Dynamical Selection of the Non-Lipschitz Critical Branch in the Stratoverso Framework

The Stratoverso Framework establishes the existence of a non-Lipschitz critical branch φc (t) ~ (t* - t) ⁰. 45. We present the first systematic numerical investigation of whether this branch is dynamically reachable, using the physically correct divergent Hubble parameter H (t) ~ (t* - t) ^-γ/2. Three definitive results emerge. (1) No power-law solution satisfies the full Klein-Gordon equation with divergent H: the residual diverges for all tested exponents, including the dominant-balance prediction αB ≈ 0. 317. (2) αB reduces the Klein-Gordon residual by up to 93× compared to αA near t*, correctly identifying the dominant-balance regime, but the acceleration term remains divergent. (3) The natural asymptotic behavior is field freezing: φ (t) → φ∞ = const ≈ 1. 86 as t → t*, driven by infinite Hubble damping. This is a new result with no analogue in standard quintessence models. This result clarifies the domain of validity of the critical solution and motivates a refined interpretation of the layered boundary conditions: the frozen field φ∞ constitutes the physical state of Layer 137 at the boundary node. A universal fixed point φ∞∞ ≈ 1. 500 is discovered for inner layers, confirmed by numerical simulation of the full 140-layer chain under Resolution C. Empirical scaling laws Hcrit ≈ H₀ and φ∞ ∝ τₖ⁰. 125 are established.