Cyclotomic Cuts and Minimal Vanishing Sums Through Weight 21

Problem and result This working paper gives a constructive algebraic classification of minimal vanishing multisets of roots of unity through weight 21. It proves that repeated terms first occur at weight 21 and restricts the equality case to four comparison patterns. Method After opposite roots are represented by signs, cyclotomic relations become vertex-potential differences on the complete bipartite graph K(3,5). Threshold cuts classify the small atoms, strong connectivity detects zero-freeness, and prime-fiber reduction yields singleton and pair replacement rules and exactly eight triple configurations. The structural classification and height theorem require no exhaustive search. Catalogue and corrections The companion catalogue expands the proved constructions into 1,198 normalized Christie–Dykema–Klep type labels and 2,062 joint height/parity records. These counts are computational; every record has an exact witness checked with a separate cyclotomic-polynomial implementation. Two explicit arguments identify an omitted weight-17 example and excess weight-21 parity claims in Appendix A of arXiv:2008.11268v2. Scope and contents The parametrization is redundant, and type labels do not count rotation orbits. Further questions concern a disjoint orbit parametrization and the new fiber configurations above weight 21. The package contains the main paper, companion catalogue, separate corrections and verification report, editable LaTeX sources, Python programs, and complete data.