Monocular outdoor navigation requires robust camera motion estimation and a dense, navigable map, where 3D Gaussian Splatting SLAM serves this task through an explicit differentiable parameterization and closed-form projection. However, typically, the geometry is recovered by associating 2D pixels with 3D points, so a modular SLAM system absorbs tracking error into the map, leading to cascading degradation. To address these gaps, we present SG^2SLAM, which couples streaming spatial geometry with a ray-conditioned 3DGS kernel. Streaming spatial geometry replaces the per- frame cross-dimensional correspondence with a chunk-level metric prior that supervises tracking and mapping from one source. A ray-conditioned kernel extends each primitive with a viewing-direction subspace that senses the monocular camera viewpoints. The joint optimization over spatial geometry and poses ensures a globally consistent map and trajectory outcome. Experiments on the KITTI and Waymo Open datasets show that our SG^2SLAM surpasses the per-frame association baselines in the monocular outdoor navigation task. Code will be made publicly available upon acceptance.

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