Galaxy-Scale Phenomenology from Entanglement-Controlled Gravitational Coupling: Environmental Dilution, Flat Rotation Curves, and Lensing Tests

We propose a scalar–tensor modification of gravity in which the effective gravitational coupling is controlled by a dimensionless “vacuum entanglement density” field Φ, with Geff (x) varying across environments. An environmental dilution closure activates (or suppresses) an effective potential sector depending on a local matter invariant, yielding a screened, GR-like core and an outer halo dilution regime. Boundary matching produces an extended Φ(r) ∝ r −1 window, naturally gener ating approximately flat rotation curves without particle dark matter. The framework predicts a sharp, falsifiable relation between lensing and dynamical responses controlled by the Brans–Dicke parameter ω, enabling joint rotation-curve and lensing tests.