Radion Thermodynamics and Dynamical Freezing in 5D Warped Extra-Dimensional Cosmology

We develop a mathematically rigorous thermodynamic formulation of a warped extra-dimensional 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 coupled to bulk fields, we derive the functional dependence of the warp factor W (r) = e^−A (r) on the extra-dimensional stress-energy tensor and show that the geometric temperature Tgeom (r) emerges naturally from the Euclidean cone regularization of the warped bulk horizon. We construct a complete Goldberger-Wise equation of state P (T, r) for the radion and demonstrate that the cosmological cycle maps onto a closed Stirling-like thermodynamic engine operating on the extra-dimensional volume, with four distinct thermodynamic phases formally identified. The scalar field freezing phenomenon is proved via two independent, fully ansatz-free Lyapunov arguments: (I) for the friction-dominated regime (γ ≥ 2), Grönwall's inequality applied to the kinetic energy Lyapunov function establishes φ̇ → 0 through the divergence of the Hubble integral; (II) for the inertia-dominated regime (γ < 2), LaSalle's invariance principle applied to the total energy Lyapunov function H (t) = ½φ̇² + W²IR U (φ) identifies the frozen state as the unique dynamical attractor via the Coleman-Weinberg corrected effective potential. For the benchmark physical parameters (γₑff = 1. 661, WIR ≈ 0. 75), the frozen field value φf ≈ 1. 500 MPl is confirmed as a stable global attractor. The framework quantitatively resolves the H₀ tension (H₀ = 72. 4 ± 1. 2 km/s/Mpc) and the S₈ tension (S₈ = 0. 778 ± 0. 015) through the radion-warped modulation of the effective Newton constant.