Platinum Icosahedral Edge Geometry as a Topological Framework for Resonant Magnetic Perturbation ELM Suppression

Abstract

The operational windows for edge-localized mode (ELM) suppression via resonant magnetic perturbations (RMPs) in tokamaks are empirically discrete — suppression occurs only when the safety factor at 95% flux surface (q₉₅) aligns with rational surfaces q = m/n, producing narrow windows of width Δq₉₅ ≈ 0.1. We propose that this discreteness reflects a deeper topological constraint: the pedestal region behaves as a rank-2 tensor boundary whose eigenmode structure is governed by the symmetry group of the edge geometry, not by continuous field amplitude. We contrast two candidate edge topologies: (i) a linear cube with eight corner singularities, which we show produces uncontrolled magnetic island nucleation and flux leakage (∮B·dA ≠ 0); and (ii) a platinum icosahedron with native golden-ratio curvature (κ = φ⁻¹ ≈ 0.618) and A₅ symmetry, which distributes resonant modes across 20 faces with no corner singularities (∮B·dA = 0). We implement a real-time visualization overlaying DIII-D's I-coil geometry (2 rows × 6 coils) onto both edge topologies, computing vacuum field gain, resonant field amplification, island width, and discrete q₉₅ resonance windows from published scaling laws (Nucl. Fusion 55, 023002; PRL 125, 045001). The simulation demonstrates that the icosahedral edge produces ELM suppression across a broader q₉₅ range than the cube for identical coil currents, because the A₅ symmetry group admits more rational-surface resonances without singularity formation. We propose that optimized stellarator coil sets (W7-X quasi-isodynamic) represent the closest existing implementation of this principle, and that future RMP coil designs should adopt polyhedral rather than toroidal-row symmetry to widen operational windows. Keywords: ELM suppression, resonant magnetic perturbations,