We introduce the Phase-Space Atlas, a novel archival format for periodically driven many-body systems that stores only initial conditions and per-segment dynamical models, enabling exact analytic reconstruction of arbitrary observables at any time. Applied to 17 days of full-cadence SDO/AIA 171~\AA\ data (129 three-hour patches), the method compresses raw imagery exceeding 14~TB to 116~kB of turning geometry (plus optional 26~MB initial images for pixel recovery) while preserving all phase-mediated observables (stiff axes, basin depth, angular Lyapunov proxies, recurrence structure) losslessly. Global concatenation confirms persistent rank-3 basin structure over weeks. The protocol supports random-access latency $<1$~ms/frame without network retrieval and scales sub-linearly with duration, rendering multi-decade archives tractable on consumer hardware.