Geometric Theory of Gravity: A Horizon-Induced Framework for Galaxy Rotation and Gravitational Lensing

This work proposes a geometric gravity framework in which galaxy rotation curves, weak gravitational lensing, and large-scale cosmological behavior emerge from a scale-dependent response of spacetime geometry rather than from particle dark matter.

The theory assumes that matter produces a local geometric depression while the full spacetime geometry responds through a compensating ring-like geometric bulge. In the weak-acceleration regime, the model undergoes an effective transition from three-dimensional to two-dimensional geometric behavior.

From these assumptions the framework derives:

• the characteristic acceleration scale
  a₀ = cH₀/(2π),
• the Variant A transition function,
• the baryonic Tully–Fisher relation,
• and a natural gravitational lensing enhancement factor
  q = 4/3.

The theory reproduces Newtonian gravity and General Relativity locally while transitioning toward MOND-like behavior on galactic scales.

A relativistic tensor–vector–scalar structure is proposed together with a geometric interpretation of weak lensing through a compensating ring-shaped curvature sector.

The framework is tested against galaxy rotation relations and weak lensing data and shows encouraging agreement with observational constraints.

This document presents the complete current formulation of the Geometric Theory of Gravity (GTG), including geometric assumptions, field equations, relativistic structure, observational tests, and a candidate horizon-driven cosmological sector.