Saddle‑Node Geometry and Invariant Structure in Neoclassical Tearing Mode Onset

The simplified two‑term modified Rutherford equation contains a canonical saddle‑node bifurcation governing neoclassical tearing mode onset [1-4]. Writing the equilibrium condition exactly yields a quadratic with two positive roots - the metastable seed threshold and the saturated island width -and an invariant W₁ W₂ = wd², independent of drive strength. This invariant isolates the stabilization scale wd as the structural link between onset and saturation. A direct test requires a matched triplet (W₁, W₂, wd) for a single discharge; no such dataset is currently available, though high‑resolution measurements [9] show that all three quantities are measurable. Extended MRE physics - curvature [4], transport [10], flow shear [8] - break the quadratic but preserve the fold geometry, yielding a generalized invariant W₁ W₂ = wd² + Δ. Normalizing gives a behaviour quotient Q = (W₁ W₂)/wd², a proposed dimensionless classifier of what we term architectural loading: the structured deformation of the minimal equilibrium architecture caused by the addition of extended physics. The framework provides a compact structural basis for future experimental verification once a matched triplet is obtained.