## 5 Experimental Framework and Results
### Case Study Analysis of CARAG
#### 5.1.1. Thematic Embedding Generation for Claim 59  
To generate a thematic embedding for Claim 59 from _FactVer_ using CEA (Section 4.1), we first needed to identify a focused subset of contextually relevant data that would form the basis of our analysis. This involved applying our SOI approach to determine the theme associated with Claim 59 (climate), then filtering the corpus to retain only instances within this theme, ensuring alignment with the claim's context. As outlined in Section 4.1, this thematic subset is then structured through clustering to organize semantically similar claims and evidence into distinct groups. To achieve this, we applied GMM-EM clustering [57; 58] within the climate theme, identifying three unique clusters: Cluster 0, Cluster 1, and Cluster 2. The selection of GMM-EM is motivated by its effectiveness in identifying underlying patterns in complex data, with prior applications in speaker identification, emotion recognition, and brain image segmentation [57; 58]. In this context, we adapt it to model the dataset as a combination of multiple thematic structures, capturing structural similarities between claims and evidence. Our methodology employs GMM in a hard clustering approach, assigning each claim and evidence item to a single cluster to ensure clear relationships and facilitate precise analysis in AFV [49]. Claim 59 is identified within Cluster 1, a dense network containing 85 nodes and 3103 edges, indicating a rich interconnection of semantically related claims and evidence. Following the methodology outlined in Section 4.1, we refined  
Figure 6: Visualization of SOI and thematic clusters within the Climate theme. (**a,c,d**) The identified clusters (Clusters 1, 0, and 2, respectively). In (**a**), Cluster 1 highlights Claim 59 for clarity, while (**b**) shows the refined SOI for Claim 59 derived from Cluster 1.  
this cluster using a cosine similarity threshold of \(\delta=0.75\) to retain thematically relevant claims and evidence. The resulting SOI dictionary for Claim 59 incorporates all the fields presented in Table 2, providing a structured foundation for embedding generation. Figure 6 provides a visualization of the thematic clusters for Claim 59. Panels (a), (c), and (d) display the three distinct clusters identified through GMM-EM clustering, Cluster 1, Cluster 0, and Cluster 2, respectively, illustrating thematic separation within the climate theme. Cluster 1, shown in panel (a), is of particular interest as it contains Claim 59 along with the most thematically relevant connections for our analysis. For this reason, we present Cluster 1 alongside its refined SOI, derived from this cluster, as bigger sub plots (panels a and b), allowing for a direct comparison between the full thematic cluster (Cluster 1) and its distilled subset (SOI). Compared to Cluster 1, Cluster 0 (panel c) is more sparsely connected, whereas Cluster 2 (panel d) is denser. This variation in density underscores the GMM-EM algorithm's flexibility in clustering, as it naturally groups conceptually related data based on thematic relevance rather than enforcing uniform cluster sizes. This approach ensures that each cluster accurately reflects the underlying thematic nuances within the broader climate context. In the SOI graph in panel (b), Claim 59 is positioned as the central node, surrounded by interconnected nodes representing the SOI components: larger teal nodes indicate annotated evidence directly related to the claim, smaller red nodes represent thematically related claims, and smaller teal nodes denote associated evidence linked to these related claims. Importantly, each component in the SOI is selectively included if relevant to Claim 59. For instance, while Evidence_59_2 and Evidence_59_3 are included, the remaining annotated evidence items (from the total of six pieces for Claim 59 in the dataset) are excluded. Similarly, for the related Claim_1, only Evidence_1_2 and Evidence_1_3 are included, while the rest of its six associated evidence pieces are excluded.