  The competition between superconductivity and localization raises profound
questions in condensed matter physics. In spite of decades of research, the
mechanism of the superconductor-insulator transition (SIT) and the nature of
the insulator are not understood. We use quantum Monte Carlo simulations that
treat, on an equal footing, inhomogeneous amplitude variations and phase
fluctuations, a major advance over previous theories. We gain new microscopic
insights and make testable predictions for local spectroscopic probes. The
energy gap in the density of states survives across the transition, but
coherence peaks exist only in the superconductor. A characteristic pseudogap
persists above the critical disorder and critical temperature, in contrast to
conventional theories. Surprisingly, the insulator has a two-particle gap scale
that vanishes at the SIT, despite a robust single-particle gap.
