A searcher moves through space looking for a target whose size and shape are unknown. Lévy flights — random walks with power-law step lengths — are a natural strategy because they are scale-free: a Lévy walk with exponent μ explores space at all scales simultaneously, spending neither too much time locally nor too much time traveling.
In three dimensions (arXiv:2603.10655), the Cauchy strategy (μ = 2) is uniquely optimal across a broad spectrum of target sizes and shapes. This is not obvious. In lower dimensions, the optimal exponent depends on the target; different targets favor different search strategies. In 3D, a geometric transition occurs: the relationship between a target's volume and surface area changes qualitatively, and this change collapses the family of optimal strategies to a single point.
The transition is geometric, not statistical. The ratio of volume to surface area scales differently in three dimensions than in two, and this scaling difference eliminates the target-dependence that exists in lower dimensions. The Cauchy exponent is not a compromise between competing objectives — it is the unique point where the geometric transition makes all targets equivalent from the searcher's perspective.
The structural claim: dimensionality can resolve a trade-off by eliminating it. In 2D, different targets demand different strategies, and any fixed strategy is suboptimal for some targets. In 3D, the geometry itself removes the variability. The searcher doesn't need to know the target because the space has absorbed the distinction.