# P3666: Balanced ground-state count for the antiferromagnetic triangular seven torus

- ID: `P3666`
- Reference: `triangular-ising-seven-balanced-ground-count`
- Page: https://theoremdb.org/statements/P3666
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/triangular-ising-seven-balanced-ground-count?representation=markdown
- Record maturity: Reviewed problem

## The problem

On the \(7\times7\) triangular torus, how many spin configurations \(\sigma:V\to\{-1,1\}\) with exactly twenty-four plus spins have Ising energy \(H(\sigma)=\sum_{uv\in E}\sigma_u\sigma_v=-49\)?

### Remarks

- **Remark.** Vertices are pairs modulo 7, with edges in the three undirected directions (1,0), (0,1), and (1,-1).
- **Remark.** Configurations are counted as labeled assignments. Translations and global spin reversal are not identified.

### What counts as a solution

- Give the exact integer count with a transfer-matrix transcript or modular residues sufficient for unique reconstruction.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

The energy optimization is already settled by a matching construction and triangle lower bound. The remaining finite count measures the frustrated model's ground-state entropy in a precisely defined slice.

- Every elementary triangle has a parallel-spin edge. Double counting triangle-edge incidences proves H at least -49, so the requested configurations are exactly the balanced ground states.
- A row transfer matrix must retain the first row to enforce vertical and diagonal wraparound. Store counts by first row, last row, plus-spin total, and accumulated frustrated edges.

- Recorded example: One ground state has plus-spin coordinates (0,4),(0,5),(0,6),(1,0),(1,1),(1,2),(1,6),(2,2),(2,3),(2,4),(2,5),(3,0),(3,1),(3,6),(4,1),(4,2),(4,3),(4,4),(5,0),(5,5),(5,6),(6,1),(6,2),(6,3).

### Computational notes

- The torus has 147 edges and 98 elementary triangles. Each triangle forces a parallel-spin edge, and every edge belongs to two triangles, so at least 49 edges are parallel and H is at least -49.
- Exact evaluation of the displayed configuration gives 49 parallel and 98 antiparallel edges, hence H=-49 and a certified nonzero target count.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `triangular-ising-seven-balanced-ground-count`, 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.