Emergent Newtonian Gravity from the Collective Acoustic Metric of a Logarithmic Superfluid Vacuum

The companion paper showed that a logarithmic superfluid vacuum requires a Painlevé–Gullstrand (PG) flow to regularise the static-metric pathology and recover the PPN parameter γ = 1. This paper addresses the open question that left behind: where the macroscopic flow comes from. Modelling matter as N vortex-Gausson solitons — the topological defects of the condensate — and coarse-graining their collective effect, the analysis works within a mean-field treatment rather than a full derivation. Two distinct densities are kept carefully apart: the order-parameter amplitude |ψ|² and the physical mass density ρ_mass = T₀₀/c², related by a dimensionful map rather than a numerical factor. The self-consistency requirement of the acoustic metric is shown to select the PG flow over the static configuration, which is mathematically ill-defined at the background equation-of-state exponent α = −1; the PG profile is the unique irrotational weak-field solution carrying a Newtonian potential, with vortical and multi-fluid configurations left open. Postulating that each soliton sources a localised perturbation of the acoustic metric (an ansatz whose first-principles derivation is identified as the central open problem), the emergent gravitational constant follows as G_eff ∼ c²/(ξ²ρ₀), universal and independent of source mass and distance; matched to Newton's constant with the natural condensate packing density, this fixes the healing length to the Planck length and the vacuum density to the Planck density. An independent Bernoulli derivation confirms that Newtonian gravity requires exactly the PG velocity profile v ∝ r^(−1/2), and that vortex or dipole flows give the wrong force law. The equation-of-state result w = −1 is obtained as the exact, coupling-independent ratio P/ε at the potential extremum, with a radiation-gas cross-check. A numerical study verifies the vortex-Gausson building block and the dynamical stability of the PG configuration under perturbation, with a grid-resolution and boundary-sensitivity convergence check; the simulation tests stability of an imposed profile, not spontaneous emergence. The continuity constraint of the PG inflow — its non-zero divergence in steady state — is discussed as an open problem with three candidate mechanisms, and the paper notes that a complete resolution may require revisiting the single-field assumption itself.