Elliptic Orbital Geometry in the COCM: Second Harmonic as an Independent Observable, CMBDR3 Connection via J, the COCM+w0wa Coexistence Scenario, and the Cosmic String Loop Hypothesis for Rn

The Concentric Orbital Cosmological Model (COCM) parametrises the Hubble rate as an oscillatory modulation at xed frequency ωz = 11.1991 in redshift space, treating the orbital geometry as circular (e = 0). This paper develops four extensions of that baseline, all traceable to results in the corpus. (1) Second harmonic. The circular orbit is the zeroth-order xed point of any smooth eccentricity distribution. For e>0, a second harmonic at 2ωz appears with amplitude ratio P(2ωz )/P(ωz ) ∝en, n ≈4.86. DESI DR3 can constrain e independently of A0 and φ. (2) CMBDR3 connection. The invariant J = (V0ε/τ)2 bridges the primordial sector (z ∼1090) and the late-time sector (z ∼1) through the bulk geometry of C0. A non-zero e implies JCMB ̸= JDR3, providing the rst direct measurement of bulk dynamics across cosmic time. (3) COCM+w0wa coexistence. The three possible DR3 outcomes are identied and formalised: COCM+w0wa coexist (Scenario 1, currently favoured, χ2 r = 4.7), COCM only (Scenario 2), and neither (Scenario 3, documented falsication). The w0wa signal is explained as the O(e2) correction to the circular-orbit baseline: the DESI DR2 w0wa detection is interpreted as an incomplete sampling of an oscillatory signal, not as a smooth dark energy component. (4) Cosmic string loop hypothesis. Rn is identied as a candidate cosmic string loop, consistent with NANOGrav constraints (Gµ∼10−11) and naturally satisfying τC0 ≫τRn . This is a working hypothesis, not a derived result. All results carry explicit claim labels. The primary prediction ωz = 11.1991, registered on 14 May 2026 (DOI: 10.5281/zenodo.20178768), is unaected.