Exact Ground States of Two Dimensional $\pm J$ Spin Glasses

We derive exact analytical expressions for the ground-state energy and entropy of the two-dimensional $\pm J$ Ising spin glass, uncovering a nested hierarchy of frustrations. Each level in this hierarchy contributes through the kernel and pseudo-determinant of effective operators, capturing the energy and entropy, respectively. At leading order, the structure coincides with geometric plaquette frustrations, while subleading corrections arise from magnetic adjacency matrices defined on percolated clusters in the dual lattice. Our results, supported by numerical simulations, provide a systematic framework for analyzing spin-glass ground states and offer new insight into glass order in finite dimensions.