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The cislunar space, governed by the circular restricted three-body problem (CR3BP), presents significant challenges for mission design due to its complex stability structure. Traditional high-fidelity numerical integration is computationally prohibitive for a systematic energy-regime census of millions of orbits. Here, we present a novel approach based on global volunteer computing via the BOINC platform to overcome this barrier. Using the public “Million Orbit” dataset from Lawrence Livermore National Laboratory, we distributed the computation of Jacobi constant time series across thousands of volunteer devices, producing over 16 billion individual values. The resulting dataset is freely available. Analysis reveals that 91.68% of orbits belong to the high-energy Region V, 8.07% to the low-energy Region I, and only 0.24% to Region III, with Region II completely absent. A single rare Region IV orbit (ID 754482) was identified and analyzed. Furthermore, we develop a lightweight deep learning surrogate that predicts whether an orbit belongs to the low-energy Region I using only the first K Jacobi constants (prefix). Our model combines an LSTM encoder with attention and an XGBoost classifier, achieving test AUC of 0.984 with K = 500 and 0.929 even with K = 10, outperforming a raw XGBoost baseline. This work demonstrates the transformative potential of volunteer computing for large-scale astrodynamics and provides an efficient machine learning tool for real-time orbit screening. Our attention analysis further reveals that the model automatically focuses on the initial transient for long sequences, quantitatively confirming the sensitive dependence on initial conditions in the Earth-Moon system.