Gravitational Curvature as Structured Entropy: Three-System Validation of the DeltaX Equation

We present observational validation of the ∆X framework across three astrophysical systems: a galaxy cluster (Abell 1689), a collisional cluster (the Bullet Cluster), and a spiral galaxy (NGC 3198). Each system is modeled using the same ∆X equation, a curvature-memory compression law derived from mass, geometry, and entropy. Observable predictions (lensing arc radius, spatial offset, and rotational velocity) are produced by a unified three-step process: field generation (∆X), projection, and physical integration. These predictions match observations using only visible mass and entropy structure, with no dark matter required. Each system employs a distinct ∆X application mode adapted to its geometry, yet all use the same locked parameters.