When can a mathematical population experience infinitely many events in finite time? In stochastic-process theory this is called explosion. It is a breakdown of the ordinary trajectory description, not a chemical detonation. This candidate studies compartments that contain substrate and a finite conserved number of catalysts. Substrate production accelerates as content grows, while fragmentation repeatedly splits a compartment and redistributes its contents. The paper identifies the exact phase boundary between these forces. If fragmentation grows faster than the critical power, the full population is conservative. If it grows more slowly, every finite state containing a catalyst explodes almost surely. At the critical power, the answer is decided by a sharp comparison between the production coefficient and the logarithmic loss at a split. The proof matters because following only one active lineage is not enough: discarded descendants can continue fragmenting and their events still count. The paper controls that passive population and then lifts the one-catalyst argument to any finite catalyst count. It also treats real growth exponents, a finer logarithmic boundary, general moment-controlled splitting and fresh random retention fractions. In that random setting the relevant quantity is the expected logarithmic loss, not the mean fraction retained; an explicit example shows that equal means can lead to different phases. Exact replay checks selected formulas in ordinary and optimized Python and deliberately corrupts inputs to verify rejection. Those checks support the calculations but do not prove the stochastic theorems. This is an unrefereed AI-assisted research candidate, not a formally verified result or external peer review. The paper, evidence and code are linked on the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.