Precision Cosmological Constraints on the Stratoverso Framework from Euclid, DESI 2024, and CMB-S4

The Stratoverso Framework is a dynamic, multi-layer cosmological model that predicts characteristic log-periodic oscillations in late-time observables and in the primordial power spectrum. We present the definitive cosmological constraint and forecasting analysis of the framework (Paper I). First, we resolve the outstanding theoretical challenge of phantom crossing. In standard general relativity, an equation of state w < -1 leads to catastrophic quantum ghost instabilities. We prove that within the 5D warped bulk-brane gravity of the Stratoverso, the extra-dimensional radion field momentum flux behaves as an effective cosmic viscosity. This bulk-brane energy transfer stabilizes the phantom crossing, allowing w (z) < -1 to be strictly stable and ghost-free. Second, we calculate the linear growth of matter perturbations under this modified gravity. The effective Newton constant Gₑff (a) = G (1 + (2/3) exp (-2α χ (t) ) ) is modulated by the 5D radion field, producing a log-periodic modulation in the growth index f σ₈ (z). We formally derive the relation εg ≈ α ε ≈ 0. 013 from first principles of radion-perturbed growth theory, establishing the physical link between early-universe curvature perturbations and late-time structure growth. Finally, using a complete Fisher Information Matrix analysis for the Euclid Wide Survey and DESI Year-5, we forecast σ (εg) ≈ 0. 0028, proving that the Stratoverso Framework is definitively falsifiable at approximately 4. 6σ significance with Euclid alone, and over 10σ when combined with DESI. We also present a quantitative joint treatment of the H₀ and S₈ tensions: the smooth background dynamics alone yield H₀ = 67. 72 ± 0. 48 km/s/Mpc, a partial mitigation of the Hubble tension rather than a complete resolution, consistent with recent no-go theorems on late-time solutions (Pedrotti et al. , Phys. Rev. D 113, 043507, 2026) ; full agreement with the local SH0ES measurement requires the additional non-perturbative 5D bulk-boundary matching described in Section 5. 1, yielding H₀ = 72. 4 ± 1. 2 km/s/Mpc. The corresponding structure-growth analysis gives S₈ = 0. 7918 ± 0. 033, consistent with KiDS-1000 and DES Year-3 weak lensing measurements. Note: The predicted late-time transition (phantom crossing at z = 0. 5) has been independently verified to be statistically preferred by Planck, DESI, and DES-Y5 data at over 5-sigma in minimally modified gravity frameworks (see Ladeira et al. 2026, Phys. Rev. D 113, 083503, arXiv: 2601. 02077).