Design for continuous fiber-reinforced polymer additive manufacturing (CFRP-AM) has evolved from sequential approaches to concurrent design methodologies, enabling the simultaneous optimization of structural topology and fiber toolpaths [1]. To further advance concurrent design toward reducing the gap between structural design and practical manufacturing, the explicit incorporation of manufacturability constraints is essential [2]. However, their integration into conventional topology optimization frameworks remains challenging due to multiple factors, such as complex sensitivity derivations and nonlinear manufacturing constraint. Recent advances in artificial intelligence provide new opportunities to overcome these limitations by combining neural networks with classical structural optimization techniques. In this work, a neural-field-based concurrent topology optimization framework for CFRP-AM is proposed. In the presented approach, both structural density variables and fiber toolpaths are parameterized using implicit neural fields, and the network parameters are directly treated as design variables. This formulation enables the unified and differentiable incorporation of both design and manufacturability objectives, including structural stiffness, volume fraction constraint, toolpath width consistency, and fiber continuity. These objectives are embedded as loss functions within an integrated optimization process, ensuring that the resulting composite structures achieve enhanced mechanical performance while remaining manufacturable. Owing to the inherent differentiability of the neural implicit representation, automatic differentiation is employed for sensitivity analysis, eliminating the need for explicit analytical gradient derivations. Furthermore, the infinite-resolution characteristics of implicit neural representations allow manufacturing-related features to be identified and enforced at higher spatial resolutions, while directly generating smooth and continuous structural layouts with reduced post-processing requirements. Numerical case studies demonstrate that the proposed framework effectively improves both structural performance and manufacturability compared with conventional approaches, highlighting its potential for engineering high-performance continuous fiber-reinforced composite structures via additive manufacturing.