The Orthic Iteration of Null Quadrangles Induces the Tetrahedral Belyi Map

This research note establishes a novel connection between the discrete geometry of orthocenters and the theory of post-critically finite (PCF) rational maps. Working within the framework of Universal Hyperbolic Geometry (UHG), we investigate the dynamical evolution of a "null quadrangle"—a configuration of four points incident to the fundamental conic. By applying the orthic construction (associating a quadrangle with the orthocenters of its sub-triangles) and utilizing a stereographic projection to track the moduli space evolution, we derive an explicit degree-4 rational map $f(\lambda)$. We provide a complete dynamical classification of this map, proving it is hyperbolic and post-critically finite with a post-critical set $\{0, 1, \infty\}$. The main result identifies $f(\lambda)$ as the Tetrahedral Belyi Map, whose monodromy group is the alternating group $A_4$. This work bridges the gap between the algebraic geometry of orthocenters and the topological structure of Grothendieck's dessins d'enfants. Included in this publication is the SymPy symbolic algebra script used to rigorously verify the rational map derivation. Key Features: Symbolic Derivation: Full algebraic proof of the degree-4 orthic map. Dynamical Analysis: Verification of critical points, orbits, and PCF status. Topological Classification: Computation of ramification indices and identification of the $A_4$ passport. UHG Implementation: Consistent use of Wildberger’s identity polarity and stereographic moduli tracking.