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

Abstract

This is a clarity revision. The physics, the derivation, and the result — the Planck length emerging as the condensate healing length by self-consistency — are unchanged. The main change tightens one claim that was overstated. The earlier version said dimensional analysis alone uniquely fixes the form of the gravitational coupling G ~ c²/(ξ²ρ₀). That isn't quite right: three dimensional quantities cannot fix a unique form by dimensions alone. The form actually follows from the linear-response structure of the condensate — G linear in the compliance 1/ρ₀, with the response range ξ entering squared through inversion of the linearized kernel — together with dimensional analysis. The revised text says this properly, and now points to the gapped amplitude dispersion of the companion paper as the origin of the ξ² factor. The result is the same; the reasoning is now honest about what carries it. I also clarified the normalization of the quantum potential (per-particle versus per-unit-mass), and adjusted the phrasing around the local phase rate μ(x) = m_eff c_s²/ℏ to state clearly that it is the chemical potential in frequency units, which is what sets the clock rate.