Orthic Quadrea Invariants of Null Tetrahedra and a Rational Cayley Correspondence with SU(2)

This paper introduces a projectively invariant quadrea associated to the orthic tetrahedron of a null tetrahedron in projective three–space, within the framework of Universal Hyperbolic Geometry. The orthic quadrea is shown to be a symmetric rational function of the null vertices, invariant under the full orthogonal group of the underlying quadratic form. Using a rational Cayley parametrization, the paper establishes an explicit algebraic correspondence between normalized null tetrahedra and elements of the Lie group SU(2)SU(2)SU(2). Under this correspondence, the orthic quadrea factors through a conjugation–invariant element of the rational function field Q(SU(2))\mathbb{Q}(SU(2))Q(SU(2)), providing a purely algebraic interpretation of a geometric invariant as a class function on a compact Lie group. The orthically regular null tetrahedron is characterized as the unique configuration at which the orthic quadrea assumes its minimal value, with extremality established by rational identities rather than analytic arguments. All constructions avoid transcendental functions, topological assumptions, and real or complex completion, remaining entirely within algebraic geometry and rational Lie theory. This work provides a new bridge between projective geometry, rational parametrizations of Lie groups, and invariant theory, and suggests a framework for studying discrete geometric dynamics and extremal configurations using algebraic calculus.