Image-goal navigation requires a global planner that connects visually specified endpoints while providing observations that a local policy can follow. Three-dimensional Gaussian Splatting (3DGS) provides map-registered RGB, opacity, and depth renderings, but does not explicitly encode navigable topology. We introduce GS-PRM, a Gaussian visual roadmap whose milestones, local connections, and visual references are constructed directly through 3DGS rendering without deriving an occupancy grid or signed distance field. We define render-based collision-free (RC-free) configurations and paths. Panoramic depth supplies conservative clearance for milestone validation and adaptive coverage, while edge-aligned orthographic rendering tests the robot's swept volume. We establish conditional soundness and sufficient conditions for deterministic query completeness and probabilistic construction completeness of GS-PRM. To validate the performance of GS-PRM, we conduct experiments in both simulated and real-world environments, demonstrating long-horizon navigation using goal-conditioned policies guided by GS-PRM. Compared with a state-of-the-art 3DGS-based navigation system, our GS-PRM-based navigation system achieves up to a twice higher average success rate and reduces travel time by up to 57.2% when both systems complete the navigation task.

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