Tactile Internet (TI) requires ultra-low latency and high reliability to ensure stability and transparency in touch-enabled teleoperation. However, variable delays and packet loss present significant challenges to maintaining immersive haptic communication. To address this, we propose a predictive framework that integrates Discrete Mode Decomposition (DMD) with Shapley Mode Value (SMV) for accurate and timely haptic signal prediction. DMD decomposes haptic signals into interpretable intrinsic modes, while SMV evaluates each mode's contribution to prediction accuracy, which is well-aligned with the goal-oriented semantic communication. Integrating SMV with DMD further accelerates inference, enabling efficient communication and smooth teleoperation even under adverse network conditions. Extensive experiments show that DMD+SMV, combined with a Transformer architecture, outperforms baseline methods significantly. It achieves 98.9% accuracy for 1-sample prediction and 92.5% for 100-sample prediction, as well as extremely low inference latency: 0.056 ms and 2 ms, respectively. These results demonstrate that the proposed framework has strong potential to ease the stringent latency and reliability requirements of TI without compromising performance, highlighting its feasibility for real-world deployment in TI systems.