# P3604: Smallest rotation-invariant saturating set in AG(3,7)

- ID: `P3604`
- Reference: `rotation-saturating-set-ag3-7`
- Page: https://theoremdb.org/statements/P3604
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: b5a83bd9bdbf7dfdc7134c15b7360f889389e7bc
- Current Markdown: https://api.theoremdb.org/v1/statements/rotation-saturating-set-ag3-7?representation=markdown
- Record maturity: Reviewed problem

## The problem

Determine the smallest subset \(S\subseteq\mathbb F_7^3\) that is invariant under cyclic coordinate rotation and has every point of \(\mathbb F_7^3\) on a line through two distinct points of S.

### Remarks

- **Remark.** Coordinate rotation sends (x,y,z) to (y,z,x).
- **Remark.** Points of S count as covered; every other point must lie on a secant of S.

### What counts as a solution

- Give an invariant saturating set of minimum size and a certificate excluding every smaller orbit union.

## Status

Status unverified. Saturating-set tables may not record this coordinate-rotation constraint.

## Research packet

### Background and intake notes

The current bounds leave four cardinalities, and each orbit choice is easy to replay.

- Search on the 119 rotation orbits, updating the set of points covered by secants after each orbit choice.
- Trap: an orbit can add three points but many duplicate secants. Counting raw pairs overstates coverage.

- Recorded example: A verified 16-set is {025,035,111,155,222,250,333,336,350,363,502,503,515,551,633,666}, with triples read as vectors over F_7.

### Computational notes

- A pair contributes at most five new interior points on its affine line, so m+5 binomial(m,2)>=343 and m>=13. Forty seeded orbit-greedy runs found the displayed 16-set; exact secant closure covered all 343 points. The certified interval is 13 through 16.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `rotation-saturating-set-ag3-7`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.