The Fractal Scale Bridge: Log-Periodic Homogenization Across the Planck–Electroweak–Atomic Hierarchy

How does the Planck scale (10¹⁹ GeV) connect to the electroweak scale (10² GeV)? The standard answer is that it doesn't — the hierarchy problem is the statement that these scales have no known structural relationship. This paper provides one. We prove a homogenization theorem for log-periodic potentials: a microscopic Hamiltonian with rapidly oscillating √2-periodic structure converges to an effective operator whose parameters are determined by a cell problem on the period [0, ln√2]. Applied to the fractal-temporal framework, this produces a geometric tower of energy scales E_n = E₀/(√2)^n that connects any two physical scales through a computable number of half-octave steps. The number of levels is not a free parameter — it is the logarithmic ratio of known scales in base √2. The Planck-to-Higgs separation requires n ≈ 113 levels, Planck-to-proton requires n ≈ 127, nuclear-to-atomic requires n ≈ 52, and proton-to-electron requires n ≈ 22 (the fractional part reflecting subleading corrections). These are not predictions but geometric identities; the prediction is that the tower is stable (convergent sum Σ2⁻ⁿ = 2), that error bounds contract geometrically per level (O(α−βn)), and that the universal log-frequency ω_f = 2π/ln√2 ≈ 18.1 appears as modulations at every scale. The physical mechanism is structured quantum decoherence through the fractal texture: at each level, a fraction Γ ≈ 0.29 (determined by the cell problem corrector, not fitted) of vibrational energy transfers to the τ-field texture, stepping down the effective energy scale. The corrector Γ connects directly to the electroweak sector of the companion paper. Testable predictions include: log-periodic spectral modulations in heavy atoms at the ~10⁻¹⁵ level (within reach of current optical clock precision ~10⁻¹⁹), cross-section modulations at colliders near the electroweak band (n = 112–115), and atom interferometry phase shifts from the nuclear-to-atomic cascade. The universality of ω_f across atomic spectroscopy, electroweak running, and QCD coupling — the same frequency in three independent sectors — is the strongest cross-check of the entire framework.