This paper introduces the Cyclic Cosmology with Informational Inheritance (CCHI), a theoretical framework proposing that the universe undergoes repeated cycles of expansion and contraction mediated by a quantum bounce, during which statistical information about the pre-bounce matter distribution survives into subsequent cycles, encoded in the post-bounce perturbation spectrum. Built upon Loop Quantum Cosmology (LQC), the framework presents three original contributions: (i) a master equation governing the cyclic evolution of the primordial power spectrum via Bogoliubov transfer coefficients, which admits a stabilized fixed point whose resonant peaks nucleate Primordial Black Holes (PBHs); (ii) a derivation of Entropic Time Dynamics (ETD) from the effective LQC metric, identifying the chrono-entropic coupling Γ(ρ) = 1 − ρ/ρ_c = g_00^(eff) with no free parameters; and (iii) a unified mechanism connecting gravitational baryogenesis, PBH dark matter in the asteroidal mass window, and the seeding of supermassive black holes observed by JWST at z > 10. The framework yields falsifiable predictions for CMB-S4, LiteBIRD, the Roman Space Telescope, and LVK O4/O5, and addresses the cyclic entropy problem through the distinction between thermodynamic entropy (diluted each cycle) and entanglement entropy (transferred as low-entropy correlations).