ODD Proof Series 2026-CP-01: Derivation of the Coulomb Potential and Validation of ODD-Derived Constants in Canonical Scattering Experiments

The Operator-Derived Dimension (ODD) framework provides a geometric origin for both the laws of interaction and the fundamental constants of nature. In this work, we first present a rigorous, first-principles derivation of the 1/r Coulomb potential. We demonstrate that this potential emerges as the unique, long-range static interaction mediated by the zero-mode of a 5D gauge field, whose Kaluza-Klein decomposition is dictated by the eigenstates of the Transverse Projection Operator. Having derived the form of the interaction, we then test the ODD-derived values for the electron charge e and mass me by using them as inputs to the standard, experimentally-verified formulas for three canonical scattering experiments: Rutherford, Mott, and Thomson. All calculations use measured experimental parameters and the ODD-derived constants without placeholders. The resulting numerical predictions show excellent agreement with experimental data, thereby providing a powerful, two-fold validation of the ODD framework’s core tenets: its ability to derive both the force laws and the physical constants that govern them.