Informational Gravity: Emergence of the Acceleration Scale from Non-linear Scalar Field Dynamics

This work presents a covariant scalar-field theory of gravity in which the characteristic acceleration scale emerges dynamically from the vacuum structure and non-linear field dynamics, rather than being introduced as a fundamental or fine-tuned parameter in the action. The model is built upon a non-linear kinetic Lagrangian that naturally recovers the AQUAL structure within the weak-field regime. In the non-relativistic limit, the derived field equations evolve into a modified Poisson equation, which yields the baryonic Tully-Fisher relation as a direct and necessary consequence of the underlying dynamics. Specifically, this research demonstrates that: The baryonic Tully-Fisher relation and the fourth-power scaling of velocity with mass emerge from first principles without phenomenological assumptions. The transition between Newtonian gravity and the modified dynamics regime is fully dynamical and determined by the field's kinetic evolution. The universal acceleration scale is not a fundamental constant of nature but arises as a derived quantity from the informational field's parameters. The theory is mathematically robust, remaining stable and free from ghost or gradient instabilities throughout its evolution. A natural theoretical connection exists between the emergent acceleration scale and the cosmological vacuum energy. By linking galactic dynamics to global cosmological properties, this approach provides a unified framework for modified gravity. The manuscript includes a comprehensive mathematical derivation, a formal stability analysis, and an extensive discussion of the physical implications for emergent gravity and dark matter alternatives.