# P2628: Zero-sum-free subsets of the unit sphere over F_7

- ID: `P2628`
- Reference: `zero-sum-free-f7-sphere`
- Page: https://theoremdb.org/statements/P2628
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the largest subset \(A\) of \(S=\{(x,y,z)\in\mathbb F_7^3:x^2+y^2+z^2=1\}\) for which no nonempty subset of distinct elements of \(A\) sums to zero.

### Remarks

- **Remark.** All coordinate arithmetic is modulo 7.
- **Remark.** Subset sums use each selected vector at most once.

### What counts as a solution

- Give a zero-sum-free subset attaining the maximum and a complete upper-bound certificate under the 42-point ground set.

## Status

An exhaustive replay certifies a 13-point construction, and Olson's exact Davenport constant for C_7^3 gives the upper endpoint. [2](#reference-2)

## Work

### Evidence for the current status

**Computation 1 (The certified interval is 13 to 18).** An exhaustive replay certifies a 13-point construction, and Olson's exact Davenport constant for C_7^3 gives the upper endpoint.

Let
\[
S=\{(x,y,z)\in\mathbb F_7^3:x^2+y^2+z^2=1\}.
\]
Direct enumeration gives \(|S|=42\). The displayed set
\[
\begin{split}
A=\{&(3,5,4),(2,4,4),(2,5,0),(4,3,5),(4,5,3),\\
&(2,0,5),(4,2,4),(5,0,5),(5,3,4),(0,0,1),\\
&(2,3,3),(0,2,5),(5,5,0)\}
\end{split}
\]
lies in \(S\). The executable verifier evaluates all \(2^{13}-1=8191\) nonempty subsets and finds no zero sum. This proves that the unknown maximum \(M\) satisfies \(M\geq13\).

Olson proved the exact Davenport constant for finite abelian p-groups. Applied to \(C_7^3\), it gives
\[
D(C_7^3)=1+3(7-1)=19.
\]
Every sequence of 19 elements of \(C_7^3\) therefore has a nonempty zero-sum subsequence. A 19-point subset of \(S\) is such a sequence, with each term appearing once, so \(M\leq18\). Hence
\[
13\leq M\leq18.
\]
The computation reported in the candidate record supplies the lower endpoint. This fixture independently replays it. The exact value remains open in this audit.

### Background and intake notes

The small sum-state space supports exact dynamic programming, while the orthogonal symmetry creates many repeated search branches worth recording.

- Original intake status: Status remains unverified. The restriction to one quadratic sphere may appear in finite-geometry or zero-sum work.
- The sphere has 42 points. Track the 343 possible subset sums as a bitset, and quotient by the orthogonal group O(3,7) during exhaustive search.
- Trap: a greedy set is valid only if all subset sums are tracked. Checking zero-sum triples or pairs alone misses longer relations.

- Recorded example: A verified 13-set is {(3,5,4),(2,4,4),(2,5,0),(4,3,5),(4,5,3),(2,0,5),(4,2,4),(5,0,5),(5,3,4),(0,0,1),(2,3,3),(0,2,5),(5,5,0)}.

### Other known results

- **Computation 2** (reproduced): Its nonempty subset sums cover all 342 nonzero vectors of F_7^3. [2](#reference-2)

### Prior approaches

- **Route 1** (inconclusive): Two of five second-point orbits returned UNSAT, while the timebox ended before three cases were certified. [1](#reference-1) [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): A dependency-free Python program checks the sphere, every nonempty subset, the full nonzero sum image, and a canonical digest. [2](#reference-2)

### Computational notes

- Direct enumeration found exactly 42 vectors on S. A seeded incremental subset-sum search found the displayed 13-set; its 8191 nonempty subsets were checked and none had sum zero. Since D(C_7^3)=19, every 19-element subset of the ambient group has a nonempty zero sum. Thus the current certified bounds are 13<=|A|<=18.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `zero-sum-free-f7-sphere`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Cosmin Pohoata and Dmitriy Zakharov, “Zero subsums in vector spaces over finite fields”. Alg. Number Th. 16 (2022) 1407-1421. DOI 10.2140/ant.2022.16.1407. arXiv:2009.08846 (2020). Cosmin Pohoata and Dmitriy Zakharov, Zero subsums in vector spaces over finite fields, Journal of the London Mathematical Society 104 (2021), 1113-1139; Oscar Ordaz, Andreas Philipp, Irene Santos, and Wolfgang A. Schmid, On the Olson and the Strong Davenport constants, Journal de Théorie des Nombres de Bordeaux 23 (2011), 715-750; timeboxed Z3 4.15.4 search on 2026-07-25 https://arxiv.org/abs/2009.08846
   - preprint; primary source; arXiv:2009.08846, version checked 2026-07-25; checked 2026-08-01
   - Source use: original_summary
   - For Zero-sum-free subsets of the unit sphere over F_7: Two of five second-point orbits returned UNSAT, while the timebox ended before three cases were certified.
2. <a id="reference-2"></a>John E. Olson, “A combinatorial problem on finite Abelian groups, I”. Journal of Number Theory 1(1) (1969), 8-10. DOI 10.1016/0022-314X(69)90021-3. John E. Olson, A combinatorial problem on finite Abelian groups, I, Journal of Number Theory 1 (1969), 8-10; lower endpoint replayed in zsf7s-artifact-thirteen-point-verifier https://doi.org/10.1016/0022-314X(69)90021-3
   - Also cited at Exact subset-sum image computed by zsf7s-artifact-thirteen-point-verifier
   - Also cited at Inline Python 3 standard-library verifier executed on 2026-07-25
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - For Zero-sum-free subsets of the unit sphere over F_7: The certified interval is 13 to 18. An exhaustive replay certifies a 13-point construction, and Olson's exact Davenport constant for C_7^3 gives the upper endpoint.
   - Source named by the research packet.
3. <a id="reference-3"></a>Oscar Ordaz, Andreas Philipp, Irene Santos, and Wolfgang A. Schmid, “On the Olson and the Strong Davenport constants”. Journal de théorie des nombres de Bordeaux 23(3) (2011), 715-750. DOI 10.5802/jtnb.784. Bounds for Olson constants of finite abelian groups https://doi.org/10.5802/jtnb.784
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For Zero-sum-free subsets of the unit sphere over F_7: Three symmetry cases remain in the 14-point search. Two of five second-point orbits returned UNSAT, while the timebox ended before three cases were certified.