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 $\phi$ to a bounded angle $\beta$ using the Gudermannian function, hyperbolic, causal, and Euclidean three-spheres are brought together into a single structure: hyper-spacetime. This structure unifies the Euclidean ${\mathbb{R}}^4$ and Minkowski ${\mathbb{R}}^{\mathrm{1,3}}$ 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.

This paper develops a geometric framework for causality and quantum observation within four-dimensional real spacetime, without invoking additional degrees of freedom or external measurement mechanisms. Promoting the bounded Lorentz angle to an intrinsic kinematic flow activates a spatial-vector ↔ temporal-bivector duality linking the Euclidean and Minkowski domains. Complementary manifolds, tangent spaces, and spinors become geometrically reciprocal, forming a closed, self-sustained intrinsic cyclic symmetry. Anchoring this flow to an observer’s local rest frame, without privileging any admissible timelike direction, establishes past-to-future causal ordering within a finite causal interaction volume whose boundary is double-covered by reflection symmetry. The geometric product of momentum and causal separation yields a Lorentz-invariant action defining a dynamical phase, motivating a 4π-periodic causal-boundary standing-wave interpretation. Spacetime event reflections generate four Dirac-related boundary eigenstates with geometric CPT closure, weighted by causal-boundary amplitudes to form a generalized wavefunction. Under semiclassical uncertainty and gravitational stability constraints, the Planck length emerges as a conditional minimum causal distance scale. Quantum observation is formulated as relational causal-boundary mode selection, registering spin orientation and matter–antimatter identity, rather than as wavefunction collapse.

In this article it will be shown that by introducing a hyperbolic rotation as function of a Euclidean rotation angle, all possible Lorentz group spacetime rotations can be performed by Euclidean rotation parameters. This unification in Lorentz group rotations parameters (all Euclidean) allows for the calculation of the volume of a spacetime region bounded by the light cone of a past event, i.e., a causality-volume with the shape of a spacetime three-sphere. As will be calculated, the surface area of the spacetime three-sphere is the volume of our space. Additionally, the unification in Lorentz group rotations parameters allows for the derivation of a Dirac spinor purely from the spacetime three-sphere symmetries. This derivation from the spacetime symmetries of a geometrical object yields the same result as solving the Dirac equation with quantum mechanical eigenvalue eigenvector complex matrix calculations.

We present a unified geometric framework for the full Lorentz group, 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 R^1,3 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.