Understanding and representing the structure of 3D objects in an unsupervised manner remains a core challenge in computer vision and graphics. Most existing unsupervised keypoint methods are not designed for unconditional generative settings, restricting their use in modern 3D generative pipelines; our formulation explicitly bridges this gap. We present an unsupervised framework for learning spatially structured 3D keypoints from point cloud data. These keypoints serve as a compact and interpretable representation that conditions an Elucidated Diffusion Model (EDM) to reconstruct the full shape. The learned keypoints exhibit repeatable spatial structure across object instances and support smooth interpolation in keypoint space, indicating that they capture geometric variation. Our method achieves strong performance across diverse object categories, yielding a 6 percentage-point improvement in keypoint consistency compared to prior approaches.