Catastrophe theory classifies the generic ways a smooth system can jump discontinuously as control parameters vary. The simplest — and most ubiquitous — is the fold catastrophe, type A₂, described locally by V(x, u) = x³/3 + ux.
The HVEH vortex transition maps onto this structure. The state variable x represents vortex coherence; u represents the ratio of curvature-gate strength to nonlinear amplification rate. The fold surface divides reachable states into a stable lower sheet (coherent helical rotation) and an unstable upper sheet (chaotic turbulence).
Correct operator order — K before F — corresponds to a path that approaches the fold from below, crossing smoothly into the stable basin. Wrong order — F before K — approaches from above, landing on the unstable sheet and remaining there.
The fold is sharp: there is no gradual transition. The system jumps. This is why the HVEH either works completely or fails completely — there is no partial vortex state. Catastrophe theory predicts this before any simulation is run.
| Normal form | V(x, u) = x³/3 + ux |
| Fold condition | ∂V/∂x = x² + u = 0 → fold at x = ±√(−u) |
| Stable sheet | K before F → path enters x < 0 (stable) |
| Unstable sheet | F before K → path stays x > 0 (chaotic) |