Robotic manipulation often requires identifying functional parts, such as a mug handle or a hammer head. However, features attached to observed 3D points can vary with viewpoint and sensor noise, giving a policy inconsistent representations of the same part. We propose an object-centric semantic field to provide more consistent part-aware features for manipulation. We use the observed object cloud to build a continuous field, then read features from this field at 3D locations independently resampled from the cloud. Each feature uses the sampled object support as context, rather than directly reusing an individual point descriptor. Part classification distinguishes functional regions, cross-instance alignment brings corresponding part features together, and perturbation consistency encourages similar features under observation changes. The queried coordinates and features form semantic point clouds that are supplied to a DP3-based policy. We evaluate the approach on four RoboTwin simulation tasks and four real-world bimanual tasks, achieving average success rates of 69.3\% and 67.5\%, respectively. These improve on Utonia Point-wise by 7.0 and 32.5 percentage points, respectively, with real-world tests on held-out objects. A point-wise control with matched part supervision scores 63.5\% in simulation, compared with our 69.3\%. These results highlight the value of stable, object-conditioned semantic fields for manipulation across object instances and varying observations. Project Page: \href{https://zainzh.github.io/beyond-point-attached-semantics}{https://zainzh.github.io/beyond-point-attached-semantics}.

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