This working paper derives the oscillation frequency ωₙ 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 ωᵦ=11. 1991. The temporal frequency uses a convention-based local mapping at zₘid=0. 8: ωₙ = ωᵦ·H (zₘid) · (1+zₘid), yielding Tₙ=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
Vázquez González José Luis (Thu,) studied this question.