A Unified Geometric-Algebraic Framework for the Fine Structure Constant: From Triangular Field Arrangements to Prime-Constrained Precision

We present a comprehensive theoretical framework that unifies geometric intuition with mathematical precision to explain the fine structure constant α through three progressive levels of understanding. At the foundational level, we establish that electromagnetic field interactions around point charges naturally organize into discrete triangular arrangements, yielding α⁻¹ ≈ T₁₆ + 1 = 137 where T₁₆ = 136 is the 16th triangular number. This geometric foundation provides intuitive physical understanding but achieves only 0.026% accuracy. We then demonstrate that mathematical enhancement through prime-constrained base-3 geometry, building upon the Isam Tayyar Formula, achieves 0.001% precision via α⁻¹ = 729/(3^(3/2) + 3^(-2)/sin(3^(1/10))). Finally, incorporating quantum geometric corrections yields α⁻¹ = 137.035999084 with error < 10⁻⁸%, matching CODATA values within computational precision. This multi-level framework suggests that fundamental constants encode hierarchical geometric-algebraic structures, with discrete triangular arrangements providing the physical foundation while prime-constrained mathematics delivers experimental precision. We extend the framework to predict other fundamental constants and discuss implications for the geometric foundations of physical law.