Dark Matter as Pressure of Closed Spatial Membranes

Within the Granular Cosmic Vacuum (GCV) framework, the galactic-halo domain P3P3 is populated by closed spatial membranes—classical geometric structures formed through a ternary composition of contracted spatial branes. These membranes carry an internal tension originating from Planck-scale physics, structurally damped across hierarchical scales, which manifests macroscopically as a pressure. This membrane pressure generates an effective stress-energy distribution that reproduces the observed phenomenology of dark matter without invoking new particles: flat rotation curves, halo mass profiles, gravitational lensing signatures, and contributions to large-scale structure. We derive the relationship between Planck tension, membrane pressure, and effective dark-matter density, compute the gravitational field of membrane halos under spherical symmetry, and identify specific observational tests—including short-range deviations from Newton’s law, characteristic halo shapes in lensing, and a small-scale cutoff in the matter power spectrum. The model provides a purely geometric, particle-free description of dark matter, structurally unified with the temporal-tension mechanism for dark energy within the same granular vacuum picture.