The Buchdahl compactness bound is traditionally derived within General Relativity as a condition preventing divergent central pressure in static, spherically symmetric matter distributions. This work derives an equivalent bound from first principles within Quantum Entanglement Spacetime Theory (QuEST), using only two postulates: finite-valence hypergraph structure and local rewrite dynamics. No spacetime metric, Einstein field equations, stress--energy tensor, pressure concept, or holographic principle is assumed. Two independent information-theoretic limits are derived: a bulk coordination constraint governing the maximal sustainable generation of distinguishability under local dynamics, and a finite boundary encoding capacity arising from bounded valence. The compactness bound appears at the point where the bulk constraint forces saturation of the boundary encoding capacity. The resulting bound reproduces the classical Buchdahl limit in the General Relativity regime and predicts a distinct compactness bound for cylindrical symmetry, for which General Relativity admits no unique analogue.