CoreLumen Fundamental III : Emergent Spacetime from the Condensate Vacuum

This work investigates the emergence of spacetime geometry from thevacuum condensate within the CoreLumen framework. Within the CoreLumen program the physical vacuum is interpreted as acondensate medium described by the order parameter Ψ = ρ e^{iθ}. CoreLumen Fundamental I established the microscopic origin of thiscondensate from a spontaneously broken complex scalar field, whileCoreLumen Fundamental II developed the quantum field theory governingthe phase sector of the vacuum medium. The present work shows how spacetime geometry can emerge from thecollective dynamics of this phase-coherent condensate. The conserved phase current Jμ = ρ² ∂μθ defines the flow structure of the vacuum medium. The propagation ofexcitations in this medium naturally defines an effective geometry,which can be interpreted as the emergent spacetime metric experiencedby long-wavelength perturbations. Within this interpretation gravity appears as a collective phenomenonarising from the dynamical response of the vacuum condensate ratherthan as a fundamental interaction. This document establishes the conceptual foundation linking themicroscopic vacuum sector of the CoreLumen framework to the emergentgravitational and cosmological descriptions developed in the laterCoreLumen series. This work is released as a Zenodo archival version as part of theCoreLumen Fundamental Physics Series.