The Transverse Slicing Operator: First Principles Derivation of Fermion Generations (Part I of the ODD Series)

We construct the Transverse Slicing Operator ˆQ, the fundamental geometric generator of spectral confinement in the Operator-Derived Dimension (ODD) framework. Rather than postulating the operator, we derive it rigorously from the Kaluza-Klein reduction of the 5D Dirac equation in a warped bulk.
The resulting transverse Hamiltonian H = ˆQ† ˆQ corresponds to a shape-invariant modified P¨oschl-Teller potential. By enforcing the spinor double-cover topological constraint, the vacuum stability index is rigidly fixed to μ = 2. The eigenvalue spectrum natively produces exactly three physical modes: two localized bound states and one threshold resonance, providing a first-principles geometric explanation for the three Standard Model fermion generations. Furthermore, we explicitly connect the geometric width of the vacuum soliton to the quantum of action (ℏ), deriving foundational quantum mechanics directly from
confinement geometry.