Derivation of the Transverse Slicing Operator: The 5D Action Principle and Spectral Stability in the ODD Framework

We present Slicer Theory, a non-perturbative five-dimensional field theory in which the observable universe emerges as a dynamically localized 4D slice. Particle masses arise as exact eigenvalues of a transverse geometric operator derived fromthe fundamental 5D action. We prove that the variational stability of the vacuum uniquely fixes the topological index to μ = 2, generating exactly three localized fermion generations and a parity-protected dark sector. The theory naturally reproduces the Standard Model mass hierarchy and provides a falsifiable prediction for a dark sector resonance at 30.2 MeV.