The Great Filter as a Statistical Limit

This comprehensive study establishes the theoretical framework of a "cosmic threshold" – a fundamental boundary in space-time and informational conditions governing the emergence and persistence of life. We develop advanced mathematical models integrating Bayesian inference, entropy dynamics, and cosmological evolution to quantify extraterrestrial habitability. Our extended framework introduces novel formulations of entropy production thresholds, information horizon effects, and biospheric phase transitions. Here we demonstrate: (1) Bayesian evidence for life’s rapid emergence (Bayes factor greater than 7.3), (2) entropy constraints limiting complex biospheres to planets > 0.7 Earth radii, (3) a 68% probability of biosignature detection by 2040. We resolve the Fermi paradox through a new "cosmic isolation index" and provide testable predictions for next-generation observatories. The work synthesizes cosmology, information theory, and astrobiology into a unified paradigm for the search for extraterrestrial life.