Effective Thermodynamic Description of the Radion and Warp Factor in Warped Extra-Dimensional Cosmology: Stirling Cycle, Lyapunov Stability, and Intrinsic Toroidal Breathing of Holographic Boundary Membranes

We develop a rigorous effective thermodynamic formulation of a discrete 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, we derive the warp factor W (r) = e^−A (r), construct the Goldberger-Wise equation of state P (T, r), and demonstrate that the cosmological cycle maps onto a closed Stirling-like thermodynamic engine with a geometric efficiency of exactly 25%, dynamically fixed by the warp factor ratio WIR/WUV = 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), LaSalle's invariance principle applied to the Coleman-Weinberg corrected potential identifies the frozen state as the unique dynamical attractor. The intrinsic toroidal breathing of the holographic boundary membranes is developed in full: the brane area expands isothermally from φ_∞ = 1. 500 MPl to φₑquatore = 2. 000 MPl and contracts back, with the warp factor e^−A = 0. 75 acting as the thermoelastic contraction coefficient of the membrane. This provides a unified geometric interpretation connecting 5D bulk geometry to the 4D scalar field dynamics. This version extends the framework with a string-theory embedding of the effective radion potential, identifying the canonical scalar field with the Kähler volume modulus of a Type IIB Calabi–Yau compactification and recovering the Goldberger-Wise potential as the large-volume limit of the flux-stabilized moduli potential. A full appendix derivation of the effective Goldberger-Wise radion potential directly from the 5D bulk-boundary action is also included, together with a holomorphic (complex-time) representation of the semiclassical waiting-time of the monodromic layer transition, giving the transition duration as a contour integral with explicit power-law dependence on the microscopic parameters gₕolo and λᵣeset. All key parameters (WIR = 0. 75, λ = 0. 300, η = 25%) are derived from the 5D bulk action without free parameters. The MCMC-constrained joint analysis (Planck + DESI DR2 + Pantheon+) yields H₀ = 67. 72 ± 0. 48 km/s/Mpc, corresponding to a partial mitigation of the Hubble tension from 5. 6σ to approximately 4. 2σ, consistent with recent no-go constraints on late-time dark energy transitions (Pedrotti et al. 2026). The framework also addresses the S₈ tension (S₈ = 0. 778 ± 0. 015). Stessa correzione fatta sulla versione branded: rimosso il claim "resolves" con 72. 4, sostituito con il dato MCMC reale e onesto.