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

In a recently submitted companion paper, we demonstrated that the Painlevé-Gullstrand (PG) inflow v = 2GM/r is the unique self-consistent weak-field solution for the acoustic metric of a logarithmic superfluid vacuum, yielding the precise gravitational light bending parameter γ = 1.However, two central open problems remained: deriving this macroscopic flow from microscopic soliton dynamics, and resolving the steady-state continuity constraint.Here, we resolve both.We derive the emergent gravitational constant G eff = ηc 2 /(4πρ 0 ) and resolve the micro-to-macro disconnect by establishing the gravitational Feynman-Onsager relation: while a single soliton produces a steeply decaying Bondi flow (v ∝ r -2 ), N solitons collectively produce the PG profile through the condensate's Bernoulli response to the meanfield potential.The first-order density correction vanishes identically (ϵ 1 = 0), ensuring the vacuum around a gravitating mass is indistinguishable from free vacuum at the 10 -12 level.A Monte Carlo simulation with N = 5000 solitons confirms the Bernoulli velocity matches the PG profile to 0.02% accuracy.Finally, we resolve the continuity constraint by demonstrating that strict spherical symmetry must be relaxed; the macroscopic inflow necessitates axial polar outflows along the rotation axis, reframing astrophysical jets as a fundamental hydrodynamic requirement of emergent gravity.