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 (HRDM) series. Paper I formulated X as a non-particle holographic residual gravitational response intended to account for the phenomena conventionally modeled as cold dark matter, and Paper II discussed galaxy-scale phenomenology. The present note focuses on merging galaxy clusters. In standard particle-CDM language, systems such as the Bullet Cluster are interpreted as collisions in which hot intracluster gas is slowed by hydrodynamic drag while collisionless dark matter remains spatially separated from the gas. The HRDM interpretation keeps the same gravitational phenomenology but changes the ontology: the lensing component is not a transported material particle fluid, but a persistent projected residual gravitational well. In a merger, collisional baryonic gas can be displaced from this pre-existing residual well, making the otherwise blended X component observationally visible through gravitational lensing. The note introduces an effective displacement variable Delta x = x_gas - x_X and a minimal response equation ddot{Delta x} + gamma dot{Delta x} + omega_X^2 Delta x = a_coll(t), where a_coll represents merger-driven hydrodynamic displacement of gas and omega_X encodes the restoring response of the residual well. The framework predicts that lensing-gas offsets in merging clusters should be transient: they can grow during the interaction, but should later saturate, decay, or appear as gas sloshing or oscillation around the residual potential. The paper is phenomenological and does not present a numerical lensing reconstruction.