Let \(\zeta=\zeta_{105}\), and identify a subset with its exponents in \(\mathbb Z/105\mathbb Z\). The equivalence relation is \(S\sim a+uS\), where \(a\in\mathbb Z/105\mathbb Z\) and \(u\) is a unit modulo 105. Exact enumeration of all nonempty inclusion-minimal vanishing subsets of weight at most 19 gives the following canonical representatives:

- weight 3: \(\{0,35,70\}\); - weight 5: \(\{0,21,42,63,84\}\); - weight 7: \(\{0,15,30,45,60,75,90\}\); - weight 14: \(\{0,1,8,22,29,30,43,45,60,64,71,75,90,92\}\); - weight 16: \(\{0,1,5,20,22,30,35,43,45,60,64,65,75,80,90,95\}\); - weight 18: \(\{0,1,3,11,16,24,26,41,42,45,46,61,63,71,76,84,86,87\}\) and \(\{0,1,3,18,22,29,30,33,45,48,60,63,64,71,75,90,92,93\}\).

The corresponding orbit sizes are 35, 21, 15, 105, 105, 315, and 210. Their stabilizer sizes in the affine group of order 5,040 are 144, 240, 336, 48, 48, 16, and 24. No orbit occurs at weights 1, 2, 4, 6, 8 through 13, 15, 17, or 19. The search uses exact remainders modulo \(\Phi_{105}\), exact Boolean constraints, and complete affine-orbit blocking. It is a bounded classification and does not settle whether further orbits occur at weight 20 or above.

Example 2.5 constructs the minimal sum \((\alpha+\cdots+\alpha^{p-1})(\beta+\cdots+\beta^{q-1})+\gamma+\cdots+\gamma^{\ell-1}=0\) for three distinct primes. With \((p,q,\ell)=(3,5,7)\), its support has \((3-1)(5-1)+(7-1)=14\) distinct 105th roots. Lower Bound Theorem 4.8 gives 14 as the minimum support of an asymmetric minimal sum in this case. Uniqueness Theorem 6.5 says every asymmetric minimal element at that support or weight is similar to \(\sigma(P_1^*)\sigma(P_2^*)+\sigma(P_3^*)\). The paper defines similarity as rotation. This supplies a source-backed uniqueness result at weight 14. In exponent notation, the construction has support \(\{7,14,15,28,30,45,49,56,60,75,77,90,91,98\}\); translation by 15 gives this packet's canonical representative.

Theorem 3.3 of arXiv:2008.11268v2 lists the types of all minimal vanishing sums of weight at most 16, up to rotation, and proves that they have height 1. The weight-16 table includes an all-odd relative-order-105 type \((R_7:1+\nu_3:(R_5:R_3))\). Section 4 and Appendix B describe an algorithmic list through weight 21. The current revision calls those later conclusions conjectural because the implementation has not been formally verified.

Table 1 cannot be used as a literal exhaustive filter for a fixed ambient conductor. Its weight-14 row places type \((R_7:(R_5:4R_3))\), including parity \((14,0)\), under displayed relative order 210, while the Lam-Leung construction realizes the relevant all-odd sum using 105th roots. The type table and this packet's fixed-conductor affine-orbit enumeration answer different classification questions.

The computation constructs the coefficient vector of each \(X^s\bmod\Phi_{105}\) and checks that the 48 by 105 matrix has rational rank 48. For each weight in increasing order, Z3 solves \(Ax=0\), \(x_s\in\{0,1\}\), \(x_0=1\), and the exact cardinality constraint. Requiring \(x_0=1\) loses no nonempty orbit because translation can move any selected exponent to zero.

Every nonminimal finite vanishing subset contains an inclusion-minimal vanishing subset of smaller weight. The solver therefore excludes a candidate exactly when it contains a previously enumerated minimal subset. When it finds a minimal subset, it blocks all affine images that contain zero. A final UNSAT result then closes that weight. The sequential replay includes every earlier orbit before moving to the next weight and ends with UNSAT at weight 19.

Two runs of the compact replay through weight 19 were byte-identical. Two earlier runs through weight 18 were also byte-identical. A standard-library-only cyclotomic construction produced the same matrix digest, and independent Gray-code scans checked all proper nonempty subsets of the weight-14, weight-16, and both weight-18 representatives. A separate targeted weight-19 solve also returned zero orbits. Its blockers included every minimal orbit through weight 16. Omitting the weight-18 blockers is harmless for that targeted check: a vanishing 19-set containing a vanishing 18-set would leave one root whose sum would have to vanish.

The audit checked Lam and Leung's paper, both arXiv source revisions of Christie, Dykema, and Klep, and the authors' public code repository. The 2020 v1 manuscript presented the implemented extension through weight 21 as a theorem. The current v2 manuscript expressly treats that extension as conjectural because the implementation was not formally verified. The public repository remains at commit b0563b270dc89ed7ca9e195535a107d5a81cc6dc from 2020-07-28 and predates the v2 correction.

Exact web searches on 2026-07-28 for conductor 105, 105th roots, distinct roots, minimal vanishing subsets, and affine or Galois equivalence found no primary source that lists every fixed-conductor affine orbit. This is a dated search result and supplies no proof of absence. Production orient resolved one published, open target, problem 2744, with zero attached research records and no stored statement-integrity advisory. Production check-plan found no stored work covering the exact matrix search, while warning that retrieval was not exhaustive.

The canonical statement omits an explicit condition that S be nonempty. Standard minimal-vanishing-sum usage has positive weight, and this packet begins at weight 1. A future resolution should add S nonempty to the statement or settle the weight-0 convention separately.

Start from the exact 48 by 105 remainder matrix and the full affine expansions of all seven certified representatives. Add the weight-20 cardinality constraint and every lower minimal-subset exclusion before solving. Canonicalize each survivor under the 5,040 affine maps, check all proper nonempty subsets exactly, and continue until the normalized solver returns UNSAT. A complete weight-20 record requires the source digest, environment, runtime, peak memory, representative and stabilizer data, and an independent replay. Compare any survivors with the conjectural weight-20 types in arXiv:2008.11268v2 as a cross-check only.

The program reduces every monomial \(X^s\), for \(0\leq s<105\), modulo \(\Phi_{105}\) over the integers. A subset vanishes exactly when its 0-1 incidence vector lies in the kernel of the resulting 48 by 105 matrix. Translation lets the solver impose \(x_0=1\). At each weight, constraints exclude every previously found lower-weight minimal subset. Every new solution is canonicalized under all 5,040 affine maps, and every normalized image in that orbit is blocked. UNSAT after orbit blocking proves completeness at that weight. A Gray-code scan checks every proper nonempty subset of each representative. The output is deterministic and records the matrix digest, ranks, representatives, orbit sizes, stabilizers, and proper-subset checks.