Ridzi Butterfly Theory: A Symmetric Visualization & Topological Approach to the Goldbach Conjecture

The Goldbach Conjecture states that every even integer greater than two can be expressed as the sum of two prime numbers. Although extensively verified computationally up to 4×10¹8, the conjecture remains strictly unproven. This work introduces a mathematical visualization framework called the Ridzi Butterfly Theory, which transforms Goldbach prime pairs into a symmetric geometric structure centered at the midpoint of an even integer. Each prime pair is represented as a parametric butterfly-like element arranged within a circular crystal configuration using a deterministic zero-gap packing algorithm. The visualization reveals symmetric distance relationships between prime pairs and provides an intuitive, topological representation of Goldbach structures. Crucially, this framework establishes a geometric equivalence: if it can be proven that for every even integer the bounding circle in this framework always contains at least one generated butterfly, then the Goldbach Conjecture is true. A Python-based software implementation, the Goldbach Dream Project, is provided for generating these crystal visualizations and large-scale statistical datasets.