NT22: Derivation of the Fundamental Mode from the DBI Action

This note derives as the fundamental oscillatory mode of the COCM ring trajectory from the DBI action, closing the Bridge 3 advanced step that presumes without derivation. The derivation proceeds in two steps: (1) the linearised equation of motion for mode yields ; (2) minimisation of the effective equilibrium energy at fixed amplitude over positive integers shows for all , making the unique minimum. The result is Closed* under the DBI action, the potential ansatz , and fixed amplitude . The selection of is conditional on the ansatz: it derives from the kinetic structure of DBI combined with the scaling of the potential curvature, not from DBI alone.