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 T_geom(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 + W²_IR U(φ) identifies the frozen state as the unique dynamical attractor via the Coleman-Weinberg corrected effective potential. For the benchmark physical parameters (γ_eff = 1.661, W_IR ≈ 0.75), the frozen field value φ_f ≈ 1.500 M_Pl 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.