Mathematical Derivations and Formula Reference for the Hasan Dipole Model (HDM) v2.0

This document serves as the complete mathematical reference for the Hasan Dipole Model (HDM), providing the formal derivations for the architectural structure of the Solar System. Key Technical Details Included: The Derivation Chain: A sequential mathematical cascade showing how Mercury's position enables the derivation of Venus and Earth with an average error of less than 0.40%. Closed-Loop Proof: Mathematical evidence demonstrating that the Solar System is a self-consistent loop, where Saturn's position can be independently derived from the positions of other planets with only 0.28% error. Boundary Constants: Precise calculations of the system's gravitational fulcrum at 7.522 AU and the emergence of the bridge constants 2.058 AU and 2.322 AU. Universal Symmetry: A detailed table of the "Perfect Torque Balance," showing that Group 1 and Group 2 planetary torques are equal to 750.928 units, balanced to six decimal places, with Pluto acting as the essential balance pin. Predictive Framework: Mathematical formulas for unknown Trans-Neptunian Objects (TNOs) and the derivation of the proposed Dark Sun's mass (0.2324 Solar Masses) through thermodynamic and gravitational symmetry. This reference is intended for researchers and astrophysicists interested in the geometric resonance and gravitational equilibrium of planetary systems.