Open-vocabulary 3D scene segmentation usually assumes RGB-D video, calibrated multi-view imagery, or a reconstructed mesh. GeoSAM-3D studies a lighter setting: a user uploads a short monocular video, clicks or names an object in one frame, and receives a propagated 3D mask over a Gaussian scene. The implementation combines frozen image and video foundation models with a monocular 3D Gaussian Splatting reconstruction and a differentiable graph-geodesic propagation kernel over Gaussian centroids. The central design choice is to propagate prompts by heat-kernel distance on the reconstructed scene graph, rather than by Euclidean nearest neighbors in 3D. This preserves continuity around curved surfaces and reduces leakage across nearby but disconnected objects. This paper describes the repository state, the mathematical kernel implemented in geosam3d.propagate, the feature head trained from Segment Anything masks, and the validation already present in the codebase. The evaluation protocol separates implementation validation, graph propagation quality, leakage control, and interactive latency.