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We propose a quantum multiverse model where universes are characterized by discrete energy levels ("tiers"), distinguished by quantum energy gaps and Hubble-scaled screening effects while sharing identical physical laws. Transitions between tiers, mediated by screened Yukawa interactions, drive cosmic evolution and generate distinct observational signatures. The model predicts a scale-invariant primordial power spectrum (n_s \approx 0.96) from inflationary transitions, high-frequency gravitational waves (\Omega_{GW} \sim\ {10}^{-15} at kHz) during reheating, and late-time phantom dark energy (w \approx -1.03) all of which are consistent with Planck and DESI 2024 data. The model provides a mechanism for baryogenesis via multiverse-mediated antimatter ejection, accounting for the observed matter-antimatter asymmetry (\eta \approx\ 6x{10}^{-10}), and unifies inflation, reheating, and dark energy through quantum transitions that preserve unitarity and the energy-time uncertainty principle. With testable predictions for CMB-S4, Einstein Telescope, and next-generation surveys, this work provides a compelling, falsifiable alternative to ΛCDM that bridges quantum mechanics and cosmology without fine-tuning.