A relativistic superfluid condensate on a Schwarzschild background: regular core, sonic point, and no-go constraints on back-reaction

We study matter infalling inside a black hole as a relativistic superfluid described by a complex scalar field in the Madelung formulation, with a Ginzburg–Landau-type potential on a fixed Schwarzschild background. The model yields exact flux conservation, a closed-form effective tension rate, and a regular condensate core despite the persistence of the background geometric singularity. The event horizon appears as a critical point of the stationary flow, where the infalling velocity reaches the speed of light and regular crossing requires a nonzero frequency. A distinct acoustic horizon is also identified for the collective phonon excitations, located just outside the event horizon and approaching it in the stiff-potential limit. The region between the two horizons is therefore supersonic but still outside the black hole, allowing high-frequency particle-like excitations while preventing phonon escape. We further establish several no-go results for gravitational back-reaction: the condensate cannot support a stationary throat, and no stationary finite-mass configuration with nonzero flux exists for a general potential under the stated conditions. The Mexican-hat potential therefore cannot provide a finite-mass stationary solution, while a massive potential permits a finite-mass branch without flux. The transverse-coherence analysis shows that the converging filament density necessarily leads to transverse merging for a broad class of coherence laws. Overall, the results define both the structural possibilities and the limitations of the model, including the fact that its weak-field galactic branch is not a universal positive correction. Negative results are retained as explicit constraints on the proposed framework.