Accurate semantic segmentation of foot ulcers is essential for automated wound monitoring, yet boundary delineation remains challenging due to tissue heterogeneity and poor contrast with surrounding skin. To overcome the limitations of standard intensity-based networks, we present LSS-LTCNet:an ante-hoc explainable framework synergizing deterministic structural priors with continuous-time neural dynamics. Our architecture departs from traditional black-box models by employing a Local Self-Similarity (LSS) mechanism that extracts dense, illumination-invariant texture descriptors to explicitly disentangle necrotic tissue from background artifacts. To enforce topological precision, we introduce a Liquid Time-Constant (LTC) refinement module that treats boundary evolution as an ODEgoverned dynamic system, iteratively refining masks over continuous time-steps. Comprehensive evaluation on the MICCAI FUSeg dataset demonstrates that LSS-LTCNet achieves state-of-the-art boundary alignment, securing a peak Dice score of 86.96% and an exceptional 95th percentile Hausdorff Distance (HD95) of 8.91 pixels. Requiring merely 25.70M parameters, the model significantly outperforms heavier U-Net and transformer baselines in efficiency. By providing inherent visual audit trails alongside high-fidelity predictions, LSS-LTCNet offers a robust and transparent solution for computer-aided diagnosis in mobile healthcare (mHealth) settings.