The Complementing Theories of Gravitational Dynamics and Resonant Structure: Conformal Cyclic Cosmology, Cosmic Algorithm Theory-12, and Ratio Theory

This paper develops a unified diagnostic framework that combines Conformal Cyclic Cosmology (CCC) and Cosmic Algorithm Theory-12 (CAT-12) to address complementary layers of physical reality: gravitational dynamics and structural resonance. CCC provides a deterministic, cyclic gravitational background through the covarying constants formulation and turn-off law (CCC+TL), fitting galactic rotation curves and outer power-law envelopes with minimal parameters and without invoking dark matter halos. CAT-12 overlays this dynamical background with a scale-free harmonic lattice, identifying low-integer PTR family ratios, Tu corridor widths, and resonance ladder placements across galactic, cosmological, and compact object domains. Through five representative figures, we show that CCC and CAT-12 are not redundant but complementary: CCC specifies how gravity behaves, while CAT-12 specifies where structure localizes. This dual framework enables two-axis falsifiability: the CCC axis tests dynamical fits, while the CAT-12 axis tests harmonic adherence. Applying this synthesis to galaxies (e.g., NGC 2775, NGC 2403, the Milky Way), CMB acoustic peaks, and black hole QPO data reveals recurring PTR(3:2), 2:1, and 5:3 families across eight orders of magnitude in scale. The paper concludes by positioning CCC×CAT-12 as a cross-scale diagnostic framework: CCC provides the dynamical backbone, CAT-12 supplies the structural tests, and together they enable rigorous, falsifiable cosmological analysis that can be applied to both modified gravity and ΛCDM contexts. This complementarity opens new pathways for testing aeon-to-aeon inheritance, harmonic persistence, and structural universality in the cosmos. Keywords Conformal Cyclic Cosmology (CCC); Cosmic Algorithm Theory-12 (CAT-12); Ratio Theory; Turn-off Law; Harmonic Profiles; PTR Families; Galactic Dynamics; Rotation Curves; Cross-Domain Resonance; Aeonic Boundaries; Modified Gravity; Cosmology; CMB; Black Hole QPOs; Resonance Lattice; Diagnostic Framework.