Existing interval carbon price forecasting methods usually estimate the upper and lower bounds independently or derive them from point forecasts, overlooking their intrinsic coupling and risking logically inconsistent outputs. We propose a constrained multi-task learning framework, CMTL-ATCN, that treats interval-valued carbon prices as coupled tasks and optimizes them jointly under logical consistency constraints. The model uses attention-enhanced temporal convolutional networks (ATCN) to extract shared and task-specific features from multi-source data, which are then fused and decoded through dual task heads; all parameters are optimized jointly with a loss function that adds a logic penalty enforcing agreement between the two bounds. On China’s national (CEA), Guangdong (GDEA), and Hubei (HBEA) carbon markets, CMTL-ATCN attains the lowest interval forecast errors among deep models and eliminates logical inconsistencies entirely with zero violations. The results indicate that the proposed framework is promising for interval-valued forecasting in complex environments.