Singularity Regularization and the Emergent Planck Scale in the Logarithmic Superfluid Vacuum

General relativity predicts spacetime singularities where curvature invariants diverge.We show that the logarithmic superfluid vacuum framework provides a natural regularization mechanism through the quantum potential (Bohm potential) Q = □ √ ρ/ √ ρ, which was neglected at macroscopic scales in companion papers but dominates at the soliton coherence scale.The healing length ξ = ℏ eff /(m eff c) of the condensate sets a minimum spatial resolution below which the hydrodynamic description breaks down and quantum pressure prevents gravitational collapse to a point.We derive ξ from the parameters fixed in the companion papers and show that self-consistency of the framework-requiring the Gausson soliton energy to reproduce Newton's constant via G = ηc 2 /(4πρ 0 )-uniquely determines ξ = ℏG/c 3 = ℓ P , the Planck length.The Planck scale thus emerges as the healing length of the vacuum condensate, not as a fundamental discreteness of spacetime.The temporal singularity is simultaneously resolved: the chemical potential µ(x) = m eff c 2 s /ℏ eff , which governs the local clock rate, is bounded by the same mechanism, ensuring that gravitational time dilation is always finite and the phase of the condensate remains well-defined everywhere.The corresponding cutoffs ∆x min ∼ ℓ P and ∆t min ∼ t P are related by c, respecting the emergent Lorentz invariance they regularize.These identifications unify ℏ, c, and G as properties of a single physical substrate: the speed of sound, the quantum coherence scale, and the emergent coupling of the vacuum condensate.