Membrane Curvature Model — A Revised First Mathematical Formulation

This preprint presents a revised first mathematical formulation of the Dark Matter Energy Theory (DMET), a speculative gravity-first cosmological framework. The paper explores whether the additional gravitational effects usually attributed to dark matter could arise from an extended geometric response to ordinary baryonic matter. It uses a nonlinear weak-field equation of AQUAL/MOND type as a phenomenological test model, deriving flat galaxy rotation behaviour and the baryonic Tully–Fisher relation while explicitly distinguishing established modified-gravity mathematics from the specifically DMET interpretation. The paper also outlines the requirements for a relativistic completion, including gravitational lensing, large-scale tidal curvature, and a speculative cross-region optical extension involving geometrical redshift and lensing. The formulation is exploratory and is not presented as a validated replacement for General Relativity or ΛCDM. Its purpose is to identify what equations, observations, and falsification tests would be required for the underlying DMET concept to be physically viable.