The Duality of Topological Saturation: Phase Transitions, Manifold Expansion, and the Phantom Metric

The Phantom Metric framework presents a parameter-free geometric resolution to the dark matter problem by treating the empirical minimal acceleration threshold (a0) as a macroscopic boundary condition of the expanding universe. In this theoretical letter, we elucidate the fundamental mechanics of the metric's topological saturation. We demonstrate that this geometric constraint manifests in two distinct physical modalities: Global Envelope Saturation, which triggers a discrete optical phase transition natively routing background photons into macroscopic Einstein rings, and Internal Kinematic Saturation (Dual Saturation), which actively expands the spatial manifold beneath rotating planar galaxies, mathematically forcing baryonic mass into geometric dispersion without localized gravitational collapse. This dual framework rigorously explains both continuous kinematic anomalies and discrete lensing observations (as well as Extreme Topological Boundary Conditions such as the CMB and unrelaxed clusters) without invoking collisionless dark matter. Note: This theoretical letter serves as a companion paper outlining the topological mechanics and phase transitions underlying the Phantom Metric framework.