Science-inspired vision models show that explicit propagation dynamics can provide structured and interpretable alternatives to conventional token mixing. Existing formulations, however, typically construct and adapt visual propagation within a particular dynamical family, while visual representations can require substantially different spatial interactions across samples, channels, and network stages. We explore cross-regime adaptive propagation and introduce TailProp, a hierarchical vision backbone built upon the Tail Propagation Operator (TPO). TPO uses Gaussian and Cauchy stable-process propagators as complementary bases with rapidly decaying and heavy-tailed spatial influence, and predicts a content-conditioned channel-wise coefficient to adaptively combine them. Because this coefficient is spatially shared, the two responses are fused directly in the DCT domain with a single DCT/IDCT pair, yielding $O(N^{1.5})$ spatial mixing for square feature maps with $N=HW$ and fixed channel width. Across image classification, object detection, semantic segmentation, robustness, and cross-backbone restoration, TailProp consistently outperforms matched propagation baselines; TailProp-B reaches 84.4% Top-1 accuracy on ImageNet-1K, 50.3/44.8 box/mask AP under the 3x Mask R-CNN schedule, and 50.8% mIoU on ADE20K. Controlled ablations further show that these gains are not explained by single-basis propagation, an additional same-family branch, or within-family adaptive order alone, supporting complementary two-basis propagation as an effective design principle for visual representation learning.

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