Confirming Alzheimer's disease (AD) typically relies on positron emission tomography (PET), which remains costly and invasive, motivating the use of structural MRI-based prescreening. Deep learning on non-Euclidean manifolds, particularly brain cortical surfaces, faces significant challenges due to the data's spherical topology. Recent surface models have enabled learning from cortical surface data; however, imposing face-based uniform patches often causes duplicate vertices at patch boundaries. In general, many surface-based models are limited in their awareness of the region of interest (ROI), which can result in non-cortical regions, such as the medial wall, being included. We propose a cortical surface tokenization that performs ROI-preserving, vertex-based, variable-sized patch partitioning. We refer to these cortical surface patches as cortical supervertices (CSVs). Building on this representation, we design the CSV Vision Transformer (CSV-ViT), a variable-size patch-tolerant Vision Transformer that uses padding and a mask-aware patch embedding. We used T1-weighted MRI and evaluated our framework by classifying AD-related status into three categories: AD diagnosis, amyloid positivity, and tau positivity. Across the experiments, CSV-ViT achieved higher classification performance than recent surface-based models. The results suggest that the proposed CSV-ViT may support MRI-based prediction of AD-related status prior to PET or CSF confirmation.