Effective Thermodynamic Description of the Radion and Warp Factor in the Stratoverso Framework: Stirling Cycle, Lyapunov Stability, and Intrinsic Toroidal Breathing of the MSO Membranes

We develop a rigorous effective thermodynamic formulation of the Stratoversal cosmological framework, in which the radion — the inter-brane distance in a warped 5D Anti-de Sitter bulk — acts as a physical thermodynamic variable. Starting from the 5D Einstein-Hilbert action, we derive the warp factor W(r) = e−A(r), construct the Goldberger-Wise equation of state P(T,r), and demonstrate that the Stratoversal cosmological cycle maps onto a closed Stirling-like thermodynamic engine with a geometric efficiency of exactly 25%, dynamically fixed by the warp factor ratio W_IR/W_UV = 0.75. The scalar field freezing phenomenon is proved via two independent, fully ansatz-free Lyapunov arguments: for the friction-dominated regime (γ ≥ 2), Grönwall's inequality establishes φ̇ → 0 through the divergence of the Hubble integral; for the inertia-dominated regime (γ < 2, physical Stratoverso case with γ_eff = 1.661), LaSalle's invariance principle applied to the Coleman-Weinberg corrected potential identifies the frozen state as the unique dynamical attractor. A new section introduces the intrinsic toroidal breathing of the MSO membranes: the brane area expands isothermally from φ_∞ = 1.500 M_Pl to φ_equatore = 2.000 M_Pl and contracts back, with the warp factor e−A = 0.75 acting as the coefficient of thermal contraction of the toroidal membrane. This provides a unified geometric interpretation of the Stirling efficiency, the frozen field value, and the layer transition mechanism. All key parameters (W_IR = 0.75, λ = 0.300, η = 25%) are derived from the 5D bulk action without free parameters. The framework resolves the H₀ tension (H₀ = 72.4 ± 1.2 km/s/Mpc) and S₈ tension (S₈ = 0.778 ± 0.015). This is Paper P of the Stratoverso series.