The Transverse Slicing Operator: Geometric Emergence of SU(2) and SU(3) Gauge Symmetries (Part III.B of the ODD Series)

We present a fully derived, parameter-free formulation of the non-Abelian SU (2) and SU (3) gauge symmetries within the Operator-Derived Dimension (ODD) framework. By starting from the hyperspherical geometry of the vacuum and the normalized P¨oschl-Teller soliton zero-mode, we explicitly construct the covariant derivatives and derive the effective4D interaction Hamiltonian.We provide the rigorous calculus proving that the 4D coupling constant projects exactly from the 5D bulk charge due to the unit normalization of the soliton. We further demonstrate via integration by parts that geometric parity symmetry forces the vanishing of mixed derivative interaction terms. Finally, the Supersymmetric algebra is shown to be analytically invariant under gauge interactions, proving the absolute stability of the vacuum geometry.