Infrared small target detection is crucial for remote sensing applications like disaster warning and maritime surveillance. However, due to the lack of distinctive texture and morphological features, infrared small targets are highly susceptible to blending into cluttered and noisy backgrounds. A fundamental challenge in designing deep models for this task lies in the inherent conflict between capturing high-resolution spatial details for minute targets and extracting robust semantic context for larger targets, often leading to feature misalignment and suboptimal performance. Existing methods often rely on fixed gradient operators or simplistic attention mechanisms, which are inadequate for accurately extracting target edges under low contrast and high noise. In this paper, we propose a novel Dual-Path Edge Network that explicitly addresses this challenge by decoupling edge enhancement and semantic modeling into two complementary processing paths. The first path employs a Bidirectional Interaction Module, which uses both Local Self-Attention and Global Self-Attention to capture multi-scale local and global feature dependencies. The global attention mechanism, based on a Transformer architecture, integrates long-range semantic relationships and contextual information, ensuring robust scene understanding. The second path introduces the Multi-Edge Refiner, which enhances fine-grained edge details using cascaded Taylor finite difference operators at multiple scales. This mathematical approach, along with an attention-driven gating mechanism, enables precise edge localization and feature enhancement for targets of varying sizes, while effectively suppressing noise. Our method provides a promising solution for precise infrared small target detection and localization, combining structural semantics and edge refinement in a unified framework.