Coordinate-Exchange Symmetry and Generator-Dependent Optimization in Two-Dimensional Fractional Random Walks with Resetting

Does square-lattice symmetry determine the search parameters selected by a fractional random walk with resetting? We compare a walk that updates one coordinate per movement with a walk generated by a fractional power of the full square-torus Laplacian. Both have the same lattice symmetry and nearest-neighbor limit. For the coordinate-updated walk, we prove that coordinate exchange maps the complete first-passage distribution to that of the parameter-swapped target problem. This identity pairs asymmetric local minima and separates the Hessian at a symmetric stationary point into even and odd sectors. For the full-torus walk, we derive the probability of simultaneous two-coordinate motion and relate it exactly to the ratio of the two lazy-chain spectral gaps. Matched searches on a 31-by-31 torus resolve one equal-index candidate for the coordinate-updated walk and two for the full-torus walk at axial target distances 3–5. One-sided derivatives support the reported boundary candidates as constrained solutions, not estimates of an unconstrained optimal index. A fixed-relative-geometry comparison at side lengths 31, 62, and 93 shows that these candidates remain sensitive to lattice resolution and the index cutoff. Thus exchange symmetry constrains relations among search problems, while the generator, encounter rule, and reset timing determine the candidates recovered in a specified finite system.