Reconstruct What the Task Needs: Progressive Task-Prior Information Bottlenecks for Accelerated MRI

Magnetic resonance imaging (MRI) reconstruction is increasingly integrated with downstream tasks to preserve task-relevant structures. However, in multi-stage unrolled networks, stage-wise task feedback can entangle newly emerging task information with redundant and task-irrelevant content. We propose Purified Task Prior Guided MRI (PTP-MRI), which represents each stage update as a history-conditioned stochastic task increment. A variational lower bound encourages the retention of newly emerging task information, while a conditional variational upper bound limits task-irrelevant residual information. The resulting constrained information objective is optimized through Lagrangian duality with an adaptive dual variable. The purified task states are progressively accumulated and decoded into spatial priors that guide reconstruction according to task relevance. Experiments on SKM-TEA at $8\times$ acceleration and AHEAD at $16\times$ acceleration show that PTP-MRI improves downstream segmentation while maintaining reasonable reconstruction fidelity. Compared with the competing baseline for each metric, PTP-MRI achieves Dice/mIoU gains of 0.78/1.28 percentage points on SKM-TEA and 0.87/1.71 percentage points on AHEAD.