The Infinite Memory of the Universe: Effective Horizons Beyond 13.8 Billion Years in a Cyclical Non-Markovian Cosmology

This work defines an effective memory horizon \( T_{\rm mem} \) for a class of non-Markovian cosmologies based on the Infinite Transformation Principle (ITP), and shows how \( T_{\rm mem} \) can be inferred from late time expansion and growth data. The key ingredient is a causal kernel that ties the present Hubble rate to the integrated history of internal energy and structure. From this kernel one can extract a characteristic timescale over which past states remain dynamically relevant. A combination of \( H(z) \) measurements, large scale structure growth constraints, and two bin ITP fits is used to constrain the delay parameter that controls the memory kernel and to map it into \( T_{\rm mem} \). In representative fits the present universe behaves as if its dynamics retains memory over effective timescales of order \( T_{\rm mem}\sim 50 \)–\( 80 \)~Gyr, several times larger than the standard \( 13.8 \)~Gyr age, without implying a literal birth time at \( t=-T_{\rm mem} \). The memory horizon is best interpreted as an operational measure of how far back the present expansion is correlated with its own past, not as a revised estimate of the age of the universe. Robustness tests with different kernel families and priors on the delay parameter show that current \( H(z)+f\sigma_8(z) \) data strongly disfavour short delay, effectively Markovian behaviour and favour a long memory regime with a conservative lower bound corresponding to several Hubble times. The picture that emerges is compatible with cyclical cosmologies in which late-time observables can carry accumulated memory from earlier phases, while remaining consistent with the empirical success of \( \Lambda \)CDM at low redshift.