Multi-task dense prediction solves complementary pixel-level tasks in a unified model, such as semantic segmentation, depth estimation, surface normal estimation, and edge detection. Existing decoder-side interactions use attention, prompts, routing, diffusion, Mamba, or bridge features to exchange task evidence, but most of them organize this evidence implicitly. They usually fuse task features by similarity or affinity, without explicitly modeling that evidence reliability varies across tasks and spatial locations. As a result, unreliable evidence may contaminate the shared representation and intensify negative transfer. We propose $\mathcal{B}^{3}$-Net, a controlled posterior bridge learning framework for multi-task dense prediction. Our method decomposes decoder-side interaction into reliability estimation, posterior bridge construction, and bounded redistribution. The Precision Field Estimator estimates patch-wise evidence precision from task-reference alignment and local variation. The Posterior Bridge Operator builds a precision-weighted posterior bridge through heteroscedastic evidence fusion, yielding a shared state more reliable than uniform or heuristic mixtures. The Contractive Dispatch Operator redistributes the bridge to each task branch through a bounded update, reducing uncontrolled feature injection. Experiments on NYUD-v2, PASCAL-Context, and Cityscapes show that $\mathcal{B}^{3}$-Net achieves competitive or superior trade-offs over representative CNN-, Transformer-, diffusion-, Mamba-, and bridge-feature-based methods. Backbone-matched comparisons and extensive analyses further verify that the gains arise from controlled posterior bridge learning rather than backbone capacity or decoder scale.