Holographic Residual Dark Matter III: Merging Clusters as Baryonic Displacement from a Persistent Residual Gravitational Well

This technical preprint presents Paper III of the Holographic Residual Dark Matter framework. Paper I introduced the core CDM-like effective component, and Paper II discussed galaxy-scale phenomenology. The present note focuses on merging galaxy clusters and the interpretation of lensing-gas offsets. In this framework, the residual component X is not modeled as a material component transported through the bulk. Instead, X is interpreted as a persistent holographic residual gravitational well. In merging clusters, collisional baryonic gas may be hydrodynamically and electromagnetically displaced from this pre-existing residual well. The lensing peak then traces the effective residual gravitational structure, while the X-ray peak traces displaced hot gas. The note formulates this picture through an effective displacement variable Delta x = x_gas - x_X and a minimal response equation d^2 Delta x/dt^2 + gamma d Delta x/dt + omega_X^2 Delta x = a_coll(t). This provides a phenomenological way to describe temporary lensing-gas offsets, post-collision relaxation, gas fallback, or damped oscillation around the residual well. The model is presented as a non-particle interpretation of merging-cluster dark matter phenomenology rather than a completed numerical lensing reconstruction.