Connectivity Geometry: Cosmological Phase Transition, and the Topological Origin of Dark Matter and Neutrino Mass

Traditional cosmology posits that the universe originated from a general relativistic "singularity" of infinite density and stagnant time, which is not only the breakdown point of classical physics but also violates the self-consistency of quantum mechanics. In the framework of connectivity geometry, this paper proves that the Big Bang is a global topological phase transition of the underlying discrete spin network. Based on the topological quench mechanism, the arrow of time is the geometric projection of the irreversible evolution of network connectivity. This paper transforms the traditional "dark energy" problem into a geometric effect, proposing that the accelerating expansion of the universe is a geometric relaxation effect of macroscopic coarse-graining after the network phase transition. We systematically examine the predictions of three mechanisms—renormalization group (RG) flow, Kibble-Zurek quenching, and topological freezing—for the macroscopic cutoff scale, pointing out that none of them fall within the required magnitude of Lambda observations, thereby positioning the cosmological constant hierarchy problem as the core open question of this framework. On this basis, this paper derives the topological origins of dark matter, extremely small neutrino masses, and black hole singularities: dark matter is a high-dimensional topological soliton lacking a 1D boundary; neutrino mass originates from the minimal eigenstate of the percolation resistance matrix, yielding a characteristic mass of 0.1 eV under model parameters (the composite suppression factor awaits numerical calibration); and there is no singularity at the center of a black hole, only a topological bounce when the network reaches its limit, with unitary information reconstruction.