Fractal Mechanics in Resistive Magnetohydrodynamics: An Analytical Nonlinear Bound on Edge-Localised Mode Energy

Type-I Edge-Localized Modes (ELMs) represent a major operational threat for ITER and future magnetic fusion reactors due to transient divertor heat loads. While linear peeling-ballooning theory predicts ELM onset thresholds, no first-principles non-linear theory currently provides an analytical upper bound on the ejected ELM energy during the crash phase. Here, we extend the Fractal Mechanics (FM) framework—recently applied to three-dimensional Navier-Stokes regularity—to resistive magnetohydrodynamics (MHD) in toroidal geometry (FM-MHD). Decomposing the Elsässer fields in a Fibonacci wavelet basis, we prove scale locality of non-linear energy transfer and global BKM regularity under a self-consistent parallel viscosity closure derived from cascade field theory. We show that toroidal geometry preserves the 1D Fibonacci cascade topology on golden-ratio safety factor surfaces (q ~ φ^k), with radial eigenfunctions taking the form of Pöschl-Teller solitons. This yields the first analytical, non-linear bound on ELM energy release: dW_ELM / W_ped <= φ * f_ped, where f_ped is the pedestal volume fraction engaged by the unstable mode and φ is the golden ratio (~1.618). The theory is validated across three independent experimental observables without free parameters: (i) global ELM energy bounds in the multi-machine ITPA database; (ii) pre-ELM precursor toroidal mode sequences n_k = {1, 2, 3, 5, 8, 13} and geometric amplitude ratios A_(k+1) / A_k <= 1/φ ≈ 0.618 on ASDEX Upgrade; and (iii) the sech² Pöschl-Teller spatial profile and flat Fourier spectrum (slt_φ = 0.89) of solitary magnetic perturbations. Active q-profile shaping strategies via ECCD in ITER hybrid scenarios (K = 1) are proposed to guarantee safe operating windows (dW_ELM <= 0.4 MJ).