A Priori Sun-Wang-Zhang Regularity, Conormal Spectral Mass Gaps, and Multi-Scale Suction Scaling in a Stiffened Compressible Superfluid

We prove four interconnected analytical results governing wave confinement, global regularity, and multi-scale boundary-layer scaling in a three-dimensional ($D = 3$) compressible Navier-Stokes fluid obeying an excluded-volume stiffened equation of state $P(\rho) = P_0 + \beta_s[\rho/(\rho_{\max} - \rho)]^\gamma$ ($\gamma > 1$) with shear viscosity bounded below by the Kovtun-Son-Starinets (KSS) inequality $\eta/s \ge \hbar/(4\pi k_B) > 0$ at an ambient thermal floor $T_0 > 0$. First, exact inversion of the barotropic enthalpy integral proves that point-mass density singularities and vacuum cavitation states are impossible for any finite potential depth $\Delta\Phi(\mathbf{x}) \ge 0$, bounding the density strictly within $0 < \rho_0 \le \rho(\mathbf{x}, t) < \rho_{\max} < \infty$ and satisfying the Sun-Wang-Zhang continuation criterion for smooth three-dimensional compressible Navier-Stokes flows. Second, as $\rho \to \rho_{\max}$, the local acoustic impedance diverges ($Z(\rho) = \sqrt{\rho K(\rho)} \to \infty$), driving the normal reflection coefficient to unity ($\Gamma \to 1$) and enforcing a conormal Neumann zero-flux boundary condition ($\lim_{\mathbf{x}\to\partial\Omega} Z(\mathbf{x})\nabla\psi\cdot\mathbf{n} = 0$) that renders the compressibility-weighted Sturm-Liouville operator strictly self-adjoint with a positive non-axisymmetric spectral mass gap ($k_0 r_0 = 1.25086 > 0$) while topologically isolating the interior saturation energy density ($\rho_E = \rho_{\max}c_0^2 \approx 8.735 \times 10^{29}\text{ J/m}^3$) from the exterior bulk pressure ($I_{\mathrm{trans}} = 0$). Third, we prove that the Prandtl Asymptotic Suction Boundary Layer (ASBL) displacement thickness ($\delta^* = \nu_A / v_s$) defines a universal Suction Reynolds Number $Re_s(L) \equiv v_s L / \nu_A = L / \delta^*$ that connects the subatomic Spindle Torus impedance ($W_\phi = R_0/r_0 = \phi^{-2}$, $\delta^*/r_e = 0.003440$, $\alpha_{\mathrm{dil}}^{-1} = 180[3-\sqrt{5}] - \sqrt{2}/3 \approx 137.03636$) to the macroscopic critical suction fixed point $Re_s(R_c) = 1$, where axisymmetric galactic disks exhibit exact flat rotation ($v_\theta(R) = V_0 = \mathrm{const}$). Fourth, matching the cyclostrophic pressure gradient $\rho^{-1}dP_{\mathrm{vortex}}/dR = V_0^2/R_c$ to Newtonian baryonic gravity at $R_c$ across the centripetal Spindle Torus acceleration scale $a_0 = \frac{c_0\mathcal{H}_{\mathrm{cmb}}}{2\pi}(1 + \phi^{-4}) = 1.194 \times 10^{-10}\text{ m/s}^2$ derives the Baryonic Tully-Fisher relation ($V_0^4 = G M_{\mathrm{baryon}}a_0$), while meridian continuity across the Spindle Torus derives the local-to-baseline Hubble velocity divergence $\mathcal{H}_{\mathrm{local}} = \mathcal{H}_{\mathrm{cmb}}(1 + \frac{1}{2}\phi^{-4})(1 + \frac{7}{3}\delta^*/r_e) = 72.90\text{ km s}^{-1}\text{ Mpc}^{-1}$ with zero adjustable parameters.