Lorentz Structure as an Extension of Quaternionic Trinity Geometry

This paper develops a geometric formulation in which Lorentzian spacetime structure can be interpreted as a natural extension of quaternionic Trinity geometry. Recent work established that the Trinity Geometric Model admits a quaternionic representation in which a trinity is described by a quaternion q = R u, where the invariant magnitude R defines expansion scale and the unit quaternion u on the three-sphere (S³) defines selector configuration. In the quaternionic formulation, the selector manifold appears naturally as the three-sphere of unit quaternions, and the group of selector rotations is isomorphic to SU(2). The quaternionic representation introduces a Euclidean invariant norm, while relativistic physics is based on the Lorentzian spacetime interval. Understanding the relationship between these invariant structures provides a geometric connection between quaternionic Trinity geometry and relativistic spacetime. This paper shows that Lorentzian spacetime structure can be interpreted as arising naturally when quaternionic selector geometry is extended from SU(2) to its complex extension SL(2,C). Spacetime events are represented by Hermitian matrices X = x^μ σ_μ constructed from the Pauli matrices. In this representation, the Minkowski spacetime interval appears as the determinant det(X) = c²t² − x² − y² − z², which remains invariant under transformations of the form X → A X A†, A ∈ SL(2,C). Rotational selector transformations correspond to the SU(2) subgroup, while Lorentz boosts correspond to non-unitary transformations within SL(2,C). The Lorentz group, therefore, appears as a natural extension of quaternionic selector symmetry. An important feature of this formulation is the distinction between two invariant quantities. The quaternion magnitude defines an invariant expansion scale associated with the Trinity primitive, while the Minkowski interval defines invariant spacetime separation. These invariants play different geometric roles and need not be identified with one another. This separation resolves the apparent tension between the Euclidean quaternion norm and the Lorentzian spacetime metric. Together with "Projective Quaternion Geometry and the Trinity Decomposition" (DOI: 10.5281/zenodo.18778999), the present work establishes a unified geometric framework in which quaternion structure and relativistic spacetime geometry arise from a common mathematical foundation.