Fractal Scale Dynamics (FSD) with Complex Phase

We present Fractal Scale Dynamics (FSD), a geometric framework that unifies quantum, cosmological, and biological phenomena through a complex fractal dimension 𝐷 ~ 𝐹 = 𝐷 𝐹 + 𝑖 𝜙 D ~ F =D F +iϕ. The theory is built on a single unifying equation: 𝑟 𝑛 = 𝑘 ⋅ 𝑛 𝐷 𝐹 ⋅ 𝑒 𝑖 𝜙 ln ⁡ 𝑛 ⋅ 𝑒 𝛾 ( 𝑛 − 1 ) r n =k⋅n D F ⋅e iϕlnn ⋅e γ(n−1) where: 𝑟 𝑛 r n is the characteristic radius or value of the system, 𝑛 n is the continuous quantum number (Time Numerator), 𝐷 𝐹 D F is the fractal dimension of spacetime (~1.948), 𝛾 γ is the rate of change of the fractal dimension (~0.35), 𝜙 ϕ is the dynamic phase, evolving over time. We demonstrate that: The fractal gradient ∇ 𝐷 𝐹 = − 𝛾 / 𝑟 ∇D F =−γ/r generates forces equivalent to gravity, friction, and inertia. Mass is replaced by wave frequency via 𝑚 = ℎ 𝜈 / 𝑐 2 m=hν/c 2 . Planetary orbits are reproduced with 0.83% error for Earth. Galaxy rotation curves are explained without dark matter ( 𝛾 ≈ 0.40 γ≈0.40). Neutrino oscillations are reinterpreted as fractal phase transitions, with the phase 𝜙 ( 𝑡 ) ϕ(t) evolving according to 𝜙 ( 𝑡 ) = 𝜙 0 ⋅ 𝑛 ( 𝑡 ) 𝛿 ⋅ 𝑒 𝜖 ( 𝑛 ( 𝑡 ) − 1 ) ϕ(t)=ϕ 0 ⋅n(t) δ ⋅e ϵ(n(t)−1) . Falsifiable weather predictions are provided for Athens and Cape Doro (2, 10, and 18 July 2026). The constant 𝛾 ≈ 0.35 γ≈0.35 appears invariant across all scales, suggesting that fractal geometry, not mass, is the fundamental source of all physical phenomena. The work includes full mathematical documentation, numerical verifications, and time-stamped predictions.