Circular and Hyperbolic Symmetry Unified in Hyper-Spacetime

A unified geometric framework for the full Lorentz group is introduced, based on a bounded angle parametrization of spacetime transformations within a hyper-spherical geometry. By mapping the unbounded hyperbolic angle φ to a bounded angle β using the Gudermannian function, hyperbolic, causal, and Euclidean three-spheres are brought together into a single structure: hyper-spacetime. This structure unifies the Euclidean R4 and Minkowski R1,3 domains, incorporates discrete symmetries in a continuous way, and removes discontinuities at the lightlike boundary. Each three-sphere carries a natural spinor set, encoding symmetry, and acting as eigen-spinors of corresponding observables. These reproduce the Dirac spectrum while confining singular behavior to a scalar factor. The bounded angle parametrization therefore provides a continuous, closed representation of the full Lorentz group and a transparent geometric basis for spacetime symmetry.