A Constitutive Framework for Weak-Field Gravity: Galaxy Rotation Curves and SDSS Weak Gravitational Lensing

We present a constitutive framework for gravity in which the gravitational field is interpreted as thenonlinear response of an effective gravitational medium, analogous in mathematical form to constitutiverelations used in electromagnetism and other field theories. The resulting weak-field equations unifygravity into a structure similar to the constitutive descriptions commonly used for the electric, magnetic,weak, and strong interactions while remaining consistent with observed galactic phenomenology.In the high-acceleration regime the framework reduces to Newtonian gravity, while in the low-acceleration regime it naturally produces MOND-like behavior governed by the acceleration scale a0.Using the same low-acceleration response structure, the framework reproduces galaxy rotation curves,the Radial Acceleration Relation (RAR), and the Baryonic Tully–Fisher Relation (BTFR) without theintroduction of explicit dark matter halos.We further extend the framework to weak gravitational lensing through a minimal relativistic weak-field metric in which the effective potential Φeff governs both particle dynamics and light propagation.Applying the resulting projection formalism to SDSS DR7 galaxy–galaxy lensing data, we perform aninitial comparison across three representative stellar-mass bins. The constitutive model reproduces theobserved excess surface-density profiles to within order unity over projected radii spanning approxi-mately 0.05–1.5 Mpc/h, while baryons-only predictions systematically underpredict the observed lensingamplitudes at large radius.The present results suggest that weak-field galactic dynamics and weak gravitational lensing mayemerge from a common constitutive gravitational response without requiring explicit dark matter halos atgalaxy scales. Although the framework remains phenomenological and restricted to the quasi-static weak-field regime, the ability of a single constitutive field structure to reproduce multiple independent galacticobservables suggests that the approach may provide a useful alternative perspective for understandinglarge-scale gravitational phenomena.The present analysis applies the weak-field relativistic extension of the framework, in which theeffective potential Φeff governs both massive particle dynamics and light propagation. Lensing predictionsare computed using the constitutive field equation∇ · (κg g) = −4πGρm, g = −∇Φeff ,together with a simplified spherical approximation to estimate the excess surface density ∆Σ(R)