Exact Correspondences Between the Hartle–Hawking and Vilenkin Boundary Conditions Under Randall-Sundrum Brane Dynamics

Through this work, we deduce a model where the Hartle--Hawking no-boundary proposal and Vilenkin's tunneling boundary condition are placed in precise structural correspondence with distinct dynamical r\'{e}gimes of Randall--Sundrum (RS) brane cosmology. Starting from the five-dimensional ADM Hamiltonian, we derive the effective brane action via the junction conditions and Gauss-Codazzi reduction, then quantize it canonically to obtain the RS-modified Wheeler--DeWitt (WdW) equation governing a Friedmann--Lema\^{\i}tre--Robertson--Walker brane embedded in an AdS\(_5\) bulk. Two qualitatively new contributions arise---a quartic correction \(\propto\rho^2/\lambda\) from the brane self-energy and a dark-radiation term \(\propto\mu/a^2\) from the Weyl projection of the bulk Riemann tensor which deform the classical turning point and alter the WKB tunneling exponent in a calculable, parameter-dependent fashion.We demonstrate that the no-boundary Euclidean saddle of the RS path integral maps to a compact Euclidean AdS\(_5\) geometry carrying a closed brane, while Vilenkin's outgoing-mode condition translates into brane nucleation from the bulk; the AdS/CFT correspondence then recasts the cosmological wave function as the partition function of a four-dimensional conformal field theory with a UV cutoff set by the brane tension. We relate our construction to earlier work on quantum cosmology in braneworld scenarios and clarify three contributions that extend and complement the earlier treatments of Hawking-Hertog-Reall and Koyama-Soda correspondence (Euclidean path integral, WKB, AdS/CFT), the definition and invariance of the phase entropy \(S_\theta\), and the phase fidelity \(\mathcal{F}_{\mathrm{bounce}}\) as a quantitative measure of quantum information preservation across cosmological bounces. Indirect observational consequences oscillatory non-Gaussianity with a specific momentum-phase signature and inter-mode entanglement extend from a Bunch Davies vacuum are identified as the only viable empirical probes of this quantum-coherent memory.