The aim of this study was to address the issue of significant performance degradation in existing defogging algorithms under extreme fog conditions. Traditional Taylor series-based deformable convolutions are limited by local approximation errors, while the heavy-tailed characteristics of the Cauchy distribution can more successfully model outliers in fog images. The following improvements are made: (1) A displacement generator based on the inverse cumulative distribution function (ICDF) of the Cauchy distribution is designed to transform uniform noise into sampling points with a long-tailed distribution. A novel double-peak Cauchy ICDF is proposed to dynamically balance the heavy-tailed characteristics of the Cauchy ICDF, enhancing the modeling capability for sudden changes in fog concentration. (2) An innovative Cauchy-Gaussian fusion module is proposed to dynamically learn and generate hybrid coefficients, combining the complementary advantages of the two distributions to dynamically balance the representation of smooth regions and edge details. (3) Tree-based multi-path and cross-resolution feature aggregation is introduced, achieving local-global feature adaptive fusion through adjustable window sizes (3/5/7/11) for parallel paths. Experiments on the RESIDE dataset demonstrate that the proposed method achieves a 2.26 dB improvement in the peak signal-to-noise ratio compared to that obtained with the TaylorV2 expansion attention mechanism, with an improvement of 0.88 dB in heavily hazy regions (fog concentration > 0.8). Ablation studies validate the effectiveness of Cauchy distribution convolution in handling dense fog and conventional lighting conditions. This study provides a new theoretical perspective for modeling in computer vision tasks, introducing a novel attention mechanism and multi-path encoding approach.