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

Why is the Weinberg angle sin²θW ≈ 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 sin²θW = 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 sin²θW (Q) with a universal, parameter-free log-frequency ωf = 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.