This record contains the mathematical note "A cyclic family arising from Larson's matroid counterexample: exact symmetric-exchange arity and a socle theorem" and a two-page theorem synopsis for quick reading. For a cyclic family M_m of rank-3m binary matroids, the note proves that the one-element symmetric-exchange graph on unordered complementary basis pairs has exactly two components for m >= 3; determines the exact simultaneous exchange arity as ceil(m/2); classifies all basis type words by rooted forests; and proves that the distinguished complementary defect is annihilated by every basis variable, equivalently (J_m : f_m) is the irrelevant maximal ideal. The all-parameter arguments are symbolic; finite computational checks are separate from the mathematical proof. The synopsis is included as a supplementary overview and is not a separate publication.