COCM Bridge 3: Effective orbital action for C₀ and the mode frequency ω_n

This working paper derives the oscillation frequency ω_n of the Concentric Orbital Cosmological Model (COCM) from an effective DBI action for the cosmological brane C₀, complementing the quantitative analysis of Vázquez González (2026e) (DOI: 10.5281/zenodo.20171261). The derivation proceeds in four steps: (1) DBI action with constant tension τ and orbital degree of freedom θ(t); (2) circular-orbit background solution; (3) linearisation with a phenomenological phase-coupling potential V(θ); (4) mode spectrum and identification of ω_z=11.1991. The temporal frequency uses a convention-based local mapping at z_mid=0.8: ω_n = ω_z·H(z_mid)·(1+z_mid), yielding T_n=2.861 Gyr as a local effective period. The key result is a consistency condition within the effective theory, not yet a first-principles prediction. The missing piece — the microscopic derivation of V(θ) from the bulk geometry of C₀ — is explicitly identified as the single remaining open problem. Status: algebraically complete, physically incomplete. Related works:P5 v6: DOI 10.5281/zenodo.20171261NT2: DOI 10.5281/zenodo.20173339P1: DOI 10.5281/zenodo.19950582