Frozen Vacuum Perturbations as a Dark Matter Analogue in the Logarithmic Superfluid Vacuum

In the standard ΛCDM model, cold dark matter serves three essential cosmological roles: providing non-baryonic gravitational potential wells, maintaining those wells through the radiation era, and producing the gravitational lensing offsets seen in cluster mergers. This paper shows that frozen perturbations of the vacuum condensate density fulfil the first two roles within the logarithmic superfluid vacuum framework, while the third — gravitational lensing — is identified as an open problem rather than a solved one. The argument has two parts. First, a no-go theorem: dark-matter phenomenology cannot arise as the static vacuum response to a baryonic source. The direct Madelung–Bohm calculation gives hydrostatic compression governed by a screened Poisson equation; the response is spatially confined to the healing length and yields Keplerian rather than flat rotation curves, so the induced response cannot masquerade as the primordial wells. Second, the actual mechanism: frozen primordial perturbations of the condensate, set during its formation. The equation of state w = −1, derived in companion papers, forces ρᵥ ∝ a⁰ and freezes the perturbation evolution — vacuum density perturbations neither grow nor decay. These structural variations in ρ₀ (x) create an intrinsic vacuum gravitational potential Φᵥ (x) = c²δᵥ (x) through the density dependence of the acoustic metric: permanent wells that, unlike matter-sourced potentials decaying as a⁻³, persist at constant depth through all epochs. Baryons oscillate in these frozen wells exactly as they would in CDM wells, producing the observed odd/even acoustic-peak asymmetry of the CMB. An inverse-problem analysis of 112 SPARC galaxies shows the required depletion amplitudes are microscopic — |δᵥ| of order 10⁻⁷ to 10⁻⁶, some five orders of magnitude below the condensate stability limit — amplified to galactic effect by the c² factor. Gravitational lensing is treated as the framework's open problem: by the conformal flatness of the static acoustic metric established in companion work, a static vacuum template modulates the orbits of massive tracers but does not by itself deflect light; reproducing cluster-merger lensing offsets such as the Bullet Cluster would require either a residual flow associated with the template or a distinct lensing channel, neither of which is currently available — a falsifiable position stated as such. Two distinguishing predictions are identified: a modified integrated Sachs–Wolfe effect and a different structure-growth history.