Counting Cosmic Cycles: Past Big Crunches, Future Recurrence Limits, and the Age of the Quantum Memory Matrix Universe

We develop a quantitative descriptive theory of contraction-expansion cycles that constitute the Quantum Memory Matrix (QMM) cosmology. In QMM, each non-singular advance adds a fixed imprint-entropy increment Delta_S_imp to a finite Hilbert-space ledger, providing an intrinsic cycle counter. By calibrating the geometry-information duality, inferring today's cumulative imprint from CMB, BAO, chronometer, and large-scale-structure constraints, and integrating the modified Friedmann ODEs with imprint back-reaction, we find that the Universe has already completed N_past = 3.6 (+- 0.4) cycles. Propagating the holographic write-rate and enforcing capacity saturation (with instability priors) implies N_future = 7.8 (+- 1.6) additional cycles. Integrating Kodama-vector proper time across all past bounces yields a total cumulative age t_QMM = 62.0 (+- 2.5) Gyr, of which 13.8 (+- 0.2) Gyr is the current phase usually labeled LambdaCDM. The framework predicts testable signatures: an enhanced faint-end UV galaxy luminosity function at redshift z >~ 12 (JWST), a stochastic gravitational-wave background with f^(2/3) scaling in the LISA band from primordial-black-hole mergers, and a nanohertz background with slope alpha approx 2/3 accessible to pulsar-timing arrays. These observations can confirm, refine, or falsify the cyclical QMM chronology with next-generation CMB and gravitational-wave probes.