Geometric Foundations of VPQW & DMQPF: The Mathematical Proof

This document provides a rigorous geometric proof of the Vesica Piscis overlapping perimeter formula and its connection to the Dark Matter Quantum Perimeter Function (DMQPF). We aim to solidify the derivations and justify all introduced equations in detail. Objectives: • Derive the perimeter formula for the Vesica Piscis overlapping region based on separation angle. • Verify the validity of the formula at key angles and justify all assumptions rigorously. • Extend this proof to dark matter structures through the DMQPF framework, ensuring all quantum corrections are fully motivated. • Establish a mathematical connection between Vesica Piscis geometry and quantum probability distributions.