The Stratoverso Connection: Unifying Holographic Scale-Space Structure and Scalar Field Dynamics

We present a rigorous mathematical and physical unification of the two primary representations of the Stratoverso cosmology: the holographic scale-space representation (Berti 2026, doi:10.5281/zenodo.18979384) and the temporal scalar field dynamical representation (Berti 2026, doi:10.5281/zenodo.20514118). By deriving the exact physical mapping between the logarithmic scale coordinate χ and cosmic time τ ≡ t* − t, we show that χ = L + (γ/2) ln(τ/τ₁₃₇), where L ≈ 140 is the total scale-space length. We demonstrate that the physical spacing of the Holographic Spherical Membranes (MSO) is δχ = ln(1/λ) ≈ 1.204, and that the observed temporal scaling factor λ ≈ 0.30 is reproduced by the effective critical exponent γ_eff = 1.661, incorporating one-loop Coleman-Weinberg corrections and gravitational backreaction, via the relation ln(1/λ) = 2/γ_eff. This yields the log-periodic frequency ω_g = π γ_eff ≈ 5.22. We establish the generating function Z[s] as the Mellin transform of the discrete MSO density, resolving the pole structure at Im(s) = n ω_g. We derive the scalar field freezing limit φ_∞ ≈ 1.500 M_Pl from the Klein-Gordon slow-roll condition and map it to the equatorial radius of the toroidal topological transition S³ → S¹ × S². A unified parameter table and a figure of the scale-time mapping complete the mathematical dictionary of the Stratoverso, establishing it as a highly predictive and falsifiable paradigm for cyclic cosmology.