The Static Density Channel Cannot Source Gravity: Four No-Go Theorems for the Logarithmic Superfluid Vacuum

Abstract

This version retracts and replaces the previous one at this DOI. The earlier version ("Frozen Vacuum Perturbations as a Dark Matter Analogue in the Logarithmic Superfluid Vacuum") proposed that frozen perturbations of the vacuum density act as a dark-matter analogue, supplying an intrinsic gravitational potential Φ_v = c²δ_v that fills two of cold dark matter's three roles, with lensing left open. That proposal was wrong, and this version proves it wrong. The error was in the acoustic metric. The intrinsic potential was read off the time-time metric component with the conformal prefactor left out. When the prefactor is derived properly — Ω² = ρ/c_s, fixed by the determinant identity with no freedom — the linear term cancels and there is no first-order potential at all. What remains at second order is six orders of magnitude too small and points the wrong way (repulsive for either sign of the perturbation). This version replaces the dark-matter claim with four no-go theorems that close the static density channel completely: a matter-induced response is spatially bounded and gives Keplerian rotation curves; no barotropic equation of state yields the general-relativistic light-bending parameter γ = 1; at the logarithmic model's operating point the static potential vanishes at first order and is repulsive at second; and static density wells are diverging, not converging, lenses — independent of the prefactor, so no metric choice can repair them. The one result from the earlier version that was correct — the screened-Poisson bound on induced response — is kept and appears here as Theorem 1. The conclusion is that gravity in this framework cannot be carried by static vacuum density. It must be carried by the macroscopic flow, which sharpens the direction of the rest of the program.