Emergent Log-Periodicity in Fractal Resonance Calculus: Scale Crystal, Bloch Mechanism, and the Dynamical Cascade

This working paper investigates whether the log-periodic structural ladder s_n = s_0 enΔ, postulated in the Theory of Fractal Resonance (TFR) and treated as a phenomenological calibration in the FRC series, can be derived from first principles. We extend the FRC action with a nonlocal, dilation-invariant scale coupling and show that (i) the convolution structure of the kernel follows from self-similarity of the S-transition cascade, and (ii) a derivative (aberration-type) inter-layer vertex generates a kernel whose Fourier transform has a minimum at finite wavenumber Q* = √(μ/ℓ), producing a modulation instability and a periodic mean field in ξ = ln s. The resulting scale Hamiltonian with periodic potential admits a Bloch–Floquet analysis: Wannier states localize on a geometric ladder of scales, while the confining tilt yields a locally equidistant (Wannier–Stark) level ladder, reconciling the linear mass spectrum of the series with the geometric structural ladder. A one-loop computation shows that the mediator mass receives a tree-level (Stueckelberg) contribution plus an infrared correction proportional to the holographic window ξ_H; the naive purely loop-generated mass is refuted. Numerical solution of the Bloch problem under a pre-registered protocol confirms localization onset above the analytical threshold V_lock and local equidistance within the window, but reveals that the Planck-scale UV barrier degrades site-locking at large potential depth: the best achieved offset is 22.9–37.2% of Δ, above the pre-registered 15% target. We conclude that the static Bloch mechanism creates preferred scale valleys but cannot alone enforce sharp layering; final sharpness must be supplied by the dynamical cascade (relaxation-oscillator model, ℓ = Δ·τ_rel/τ_grow). The parameter Δ therefore remains a calibration, now with an identified structural origin and a falsifiable dilemma arbitrated by the comb shape in P(k). Status note: Hypothesis with partial numerical confirmation. Pre-registered protocol 4b results reported, including refuted sub-claims (sharp site-locking P1). This document is a boundary-research supplement to the core TFR/FRC series; the core remains unaffected by the outcomes herein. Supplementary materials included in this record:• Emergent_Log_Periodicity.pdf — main working paper• Emergent_Log_Periodicity.tex — LaTeX source code• protocol_4b_final.py — Python script for pre-registered numerical protocol 4b (Bloch mechanism verification)• protocol_4b_plots.zip — generated plots (8 PNG files) visualizing Wannier localization and UV-barrier interference effects All code and data provided for full reproducibility.