Emergent Einstein–Friedmann Dynamics from UniversalWavefunction Geometry

We develop a geometric framework in which an effective spacetime description and an Einstein--Friedmann–type structure emerge from the geometry of a universal wavefunction, without postulating gravitational field equations or introducing matter fields as independent degrees of freedom. Starting from a conserved current associated with the wavefunction, we define a flux hypersurface embedded in a higher-dimensional ambient space and show that, under minimal assumptions of homogeneity and isotropy, its induced Lorentzian geometry is necessarily of Friedmann--Robertson--Walker type. The intrinsic curvature of the induced metric is fixed by the embedding geometry itself. A maximally symmetric hyperboloid corresponds to exact de Sitter spacetime, while more general, physically admissible, and normalisable wavefunction envelopes give rise to a time-dependent curvature scale. In this case, the effective cosmological term is approximately constant only in a narrow intermediate regime, where the expansion is transientlyquasi–de Sitter, and evolves away from this limit at both early and late times. By identifying a conserved, potential-like geometric invariant inherited from the universal wavefunction, we obtain an effective Einstein--Friedmann structure on the hypersurface without invoking gravitational dynamics. This invariant fixes the scaling of the dominant effective density ρ ~ 1/a2 and determines the effective gravitational coupling. For closed spatial slicing, this matter-like contribution cancels identically against the spatial curvature term in the Friedmann equation, leaving a purelygeometric constraint relating the Hubble rate to a residual, time-dependent vacuum-like sector. We show that the apparent tension between a de Sitter–like Friedmann constraint and a nonvanishing is resolved once the effective continuity equation is taken into account: the expansion rate is fixed algebraically at eachinstant, while its time evolution is governed by the slow variation of the effective cosmological term. As a result, the cosmological evolution exhibits three distinct regimes: a strongly non–de Sitter early-time phase, a transient quasi–de Sitter regime, and an asymptotically coasting late-time expansion with w→-1/3 emerging dynamically as an attractor. These results position general relativity as an effective geometric description arising from a deeper,wavefunction-based structure, in which spacetime curvature, expansion, and cosmological dynamics are emergent properties of the underlying quantum geometry.