Action-Rank Stratification and Orbital Fourier Transforms in Type B2 over Finite Fields

For an odd finite field, we study the symmetric-pair commutator map from pairs of symmetric matrices to alternating matrices. In the five-dimensional case, its Fourier transform is governed by an action rank on the type-B2 Lie algebra. We prove the resulting action-rank stratification with ranks 0, 8, 10, and 12, and isolate the rank-ten rational long wall. A direct SO5 eigenplane calculation gives an exact wall-incidence formula and the corresponding rank-ten wall Fourier value. Using the accidental isomorphism between sp4 and so5, we complete the remaining rank-twelve targets through seven rational and Jordan transform types, including nonsemisimple cases. These exact transform values show that the rank-ten stratum is the unique action-rank stratum with a depth-dependent sign transition: its normalized point mass is above uniform at depth one and below uniform at every later depth. Finite Lie-algebra Fourier and Green-function theory provides the ambient framework, while the repeated-root rational wall union and its incidence correction are evaluated directly. This record contains a preprint version of the manuscript.