Fractal Electroweak Unification: The Weinberg Angle, Hierarchy Resolution, and the√2-Cascade from Atomic to Electroweak Scales

Why is the Weinberg angle sin²θ_W ≈ 0.231 and not some other value? The Standard Model takes it as a free parameter. This paper proposes a structural answer. In the fractal-spectral framework, the electroweak mixing is determined by the geometry of the fractal texture tower with scaling ratio √2. At leading order, the ratio of U(1)_Y and SU(2)_L couplings at the fractal pivot gives sin²θ_W = 1/(1+(√2)²) = 1/3 — a parameter-free prediction from the fractal structure alone. Computable radiative corrections from the texture tower (a convergent series in 2⁻ⁿ) bring the value to 0.2312 ± 0.0002, matching the experimental measurement. One coefficient (c₁ ≈ 0.57) is fitted; its first-principles derivation from fractal vacuum polarization remains an open problem. The hierarchy problem is addressed through the same mechanism: quantum corrections to the Higgs mass from each fractal level are suppressed by 2⁻ⁿ, producing a convergent sum (Σ 2⁻ⁿ = 2) instead of a quadratically divergent integral. No supersymmetry, extra dimensions, or fine-tuning is required — the suppression follows directly from the kinetic weights (√2)⁻²ⁿ in the fractal-temporal Lagrangian. The scale gap between atomic physics (10²³ Hz) and the electroweak scale (10¹⁶ Hz) is bridged by exactly 23 levels of √2-scaling, derived through log-periodic homogenization with rigorous error bounds. This is not a numerical coincidence: it is the number of half-octave steps separating the two energy scales in a base-2 universe. The most distinctive prediction is log-periodic oscillations in the running of sin²θ_W(Q) with a universal, parameter-free log-frequency ω_f = 2π/ln√2 ≈ 18.1 and estimated amplitude ε ~ 10⁻⁵–10⁻⁴. The Standard Model predicts pure logarithmic running; the fractal tower predicts a superimposed oscillatory residual with discrete scale invariance under Q → Q√2. Detection requires measurements at multiple energy scales spanning at least one full period (~91 to ~129 GeV). The FCC-ee baseline program includes off-peak runs sufficient to resolve this signature.