A Minimal Divergence-Free Extension of General Relativity with Natural Cutoff, Galactic Scaling, and Singular-Free Strong-Field Structure The model introduces an elastic response in the spaetime manifold, producing a divergence-free field that supplements the Einstein tensor without altering the classcal GR limit. The theory naturally reproduces large-scale structure formation and yields a µCDM-like behavior without invoking particle dark matter. Numerical results demonstrate that the elastic field generates filamentary cosmic-web structures consisent with observations. This work provides a compact, parameter-sparse alternativ to modified gravity and dark matter models, offerig a unified geometric mechanism for cosmic structure formation.