T. Wevers, T. J. Maccarone, A. Palmese, D. J. Sand, S. Gezari, L. Rivera Sandoval, J. DiPalma, G. Hosseinzadeh, T. J. Hoyt, M. Karmen, J. Kennea, K. Kunnumkai, S. Perlmutter, M. Rigault, A. Roy, G. Stefansson, F. Yang Yang

Advancing time-domain and multi-messenger astronomy requires a multi-wavelength network of observatories capable of rapidly discovering, classifying, and characterizing transient phenomena. A critical gap in current capabilities is the inability to follow up faint, fast-evolving transients with sensitive, wide-band imaging and spectroscopic observations from space on timescales of minutes to hours. We discuss how the Lazuli Space Observatory will address this gap through a large collecting area, optical/NIR photometry and low-resolution integral field spectroscopy, and a rapid-response architecture with a mission requirement of $<$4 hours from trigger to first photon. Based on a latency analysis, we find a credible path to realizing response times well below this requirement, with best-case scenarios below 90 minutes under favorable conditions. We highlight extragalactic science opportunities in currently un(der)explored parts of parameter space, including gravitational wave follow-up, kilonova characterization, supernova progenitor physics, and a wide variety of fast-evolving transients and high redshift events. We further outline new observational capabilities for Galactic time-domain science, including high frequency variability in accreting systems, precision astrometry of compact objects, and the detection of compact and ultracompact binaries, enabled by high-frequency, diffraction-limited imaging and astrometry. Together, its capabilities - combining flagship sensitivity with response times one to two orders of magnitude faster than existing large space observatories - position Lazuli to make transformative contributions across time-domain and multi-messenger astrophysics.