Relativistic Informational Gravity: Field Equations, Time Emergence, and Cosmological Consistency

We propose a thermodynamic framework in which gravity and time emergence arise from a fundamental informational energy scale η. This parameter represents the minimal energy required to localize nonlocal quantum correlations into classical spacetime structure. We construct a covariant scalar–tensor theory where η acts as an infrared cutoff scale. The resulting field equations reproduce General Relativity in the strong-field regime and MOND-like dynamics in the weak-field limit. We further show that η links the galactic acceleration scale and the cosmological constant, and leads to a modified description of time as an informational throughput. A redshift-dependent correction F(z) is derived, including a numerical estimate providing a falsifiable prediction. The framework is mathematically consistent, stable, and provides a unified infrared description connecting quantum information, gravitation, and cosmology.