We propose a universal electronic Hamiltonian model based on HamGNN1,2, which parameterizes the Kohn-Sham DFT Hamiltonian using equivariant graph neural networks, enabling electronic structure prediction for arbitrary elemental combinations without additional training. The model demonstrates strong transferability across complex systems including solid-state electrolytes, moiré heterostructures, and metal-organic frameworks. High-throughput calculations on ~380,000 GNoME crystals identified 3,940 direct-bandgap materials and 5,109 flat-band systems, achieving two orders of magnitude speedup over conventional DFT2. We further developed machine learning frameworks for electron-phonon coupling3 and nonadiabatic molecular dynamics4. Additionally, the Uni-HamGNN5 model was developed for spin-orbit coupling using Hamiltonian decomposition and delta-learning, identifying 138 topological insulator candidates, and accurately predicting valley polarization and twist-angle-dependent electronic structures in two-dimensional heterostructures. We establish a new paradigm for full-periodic-table electronic-structure-based materials screening, with broad applications in semiconductors, energy, and quantum materials.