φ-Adic Numbers: A Golden Ratio Completion of ℚ(√5)

We construct a novel non-archimedean completion of the quadratic number field ℚ(√5) based on the golden ratio 𝜑 = (1 + √5)/2. Unlike classical 𝑝-adic completions associated with prime ideals, our 𝜑-adic completion exploits the Fibonacci recursion inherent in 𝜑’s defining equation 𝜑² = 𝜑 + 1. We prove that the resulting complete field ℚ_𝜑 possesses an ultra-metric topology with a Cantor set structure, and we characterize its Bruhat-Tits tree with Fibonacci branching. This construction provides a natural arithmetic framework for dimensional cascade in theoretical physics, where Fibonacci dimensions play a central role. We derive convergence criteria for 𝜑-adic series and establish connections with classical 𝑝- adic theory through Ostrowski-type classification theorems. Physical applications include particle mass hierarchies, quantum uncertainty bounds, and gravitational quantization.