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Current deep learning-based object detection for Synthetic Aperture Radar (SAR) imagery mainly adopts optical image methods, treating targets as texture patches while ignoring inherent electromagnetic scattering mechanisms. Though scattering points have been studied to boost detection performance, most methods still rely on amplitude-based statistical models. Some approaches introduce frequency-domain information for scattering center extraction, but they suffer from high computation cost and poor compatibility with diverse datasets. Thus, effectively embedding scattering topological information into modern detection frameworks remains challenging. To solve these problems, this paper proposes the Physics-Aware Scattering Topology Embedding Framework (PASTE), a novel closed-loop architecture for comprehensive scattering prior integration. By building the full pipeline from topology generation, injection to joint supervision, PASTE elegantly integrates scattering physics into modern SAR detectors. Specifically, it designs a scattering keypoint generation and automatic annotation scheme based on the Attributed Scattering Center (ASC) model to produce scalable and physically consistent priors. A scattering topology injection module guides multi-scale feature learning, and a scattering prior supervision strategy constrains network optimization by aligning predictions with scattering center distributions. Experiments on real datasets show that PASTE is compatible with various detectors and brings relative mAP gains of 2.9% to 11.3% over baselines with acceptable computation overhead. Visualization of scattering maps verifies that PASTE successfully embeds scattering topological priors into feature space, clearly distinguishing target and background scattering regions, thus providing strong interpretability for results.