## 4 Methodology
### Thematic Embedding Generation  
The first step in our methodology is to generate thematic embeddings by leveraging the Subset of Interest (SOI), a concept introduced to find non-local context of a claim under investigation [49]. In that work, the SOI was utilized for cluster visualization, offering insights into both the claim's annotated evidence and also its broader, non-local context. The SOI generation process starts by identifying the theme of the selected claim (e.g., climate change) from the fact-verification dataset. The dataset is then filtered to retain only claims and evidence relevant to the identified theme. This thematic subset is then structured through clustering, organizing semantically similar claims and evidence into distinct groups based on their embeddings, as further elaborated on in Section 5.1.1. The cluster containing the selected claim is then identified, and all items within this cluster are extracted to form an initial set. This set includes the following: (i) the selected claim, (ii) its directly annotated evidence (if any in the same cluster), (iii) other claims within the identified cluster (hereafter referred to as related claims), and (iv) the annotated evidence of these related claims (if available within the same cluster, and hereafter referred to as thematic cluster evidence). Importantly, not all annotated evidence of the selected claim or related claims will necessarily be present in the identified cluster, as it is the result of an unsupervised clustering process. Next, cosine similarity is calculated individually between the embedding of the selected claim and the embedding of each item in this initial set. Items that do not meet the empirically chosen similarity threshold (\(\delta=0.75\)) are excluded. This threshold was selected as a balanced criterion to filter out loosely related instances while retaining a thematically relevant subset of the fact-verification corpus for a given claim. The resulting refined subset forms the SOI of the selected claim. The SOI is stored in a dictionary format, as shown in Table 2, containing the claim details, its directly annotated evidence pieces, related claims, thematic cluster evidence, and cosine similarity scores quantifying the relevance of each evidence item or claim to the selected claim. (The complete algorithm for SOI generation is provided in Appendix B for those interested in the technical details). \[\text{Thematic\_Embedding}=\frac{1}{n}\sum_{i=1}^{n}\text{Embedding}(e_{i}), \quad e_{i}\in \begin{cases}\text{SOI}[\text{'annotated\_evidences'}],\\ \text{SOI}[\text{'related\_claims'}],\\ \text{SOI}[\text{'thematic\_cluster\_evidences'}]\end{cases}  \tag{1}\]  
However, while the possibility of leveraging the SOI of a claim for evidence retrieval or inference mechanisms within the AFV pipeline was highlighted as a future direction in prior work [49], it was not implemented. In this research, we extend the utility of SOI by integrating it into the AFV pipeline for the first time, moving beyond its visualization  
Figure 3: Overview of the methodology components. purpose. This transformation evolves the SOI from a static visual representation into an active component of CARAG. To implement this SOI integration into CARAG, selected elements from the SOI dictionary are extracted, specifically, SOI['annotated_evidences'], SOI['related_claims'], and SOI['thematic_cluster_evidences']. Each item in these fields is then passed through a Sentence Transformer model (SBERT, specifically the all-mpnet-base-v2 variant), which converts the text into a numerical vector (embedding) that captures the semantic meaning and contextual relevance of the text. Next, as represented in Equation (1), a single unified thematic embedding is generated by calculating the element-wise average of embeddings corresponding to these specific elements within the SOI. This involves summing the element-wise numerical components of the embeddings and dividing the result by the total number of embeddings.