This work establishes geometric (exponential) growth within the Quantum Entanglement Spacetime Theory (QuEST), following the QuEST Execution Protocol (QEP). The result is derived using only two postulates: (1) a bounded-valence quantum hypergraph substrate, and (2) unitary local Pachner 2→3 dynamics. A two-sided face-density condition is introduced, requiring that every reachable triangulation contains a sufficient number of internal triangular faces shared by exactly two tetrahedra. From these ingredients, a constructive recurrence is obtained that guarantees multiplicative growth of the tetrahedron count across dynamical steps. This implies a lower bound corresponding to exponential growth of the QuEST-native scale factor. The analysis is self-contained, employs no external gravitational equations or hidden assumptions, and does not rely on Postulates 3 or 4.