The history and dynamical evolution of the Kreutz group of sungrazing comets can be investigated by modeling the observed temporal and spatial distributions of the group's members, more than 250 of which were known by the end of 2000. This paper presents the results of an early phase of this modeling effort, which is based on (i) a long-standing consensus that all comets belonging to the Kreutz group are fragments of one original, massive comet; and (ii) a breakthrough in the understanding of the sungrazers' pairs and tight clusters detected in large numbers with the coronagraphs onboard SOHO. All these objects are products of progressive fragmentation that typically begins with a tidally-triggered (or tidally-assisted) breakup near perihelion, in the immediate proximity of the Sun. However, instead of terminating at that point - as postulated in all previous hypotheses - the process is shown to continue in a cascading fashion throughout the orbit about the Sun.