Quadratic symmetric-exchange defects in a cyclic family of binary matroids: A multiplicity-free classification and a trace formula

This record contains the preliminary v0.3.9 mathematical note "Quadratic symmetric-exchange defects in a cyclic family of binary matroids: A multiplicity-free classification and a trace formula" and a two-page theorem synopsis for quick reading. It is a follow-up to the cyclic-family study archived at https://doi.org/10.5281/zenodo.21706593. For the rank-3m cyclic affine-line matroids M_m, the note proves that every quadratic multidegree fibre is empty, connected, or has exactly two connected components. It classifies quadratic symmetric-exchange defects by an explicit 89-element noncommutative transfer monoid and a trace function, shows that the degree-two quotient (I_m/J_m)_2 is multigraded multiplicity-free, and derives a rational generating function and exponential growth rate for its dimension. The manuscript is explicitly preliminary and computer-assisted. Finite computations establish the local affine-orbit data and transfer-monoid structure, while the all-parameter consequences are mathematical deductions from that finite classification. The synopsis is included as a supplementary overview and is not a separate publication.