Finding the fuel-optimal aircraft route through a wind field is a continuous optimization problem. The wind bends the effective cost of travel — flying with the wind is cheap, flying against it is expensive — and the optimal path is generally not a straight line. The Hamilton-Jacobi-Bellman equation describes the cost field, and the optimal trajectory follows its gradient. Standard eikonal solvers handle this efficiently. The difficulty is at the cut locus.
The cut locus is where two or more optimal paths arrive at the same point with the same cost but from different directions. At the cut locus, the cost field is non-differentiable — the gradient has a discontinuity — and numerical solvers cannot distinguish between the true global optimum and a nearby suboptimal path. Small discretization errors near the cut locus are amplified into convergence to the wrong solution.
Borndörfer, Jocas, and Weiser (arXiv:2603.11830, 2026) observe that the cut locus, while dangerous for the solver, is identifiable in advance. Finite element error analysis provides bounds on how close a destination can be to the cut locus before the discretization error exceeds the cost difference between competing paths. This defines a protective region: destinations outside the region are guaranteed to be decoded correctly by the solver; destinations inside require finer resolution or alternative treatment.
The structural point: the pathology advertises itself. The cut locus is exactly the set of points where two optimal paths tie — and the existence of a tie is computable from the cost field before any trajectory is traced. The optimizer does not stumble into the trap unknowingly. The trap's location is encoded in the same structure (the cost field) that the optimizer uses to find paths. The failure mode and the detection method share a substrate. Rather than making the solver robust everywhere, the approach makes the danger zone visible and treats it differently — precision where the problem demands it, efficiency where it doesn't.