On-sky trajectory evolution as a function of spherical orbital parameters. Each panel shows all detections in the trajectory of a particularly bright implanted object (color-coded by time with a baseline of 80 days) for a specific combination of orbital parameters. Left: variation across barycentric distance r (rows) and tangential speed v_Ω (columns), holding v_r and ψ fixed. Right: variation across r (rows) and tangential direction angle ψ (columns), holding v_r and v_Ω fixed. Trajectory morphology changes systematically with orbital parameters: trajectories further from the true orbital elements of the object trace longer, more curved arcs, while the direction angle ψ controls the orientation and curvature of the path. The center tile in each plot represents the true orbital elements of the fake, at which point the trajectory collapses onto itself as all detections are coincident. These patterns illustrate how small changes in the spherical basis parameters produce distinct on-sky signatures, motivating the adaptive grid refinement strategy. Note that these panels span a longer baseline than the grid is constructed to certify, and several columns/rows correspond to cells increasingly distant in parameter space from the object’s home cell; the resulting angular spread is in most cases far too large to satisfy the initial clustering tolerance ε (Section 3.1.1), let alone τ_∠, and so most of the depicted arcs would not themselves form valid trajectories at the tolerances used in the search. The figure is intended only to illustrate that residual structure remains ordered even well outside the cell where a real trajectory would actually be recovered.