Quantum Loops and Emergent Constants in the Operator-Derived Dimension (ODD) Framework

We present the ”First Four Quantum Loops” of the Operator-Derived Dimension (ODD) framework. Unlike perturbative loops in standard quantum field theory, these are topological integrals over the closed geometry of the 5th dimension. We demonstrate how the fundamental constants (ℏ, GN ), the fermion mass hierarchy, and the structure of the dark sector emerge naturally from these four geometric cycles. The loops are constructed from the eigenstructure of the Transverse Slicing Operator ˆQ acting on a sphere-on-sphere intersection topology.