The Geometric Identity of Dark Matter: Cosmic Stiffness Defects and the Takahashi Inversion

The galaxy rotation curve problem and the dynamics of the Bullet Cluster have long been interpreted as evidence for the existence of unknown elementary particles (Dark Matter). However, despite 70 years of searching, no such particles have been detected. Bulk Network Cosmology (BNC) proposes a solution to this problem based not on "Matter" (particles) but on "Geometry". This paper redefines the Dark Matter phenomenon as spatial "Stiffness Defects" or "Ridgelines of the Cosmos" resulting from phase transitions (Kibble-Zurek mechanism) in the early universe. Using a novel analytical method called the "Takahashi Inversion", we derive the geometric stiffness density distribution (rho proportional to 1/r^2) directly from observational data, proving that it aligns more consistently with observations than conventional particle models. This study elucidates Dark Matter not as elusive particles, but as topological defects (networks) inherent in the structure of the cosmos itself.