Accurate liver and tumor segmentation on abdominal CT images is critical for reliable diagnosis and treatment planning, but remains challenging due to complex anatomical structures, variability in tumor appearance, and limited annotated data. To address these issues, we introduce Hyperbolic-convolutions Adaptive-temporal-attention with Neural-representation and Synaptic-plasticity Network (HANS-Net), a novel segmentation framework that synergistically combines hyperbolic convolutions for hierarchical geometric representation, a wavelet-inspired decomposition module for multi-scale texture learning, a biologically motivated synaptic plasticity mechanism for adaptive feature enhancement, and an implicit neural representation branch to model fine-grained and continuous anatomical boundaries. Additionally, we incorporate uncertainty-aware Monte Carlo dropout to quantify prediction confidence and lightweight temporal attention to improve inter-slice consistency without sacrificing efficiency. Extensive evaluations of the LiTS dataset demonstrate that HANS-Net achieves a mean Dice score of 93.26%, an IoU of 88.09%, an average symmetric surface distance (ASSD) of 0.72 mm, and a volume overlap error (VOE) of 11.91%. Furthermore, cross-dataset validation on the AMOS 2022 dataset obtains an average Dice of 85.09%, IoU of 76.66%, ASSD of 19.49 mm, and VOE of 23.34%, indicating strong generalization across different datasets. These results confirm the effectiveness and robustness of HANS-Net in providing anatomically consistent, accurate, and confident liver and tumor segmentation.