Galaxy-Scale Gravity Without Dark Matter: Emergent Gravitational Coupling from Discrete Field Geometry

This work presents a relativistic, galaxy-scale gravity framework based on a lattice field formulation with an emergent gravitational coupling. Building on prior results demonstrating successful prediction of galaxy rotation curves without invoking dark matter, this paper introduces a χ-dependent gravitational coupling that remains fixed across all tests. With a single governing equation and one universal calibration constant, the framework simultaneously accounts for: • galaxy rotation curves• galaxy–galaxy weak gravitational lensing• strong-lensing time delays using baryonic matter only and without auxiliary fields. A key result is the demonstration of cross-consistency between dynamics and lensing: the same geometric field inferred from galaxy kinematics predicts observed lensing signals without re-inference or tuning. The framework is shown to be relativistically viable, distinguishing it from MOND-like theories, which require additional fields to reproduce lensing and time-delay phenomena. The paper explicitly derives the effective Poisson equation, rotation law, lensing potential, and closure relations implied by the governing field equation, establishing priority and completeness at galaxy scales. The scope of validity is restricted to galactic systems; no claims are made regarding clusters or cosmology. This work demonstrates that dark matter is not required to explain galactic gravity once gravitational coupling is allowed to emerge from geometric field structure.