Exact Non-Singular $\Gamma \to 1$ Acoustic Horizons and Poloidal-to-Radial Eigenmode Transitions in a Compressible Spindle Torus

We prove two mathematical results governing wave confinement in a compressible fluid with an excluded-volume maximum density threshold $\rho_{\max}$. First, for a barotropic fluid obeying the stiffened equation of state $P(\rho) = P_0 + \beta_s[\rho/(\rho_{\max} - \rho)]^\gamma$ ($\gamma > 1$) in an attractive potential $\Phi(R)$ that diverges as $R \to 0$, exact analytical inversion of the Euler hydrostatic enthalpy integral proves that point-mass density singularities are impossible: for $\gamma = 2$, the density saturates smoothly in closed form as $\rho(R)/\rho_{\max} = 1 - [1 + 2\tilde{\Phi}(R)]^{-1/2} < 1$, while the local bulk modulus $K(\rho) = \rho\,dP/d\rho$ and acoustic impedance $Z(\rho) = \sqrt{\rho K(\rho)}$ diverge as $[1 + 2\tilde{\Phi}(R)]^{3/4} \to \infty$. Consequently, the acoustic reflection coefficient approaches strict unity ($\lim_{R\to 0}\Gamma(R) = 1$), dynamically enforcing a self-forming homogeneous Neumann boundary condition ($\nabla\Psi \cdot \mathbf{n} = 0$). Second, solving the exact three-dimensional cylindrical Helmholtz equation via conservative finite-volume discretization inside the non-convex exterior crescent domain of a self-intersecting Spindle Torus ($W = R_0 / r_0 = \phi^{-2} \approx 0.3820$) bounded by $\Gamma = 1$ reveals a topological mode-branch transition: the cavity supports exactly three excited poloidal arc harmonics ($n = 1, 2, 3$) before the fourth mode ($n = 4, k_4/k_0 = 3.9866 \approx 4$) switches orthogonally into the first radial shell excitation. Fourth-order Runge--Kutta integration matches the analytical singularity-arrest theorem to $2.75 \times 10^{-12}$.