An Orthic Mass Gap for Null Tetrahedra in Universal Hyperbolic Geometry

This research identifies and proves the existence of a universal spectral range for the face quadrea of orthic tetrahedra in three-dimensional Universal Hyperbolic Geometry (UHG). While classical hyperbolic geometry often relies on transcendental measures, this work utilizes the rational framework of UHG to establish algebraic invariants that govern the "ground state" and "asymptotic limits" of null configurations. Key Contributions: The Orthic Mass Gap: We prove that for any nondegenerate null tetrahedron, the face quadrea of the associated orthic tetrahedron is strictly bounded below by $5/54$. This minimum is attained if and only if the tetrahedron is regular. Dimensional Collapse & Supremum: We establish a supremum of $16/27$, approached as the 3D tetrahedron degenerates into a 2D planar quadrangle. This result anchors 3D orthic geometry to the established 2D invariants found in previous studies. Symbolic Rigor: The proof utilizes a surjective rational parametrization of the absolute quadric $[x^2 + y^2 + z^2 - w^2 = 0]$ and is verified through symbolic Hessian and gradient analysis (SymPy). Methodology: The study employs projective exhaustion to fix a standard frame for null vertices, reducing the configuration space to two degrees of freedom. By analyzing the quadrea function across the absolute quadric, we demonstrate that the regular tetrahedron represents a unique, stable local minimum, while planar configurations represent an unreachable boundary. Context: This work extends the trigonometry of Universal Hyperbolic Geometry from 2D quadrangles to 3D tetrahedra, providing a robust example of rigidity and extremal phenomena in projective metrical geometry.