[29]Connectivity Geometry: k=3 Topological Locking and the Algebraic Necessity of 3+1 Spacetime

The compactification mechanism of higher-dimensional spacetime has long been an unsolved mystery in theoretical physics. Based on the framework of connectivity geometry, this paper demonstrates that the dimensionality of macroscopic spacetime is not an arbitrary postulate, but is uniquely locked by the topological self-consistency of the underlying discrete spin network. The study shows that in a non-local jumping network with a step size of k=3, to ensure phase closure (eliminating half-integer vortex divergence) and maintain a true hyperbolic spacetime signature (1, 3), the spatial dimension must be strictly locked to 3. This derivation from the first principles of pure geometry completely rules out the possibility of higher-dimensional collapse, providing extremely rigorous algebraic proof for the origin of cosmic dimensions. Furthermore, this dimensional locking mechanism is completely self-consistent at the topological root with the 1.391 prediction of lattice QCD under the k=3 network (Ref. [2]).