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 — a complex scalar field in the Madelung formulation, with a Ginzburg–Landau-type potential, on a fixed Schwarzschild background. From first principles we obtain: flux conservation as a theorem of the model; a geometric criterion whereby anisotropy dominated by radial tension is possible only inside the horizon; the coefficient kappa_eff = 4/r + f'/f + rho'/rho in closed form, validated numerically within a declared parameter domain; and the identification of the horizon as the sonic point of the flow (an acoustic analogue in the sense of Unruh), whose regular crossing requires omega ≠ 0. The core turns out finite and regular, but its regularity leaves a dimensionless parameter R_c^2 free, whose irreducibility within the stationary model we establish along three independent routes. The heart of the work is a pair of no-go theorems on back-reaction: the condensate cannot support a throat (the null energy condition, always satisfied, is incompatible with the flare-out conditions), and it admits no finite-mass configurations with the symmetry-breaking potential (the energy diverges, and this is incompatible with the sonic condition omega ≠ 0). It follows that a double-lobed geometry does not emerge from this matter. The transverse-coherence analysis shows that the comoving density of the filaments obeys the universal kinematic law n = K x^(-3/2), independent of the potential: transverse merging is therefore inevitable for any coherence length xi ~ n^(-s) with s < 2/3, the potential controlling only its margin and radius. A massive potential with omega < m remains the only constructive route to a finite mass. The galactic weak-field branch is reported as closed in the negative. Negative results are presented with the same prominence as positive ones, as a rigorous delimitation of the model.