We demonstrate that gravitational and quantum dynamics can be unified as emergent consequences of strictly bounded computational bandwidth within a discrete generative spacetime (the Discrete Emergent Universe). In this framework, the fundamental phenomena of physics arise from scheduler load: persistent topological defects impose chronic bandwidth demand (analogous to gravitational mass), while acute logical conflicts trigger saturation-limited resolution events (analogous to quantum state collapse). Through pre-registered experiments on a growing triangulated foam, we derive a unified two-queue ledger law---governing forced backlog and standing vacuum demand---that dictates this load response with zero fitted parameters. This single internal law accurately brackets the chronic starvation collapse threshold (a gravitational-regime constant of x50 = 1.30 and completely reproduces acute Zeno-pattern suppression in temporally structured loads. While the central claim is a model-internal unification of these two operating regimes, we provide external bridge diagnostics---including a CMB amplitude basin, preliminary GW150914 ringdown-compression signatures, and formal equivalence to standard quantum and relativistic clock laws---as falsifiable targets for physical equivalence.