Isomorphism Between Golden Logarithmic Spiral and Wilsonian Renormalization Group Dynamics: What is "1"?

This work presents a rigorous mathematical derivation proving structural isomorphism between the Golden Logarithmic Spiral phase flow in complex space and Wilsonian Renormalization Group (RG) dynamics (PSpiral MRG). Key theoretical contributions include: 1. Exact Derivation of Spiral Expansion Factor: Derived from the local field potential f(r) = r⁻² - r², the natural spiral expansion factor is locked to k = lnφ/2π, yielding the linearized Wilsonian Beta function β(g) = Δeff · g with complex anomalous scaling dimension Δeff = lnφ + i2π.2. Taiji Transformation & Base Duality: Establishes the dual mapping g(x) = x⁻ᵖ = φᵏ (where k = -p ln x/lnφ), mapping any power-law field potential directly to the Golden Ratio scale ladder (φᵏ e⁻ᵖᵘ).3. Discrete Scale Invariance (DSI) & Quantum Oscillations: Under $U(1)$ phase closure (ei2π = 1), continuous scale invariance breaks down to Discrete Scale Invariance with exact scaling multiplier λ = φ. The imaginary term i2π generates pure phase oscillations around the fixed-point anchor $r = 1$, providing a geometric foundation for quantum amplitude behavior.4. Resolution of Coordinate Singularities: Demonstrates that microscopic limits (x → 0⁺) represent smooth scale ascension along the golden scale ladder rather than physical divergent singularities (g(x) → ∞).