We present an extension of the Wave 3 framework in which spacetime is not fundamental but emerges from a nonlinear scalar wave field representing a continuous informational medium. A disformal metric construction ensures a full-rank emergent geometry, while a covariant action formalism guarantees consistency with general relativity in the appropriate limit. Dissipative dynamics are derived from an effective field theory treatment of coarse-grained degrees of freedom, leading to an effective spacetime viscosity and an emergent arrow of time. We develop the cosmological implications of the model, showing that the modified scalar dynamics can influence large-scale expansion without introducing an explicit cosmological constant. The framework predicts a distinctive observational signature: redshift-dependent attenuation of gravitational waves without modification of propagation speed. We derive the corresponding damping relation and provide order-of-magnitude constraints using current gravitational wave observations, demonstrating partial testability of the model. This work positions the Wave 3 framework within the broader class of scalar–tensor and emergent spacetime theories while introducing a physically motivated dissipative sector and a clear pathway toward experimental validation.