankone}0.7641} & \textbf{\cellcolor{rankone}0.7028} & \cellcolor{ranktwo}0.7413          & \cellcolor{ranktwo}0.7061          \\ \bottomrule
\end{tabular}
}%
\end{table}
% You can have as much text here as you want. The main body must be at most $8$
% pages long. For the final version, one more page can be added. If you want, you
% can use an appendix like this one.

% The $\mathtt{\backslash onecolumn}$ command above can be kept in place if you
% prefer a one-column appendix, or can be removed if you prefer a two-column
% appendix.  Apart from this possible change, the style (font size, spacing,
% margins, page numbering, etc.) should be kept the same as the main body.

\section{Additional Hallucination Detection Results}
\paragraph{Generalization Across Scales and Tuning Strategies}
To evaluate the generalizability of FEPoID beyond instruction-tuned models and standard model scales, we conduct experiments on LLaMA-3.1-8B (base), LLaMA-3.2-3B, and LLaMA-3.2-1B. Results are reported in Tables~\ref{tab:model_8B_base} and~\ref{tab:model_scales}, in which FEPoID consistently selects high-
performing layers and outperforms all baselines.

\paragraph{First-Sentence Truncation Analysis}
To better understand the impact of token-position choice, we report detailed hallucination detection results with first-sentence truncation across datasets in \cref{tab:first_sentence_truncation}. 
Overall, under the FST setting, \ac{our_method} continues to achieve the best average performance across different models.
In addition, \cref{fig:layer_selection_FST} presents layer-wise AUROC curves together with intrinsic dimension estimates when representations are extracted at the last token of the first generated sentence.
Compared to \cref{fig:layer_selection_no_truncation}, the AUROC curves of the two models become noticeably more aligned after applying FST. This observation suggests that different instruction-tuned models tend to capture more consistent and useful information at earlier stages of generation, leading to improved consistency in detection performance.


\paragraph{Sensitivity to Forward Horizon $w$}
Finally, we examine the sensitivity of FEPoID to the forward horizon parameter $w$ in \cref{fig:w}.
The results demonstrate that FEPoID is robust to a wide range of $w$ values across datasets.
% \paragraph{Model Scale}


\input{tables/firstsentence.tex}

\begin{figure*}[t]
\centering
\includegraphics[width=0.75\linewidth]{figures/ID_AUROC/without_FST/legend.pdf}
\captionsetup[subfigure]{font=footnotesize, labelfont=footnotesize}
% 控制 tabular 默认列间距（越小越紧）
\setlength{\tabcolsep}{1pt}
\resizebox{\linewidth}{!}{%
  \begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/ID_AUROC/with_FST/auroc_label.pdf}} &
    \subcaptionbox{CoQA}{\includegraphics[height=3.7cm,keepaspectratio]{figures/ID_AUROC/with_FST/coqa.pdf}} &
    \subcaptionbox{SQuAD}{\includegraphics[height=3.7cm,keepaspectratio]{figures/ID_AUROC/with_FST/squad.pdf}} &
    \subcaptionbox{HotpotQA}{\includegraphics[height=3.7cm,keepaspectratio]{figures/ID_AUROC/with_FST/hotpotqa.pdf}} &
    \subcaptionbox{TriviaQA}{\includegraphics[height=3.7cm,keepaspectratio]{figures/ID_AUROC/with_FST/triviaqa.pdf}} &
    \subcaptionbox{PsiLoQA}{\includegraphics[height=3.7cm,keepaspectratio]{figures/ID_AUROC/with_FST/psiloqa.pdf}} &
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/ID_AUROC/with_FST/ID_label.pdf}}
  \end{tabular}%d
}

\caption{Layer-wise AUROC and Intrinsic Dimension across QA datasets with FST. Diamond markers indicate the layers selected by FEPoID, and star markers denote the oracle best-performing layers in terms of AUROC. The representations are extracted at the last token of the first generated sentence.}
\label{fig:layer_selection_FST}
\end{figure*}




\begin{figure*}[t]
\centering
\includegraphics[width=0.55\linewidth]{figures/windowsize/legend.pdf}

\captionsetup[subfigure]{font=footnotesize, labelfont=footnotesize}
\setlength{\tabcolsep}{1pt}

\resizebox{\linewidth}{!}{%
  \begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/windowsize/llama_wo_FST_y.pdf}} &
    \subcaptionbox{LlaMA-Instruct (w/o FST)}{\includegraphics[height=3.7cm,keepaspectratio]{figures/windowsize/llama_wo_FST_result.pdf}} &
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/windowsize/mistral_wo_FST_y.pdf}} &
    \subcaptionbox{Mistral-Instruct (w/o FST)}{\includegraphics[height=3.7cm,keepaspectratio]{figures/windowsize/mistral_wo_FST_result.pdf}} &
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/windowsize/llama_w_FST_y.pdf}} &
    \subcaptionbox{LlaMA-Instruct (w/ FST)}{\includegraphics[height=3.7cm,keepaspectratio]{figures/windowsize/llama_w_FST_result.pdf}} &
    \raisebox{0pt}[0pt][0pt]{\includegraphics[height=3.7cm]{figures/windowsize/mistral_w_FST_y.pdf}} &
    \subcaptionbox{Mistral-Instruct (w/ FST)}{\includegraphics[height=3.7cm,keepaspectratio]{figures/windowsize/mistral_w_FST_result.pdf}}
  \end{tabular}%
}

\caption{AUROC versus forward horizon $w$ for \ac{our_method} across QA datasets, with and without FST. Results show strong robustness to $w$, with slightly improved performance under FST for larger horizons on some datasets.}
\label{fig:w}
\end{figure*}

% \begin{figure}[t]
%     \centering
%     \includegraphics[width=\textwidth]{figures/pdfs/windowsize.pdf}
%     \caption{AUROC versus forward horizon $w$ for \ac{our_method} across QA datasets, with and without FST. Results show strong robustness to $w$, with slightly improved performance under FST for larger horizons on some datasets.}
%     \label{fig:w}
% \end{figure}
\section{Linear Probe}
\label{linear_probe}
To support that the FEPoID-selected layer encodes abstract semantic information, we additionally run two sets of linear probing experiments using logistic regression on the frozen hidden states, comparing the FEPoID-selected layer (index 0) against its neighboring layers (index $\pm1$, $\pm2$, $\pm3$). 
% We provide the results below which further validate our statements.
\paragraph{Factual Correctness Probing}
We train a logistic regression probe at each neighboring layer to predict whether the model's generated answer is factually correct, using the binary correctness labels already available in our experimental pipeline. We report AUROC across all five QA datasets for LLaMA-3.1-8B in \cref{tab:linear_probing}. 
To further assess whether the FEPoID-selected layer encodes abstract semantic categories beyond task-specific signals, we consider three external benchmarks:
\begin{enumerate}
    \item Odd Man Out~\citep{conneau-etal-2018-cram}: A binary semantic coherence task where each sentence is labeled as either Original (well-formed) or Changed (a noun or verb has been replaced by a random word of the same part of speech). Correctly distinguishing O from C requires genuine semantic world knowledge rather than surface-level features.
    \item AG’s News~\citep{zhang2015character}: A 4-class topic classification task over news articles (World, Sports, Business, Science and Technology), probing whether the layer encodes semantic topic categories.
    \item DBPedia~\citep{zhang2015character}: A 14-class ontology classification task over Wikipedia entity descriptions, requiring the probe to distinguish fine-grained semantic categories such as Artist, Athlete, Animal, Building, and Film, categories that share surface-level features but differ in semantic type.
\end{enumerate}
For all three tasks, we feed each sentence directly into the frozen LLaMA-3.1-8B and conduct probing using logistic regression. 

We report the AUROC results for both sets of experiments in \cref{tab:linear_probing}. The FEPoID-selected layer (index 0) consistently achieves the highest or near-highest AUROC across datasets, while performance consistently degrades as we move to deeper or shallower neighboring layers. 
Both the factual correctness probing and semantic category probing results confirm that the FEPoID-selected layer is not merely a coincidental choice for hallucination detection, but is the layer where abstract semantic information is encoded. We hope you can take our response into account and consider raising your score in the final assessment.
\begin{table}[ht]
\centering
\caption{Linear probing accuracy across layers. The FEPoID-selected layer (index 0) is compared against neighboring layers. Best results per column are in bold.}
\label{tab:linear_probing}
\footnotesize
\begin{tabular}{ccccccccc}
\toprule
Layer & CoQA & SQuAD & HotpotQA & TriviaQA & PsiLoQA & Odd Man Out & AG's News & DBPedia \\
\midrule
$-3$ & 0.7505 & 0.8250 & 0.7635 & 0.8494 & 0.8293 & 0.8180 & 0.9830 & 0.8318 \\
$-2$ & 0.7870 & 0.7824 & 0.7722 & 0.8543 & 0.8157 & 0.8207 & 0.9836 & 0.8227 \\
$-1$ & 0.8017 & 0.8308 & 0.7604 & 0.8536 & 0.8692 & 0.8217 & 0.9835 & \textbf{0.8398} \\
0 (FEPoID) & 0.8109 & \textbf{0.8582} & \textbf{0.7791} & 0.8555 & \textbf{0.8849} & \textbf{0.8252} & \textbf{0.9837} & 0.8386 \\
1 & \textbf{0.8130} & 0.8313 & 0.7761 & \textbf{0.8560} & 0.7920 & 0.8226 & 0.9833 & 0.8175 \\
2 & 0.8066 & 0.8419 & 0.7722 & 0.8533 & 0.8555 & 0.8185 & \textbf{0.9837} & 0.7969 \\
3 & 0.7832 & 0.8566 & 0.7568 & 0.8466 & 0.8726 & 0.8170 & 0.9828 & 0.7748 \\
\bottomrule
\end{tabular}

\end{table}
\section{Generalization to Vision Tasks}



\begin{table}[t]
\centering
\caption{Results on the \textbf{LlaMA-3.1-8B base} model with $w=7$. The representations are extracted with FST. FEPoID maintains strong performance on a non-instruction-tuned 
model, demonstrating that our method \emph{generalizes} across both base and 
instruction-tuned models. }
\label{tab:model_8B_base}
\begin{tabular}{ccccccc}
\toprule
          & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                \\ \midrule
RankME    & 0.7624                             & 0.8540                             & 0.6856                             & 0.6612                             & 0.8398                             & 0.7606                             \\
Curvature & \cellcolor{rankthree}0.7951        & \cellcolor{rankthree}0.8613        & \cellcolor{ranktwo}0.7839          & \cellcolor{rankone}0.8596          & \cellcolor{ranktwo}0.9049          & \cellcolor{rankthree}0.8410        \\
Val Loss  & \cellcolor{ranktwo}0.8162          & \cellcolor{ranktwo}0.8870          & \cellcolor{rankthree}0.7735        & \textbf{\cellcolor{rankone}0.8799} & \textbf{\cellcolor{rankone}0.9051} & \cellcolor{ranktwo}0.8523          \\
RGN       & 0.7285                             & 0.7305                             & 0.7499                             & \cellcolor{rankthree}0.7684        & 0.8583                             & 0.7671                             \\
SNR       & 0.6565                             & 0.7230                             & 0.6552                             & 0.6370                             & 0.7999                             & 0.6943                             \\
ID        & \textbf{\cellcolor{rankone}0.8264} & \textbf{\cellcolor{rankone}0.8884} & 0.7232                             & \cellcolor{rankthree}0.7684        & \cellcolor{rankthree}0.8985        & 0.8210                             \\
FEPoID    & \textbf{\cellcolor{rankone}0.8264} & \textbf{\cellcolor{rankone}0.8884} & \textbf{\cellcolor{rankone}0.7972} & \textbf{\cellcolor{rankone}0.8799} & \cellcolor{rankthree}0.8985        & \textbf{\cellcolor{rankone}0.8581} \\ \bottomrule
\end{tabular}
\end{table}



\begin{table}[t]
\centering
\caption{Results on different model scales, where representations are extracted with FST. Forward horizon is set to $w=7$ for \textbf{LlaMA-3.2-3B} and $w = 3$ for \textbf{LlaMA-3.2-1B}. Top-3 results are highlighted, with darker color indicating better performance. FEPoID achieves consistently strong performance across both model scales, demonstrating its \emph{generalizability} to models of varying scales. 
}
\label{tab:model_scales}
\footnotesize
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccccccc}
\toprule
          & \multicolumn{6}{c}{LlaMA-3.2-3B}                                                                                                                                                                                   & \multicolumn{6}{c}{LlaMA-3.2-1B}                                                                                                                                                                                   \\
          & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                & CoQA                               & SQuAD                              & HotpotQA                           & TriviaQA                           & PsiloQA                            & Avg                                \\ \midrule
RankME    & 0.6265                             & 0.7338                             & 0.7083                             & 0.6798                             & 0.7950                             & 0.7087                             & 0.5813                             & 0.6288                             & 0.7009                             & 0.6547                             & 0.7151                             & 0.6561                             \\
Curvature & \cellcolor{ranktwo}0.7737          & \cellcolor{rankthree}0.8163        & \cellcolor{rankthree}0.7779        & 0.8200                             & \cellcolor{rankthree}0.8687        & \cellcolor{rankthree}0.8113        & \cellcolor{rankthree}0.7190        & \cellcolor{ranktwo}0.7340          & \cellcolor{ranktwo}0.7957          & \cellcolor{ranktwo}0.7876          & \cellcolor{ranktwo}0.8097          & \cellcolor{rankthree}0.7692        \\
Val Loss  & \textbf{\cellcolor{rankone}0.7919} & \textbf{\cellcolor{rankone}0.8534} & \cellcolor{ranktwo}0.7900          & \cellcolor{ranktwo}0.8441          & \cellcolor{ranktwo}0.8822          & \cellcolor{ranktwo}0.8323          & \cellcolor{ranktwo}0.7329          & \textbf{\cellcolor{rankone}0.7566} & \textbf{\cellcolor{rankone}0.8030} & \cellcolor{rankthree}0.7750        & \textbf{\cellcolor{rankone}0.8323} & \cellcolor{ranktwo}0.7800          \\
RGN       & 0.6786                             & 0.6333                             & 0.7493                             & 0.7027                             & 0.8295                             & 0.7187                             & 0.6421                             & 0.6499                             & 0.7485                             & 0.7199                             & 0.7788                             & 0.7078                             \\
SNR       & \cellcolor{rankthree}0.7639        & 0.6333                             & \cellcolor{rankthree}0.7779        & \cellcolor{rankthree}0.8337        & 0.7665                             & 0.7550                             & 0.6859                             & 0.5973                             & \cellcolor{rankthree}0.7694        & 0.6547                             & \cellcolor{ranktwo}0.8097          & 0.7034                             \\
ID        & \textbf{\cellcolor{rankone}0.7919} & \cellcolor{ranktwo}0.8461          & 0.7309                             & 0.7027                             & \textbf{\cellcolor{rankone}0.8843} & 0.7912                             & \textbf{\cellcolor{rankone}0.7356} & \cellcolor{rankthree}0.7074        & 0.7485                             & 0.7199                             & \cellcolor{rankthree}0.7897        & 0.7402                             \\
FEPoID    & \textbf{\cellcolor{rankone}0.7919} & \cellcolor{ranktwo}0.8461          & \textbf{\cellcolor{rankone}0.7992} & \textbf{\cellcolor{rankone}0.8479} & \textbf{\cellcolor{rankone}0.8843} & \textbf{\cellcolor{rankone}0.8339} & \textbf{\cellcolor{rankone}0.7356} & \textbf{\cellcolor{rankone}0.7566} & \textbf{\cellcolor{rankone}0.8030} & \textbf{\cellcolor{rankone}0.8025} & \textbf{\cellcolor{rankone}0.8323} & \textbf{\cellcolor{rankone}0.7860} \\ \bottomrule
\end{tabular}
}%
\end{table}
We evaluate \ac{our_method} on CIFAR-10 using an ImageNet-pretrained ViT. For each layer, we use the \texttt{[CLS]} token representation to train and evaluate the downstream MLP.

% Following the hidden-state probing paradigm, the ViT backbone is kept frozen, and downstream classifiers are trained on representations extracted from individual transformer blocks.
As shown in \cref{fig:image_task}, test accuracy increases monotonically with network depth and reaches its maximum at the final transformer block.
This behavior contrasts with hallucination detection, where intermediate layers often yield the strongest probe performance (\cref{fig:motivation,fig:layer_selection_no_truncation}).

Consistent with this trend, the estimated intrinsic dimension increases steadily across layers and peaks at the penultimate layer.
Importantly, with $w\in[2,5]$, \ac{our_method} accurately selects this layer, closely matching the oracle best-performing layer.
These results demonstrate that \ac{our_method} reliably identifies informative layers for vision tasks, extending its applicability beyond language-based settings.
\begin{figure}[t]
    \centering
    \includegraphics[width=0.5\linewidth]{figures/pdfs/image_classification.pdf}
    \caption{Layer-wise accuracy and intrinsic dimension for image classification. \ac{our_method} picks the last second layer with $w\in \{2,3,4,5\}$.}
    \label{fig:image_task}
\end{figure}