arXiv:2610.34479v1 Announce Type: new Abstract: Stacked intelligent metasurfaces (SIMs) cascade programmable layers in front of a feed. In the SIM designs reported so far, including those proposed for low-Earth-orbit (LEO) satellites, the layer coefficients are space-only controls or are varied randomly in time to support opportunistic scheduling, so that every radiated beam carries the frequency of the feed. This paper introduces space-time SIMs (ST-SIMs), whose layers are modulated in time and use the temporal dimension of the stack deterministically: each spatial mode of the aperture is paired with its own prescribed frequency translation. The application is LEO broadcasting, where a common waveform reaches terminals with different Doppler shifts, and any common Doppler correction leaves residuals of up to half the Doppler span. We derive the harmonic-domain model of the stack, solve the minimum-energy space-frequency synthesis in closed form, and prove that full column rank of the steering matrix is necessary and sufficient for decoupling K terminal-specific translations, a result that extends to groups of terminals. We further propose a phase-only synthesis algorithm with guaranteed stationarity, together with quantization and control-rate laws. Simulations at 20 GHz over 50 random drops show that a four-layer ST-SIM with a single radio-frequency chain removes the Doppler floor of a static SIM for footprint radii above about 45 km and, under the ideal phase-only model and at equal radiated peak energy, approaches ideal digital per-terminal precompensation within 0.5 dB for resolvable terminals, whereas, in the evaluated geometries, a single time-modulated layer cannot synthesize the required modes.