Ball-end milling path planning on multiply connected freeform surfaces is pivotal for high-quality and efficient machining of components in automotive and aerospace manufacturing. Although scalar-field-based optimization provides a unified framework for multi-objective toolpath generation, maintaining boundary conformity while eliminating zero-gradient singularities that cause iso-curve branching or termination and disrupt toolpath continuity remains challenging on multiply connected surfaces. We propose an efficient strategy to robustly enforce these constraints throughout optimization. Conformal slit mapping is employed to construct a feasible, singularity-free initial scalar field. The optimization is reformulated as a topology-preserving mesh deformation governed by boundary-synchronous updates, enabling globally optimized spacing, scallop-height uniformity, and smooth trajectory transitions. Consequently, the toolpaths are continuous, boundary-conforming, and free of self-intersections. Milling experiments demonstrate that, compared with a state-of-the-art conformal slit mapping-based method, the proposed approach increases machining efficiency by 14.24%, improves scallop-height uniformity by 5.70%, and reduces milling impact-induced vibrations by over 10%. The strategy offers broad applicability in high-performance machining scenarios.

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