Hadronic Spectroscopy from the Geometry of Singular Distributions: \break A First-Principles Calculation of the Mass Gap

We present a geometric derivation of the hadronic mass spectrum based on the theory of Singular Distributions on Tubular Neighborhoods. We demonstrate that the "Double Layer" distribution $\delta^{(1)}$, which arises rigorously at the Gribov Horizon, acts as an isotropic scalar stiffness. By virtue of transverse symmetry (Curie's Principle), this boundary acts as a projection operator, filtering out anisotropic modes for matter fields. By identifying the tubular thickness $\langle \lambda \rangle$ as the fundamental scale of confinement, we derive a spectral formula dependent on the zeros of Bessel functions. Calibrating the theory with Lattice QCD data for the scalar glueball ($0^{++}$), treated as a quadrupolar deformation of the geometry itself, we determine the tubular thickness to be $\langle \lambda \rangle \approx 0.586$ fm. Using this single parameter and the isotropic projection for vector fields ($J_0$), we predict the mass of the $\rho$ meson ($1^{--}$) to be $809$ MeV. The small deviation ($4.3\%$) from the experimental value ($775$ MeV) is interpreted as the signature of the finite elasticity and surface tension of the tubular boundary. Finally, we propose that this geometric stiffness provides a deterministic origin for the Pauli Exclusion Principle via the non-orientability of the tubular manifold.