Ultrasound-guided interventions can require localization of an untracked 2D frame within a 3D anatomical reference. Rigid slice-to-volume registration (SVR) estimates this six-degree-of-freedom pose but remains challenging because of limited anatomical context, acoustic artifacts, and view-dependent appearance. Existing methods often use dense cross-attention, whose cost scales with the product of slice and volume token counts, or direct pose regression without explicit correspondence constraints. We introduce SCoPE-Reg, combining state-space slice--volume interaction, dense 3D coordinate prediction, and parameter-free weighted Kabsch estimation. On SVR tasks from CAMUS and $μ$-RegPro, SCoPE-Reg yields mean target registration errors of $0.73$ mm and $2.27$ mm against $1.24$ mm and $2.63$ mm for the state of the art (SOTA), reduces peak error on CAMUS by 56% below SOTA ($12.5\!\to\!5.5$ mm), and registers $100\%$ and $80\%$ of frames within $3$ mm. On CAMUS at $128^2$ it retains the lowest error at increasing pose-perturbation magnitude. It holds $6.49$ M parameters independent of resolution, sustaining $51$ FPS at $512^2$. SCoPE-Reg establishes a SOTA in rigid ultrasound SVR: by coupling correspondence-based accuracy with bounded worst-case error and resolution-independent cost, it becomes viable at native acquisition resolution during intervention, where prior methods trade accuracy, reliability, or frame rate against one another. Supplementary code provided and will be open-sourced upon acceptance.

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