The Goldbach conjecture states that every even number greater than 2 is the sum of two primes. This paper explores the geometric structure of prime pairs through the φ-eigengeometry framework. The transfer matrix for prime pair distributions has eigenvalue φ³ = φ² + φ = 2φ + 1. This connection between primes and the golden ratio is not accidental — both arise from the discriminant-5 structure of ℚ(√5). The paper establishes a link between Pentagon Physics and number theory through the shared algebraic foundation.