Minimal vanishing sums of roots of unity: counts to weight 16, provisional counts to weight 21, and repeated-root witnesses at conductor 210

A vanishing sum of roots of unity is minimal if no proper non-empty sub-sum vanishes. The number a(n) of minimal vanishing sums of weight n, up to rotation, was known through n=12 (OEIS A273096, a(12)=69). We report computationally supported counts a(13)=220, a(14)=518, a(15)=1126 and a(16)=2886: two sessions enumerated complementary sets of conductors and verified each other's lists entry by entry with their own exact arithmetic; completeness is supported by complete Hilbert bases at six conductors (30,42,66,70,78,110), by constraint-solver INFEASIBLE verdicts (conductor 210 only through weight 12), by SAT proofs checked with drat-trim (conductors dividing 105 at weights 13 to 16, and conductors 30, 42, 66, 70, 78, 105 from weight 2 through 12), and by a separately implemented enumerator in the same group, which reproduced as sets our lists for conductors 210,42,70,105 at weights 13 to 16 but shares the class decomposition, the exact-conductor test and the rotation quotient with ours and so does not establish completeness; completeness is not proved. Provisional counts to weight 21 combine a counting formula for sums that have a class of size one at an odd top prime with a restricted enumeration of the rest. We prove that such sums are exactly the sums (R_p:T_1,, ...,T_j) of Poonen and Rubinstein at every weight, that no sum all of whose classes have size at least two has weight exactly 2p (a version of this is also checked in Lean 4, with the class decomposition as a hypothesis), and that the rational dimension d=|support|-rank_Q propagates through the construction. We exhibit 336 verified minimal vanishing sums of weight 21 and conductor 210 with a repeated root (witnesses, not a complete count), consistent with the conjectured first weight for repeated roots and reproduced as a set by the third session's enumeration of the sums with all classes of size at least two, and a lifted example of weight 30 and conductor 2310. We state plainly what is not established: completeness beyond weight 16, completeness of the number 336, and any proof about the dimension d (the pattern d=1 fails at weights 22 and 23, in 48 and 768 sums found by one session and verified exactly by a second; these are witnesses, not a complete count). The SAT runs produced proofs that drat-trim accepted; the encodings were written by one session, the proofs were not retained, and the archive holds the verdicts, the CNF hashes and the commands to regenerate and check them. The Lean 4 statements are conditional on the class decomposition. None of the counts is machine-checked. The archive (vanishing-roots-v1.0.zip) holds the lists, code, claims index, notes and the files of other sessions of the team, with credit and hashes; README.md and REPRODUCE.md describe what can be replayed quickly and what cannot.