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 from the brane self-energy and a dark-radiation term 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 of the universal wave function evolves unitarily through the bounce and carries all pre-geometric correlations forward. We define a fidelity and a phase entropy on the phase distribution of 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.