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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 régimes of Randall--Sundrum (RS) brane cosmology. The Wheeler--DeWitt (WdW) equation governing a Friedmann--Lemaître--Robertson--Walker brane embedded in a five-dimensional anti-de~Sitter (AdS\( _5 \)) bulk acquires two qualitatively new contributions---a quartic correction $\propto a^4/\ell_{RS}^2$ 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. Exploiting this unification, we address whether cyclic bounces can retain information across the quantum-gravity epoch. An entropy argument distinguishes---categorically and not merely in degree---compression within spacetime from compression of spacetime itself: classical observables (cosmic microwave background, primordial gravitational waves) are erased at the bounce because the geometric substrate that supports them ceases to exist, whereas the phase $\theta[h_{ij},\phi]=\arg(\Psi[h_{ij},\phi])$ of the universal wave function evolves unitarily through the bounce and carries all pre-geometric correlations forward. We define a fidelity $\mathcal{F}_{bounce}$ and a phase entropy $S_\theta$ on the phase distribution of $\Psi$ over superspace, prove their invariance under any unitary bounce operator, and contrast their behaviour with thermal and Weyl entropies, both of which are extinguished at the bounce. Indirect observational consequences---oscillatory non-Gaussianity with a specific momentum-phase signature and inter-mode entanglement absent from a Bunch--Davies vacuum---are identified as the only viable empirical probes of this quantum-coherent memory.