Finite branching bound in a complete market ($\#\mathrm{atoms}({\mathcal{F}}_t)\le n^t$)

With $n$ assets and a trivial initial sigma-field, market completeness forces at most $n$ positive-probability successors per current atom. Indicators of disjoint successor events are independent payoffs, whereas terminal holdings on a parent atom span at most $n$ asset-price functions. Iterating yields at most $n^t$ atoms at time $t$.