Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes

Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent L\'evy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (L\'evy exponent $\mu = 2$) uniquely achieves scale-invariant, near-optimal detection across a broad spectrum of target sizes and shapes. In a domain of volume $n$ with boundary conditions, expected detection time for a convex target of surface area $\Delta$ optimally scales as $n/\Delta$. Conversely, L\'evy strategies with $\mu < 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $\mu > 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $\mu = 1$, ceding dominance to surface area as $\mu \to 2$, after which surface area and elongation couple to govern detection. Ultimately, 3D search introduces a pronounced sensitivity to target shape that is absent in lower dimensions. Our work provides a rigorous foundation for the L\'evy flight foraging hypothesis in 3D by establishing the scale-invariant optimality of the Cauchy walk. Furthermore, our results reveal dimensionality-driven shape vulnerabilities and offer testable predictions for biological and engineered systems.