# P3706: Zero-flux square-ice configurations on the twelve torus

- ID: `P3706`
- Reference: `square-ice-twelve-torus-zero-flux-count`
- Page: https://theoremdb.org/statements/P3706
- 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/square-ice-twelve-torus-zero-flux-count?representation=markdown
- Record maturity: Reviewed problem

## The problem

How many orientations of the edges of \(C_{12}\square C_{12}\) have indegree and outdegree \(2\) at every vertex and horizontal and vertical winding flux both equal to \(0\)?

### Remarks

- **Remark.** Horizontal flux is right-pointing minus left-pointing arrows across any fixed vertical cut; the ice rule makes it cut-independent. Vertical flux is defined analogously.
- **Remark.** Edges and vertices are labeled. Translations and arrow reversal are not identified.

### What counts as a solution

- Give the exact integer count with sector-resolved transfer-matrix residues and a bound proving unique reconstruction.

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Background and intake notes

Fixing both winding fluxes removes the large sector ambiguity that otherwise makes finite torus counts hard to compare.

- Use a row transfer indexed by the 4096 vertical-arrow masks and track horizontal winding. Restrict to masks with six up arrows for zero vertical flux.
- Compute modulo independent primes and verify that row-transfer traces agree under arrow reversal and cyclic shifts.

- Recorded example: Choose six rows whose horizontal arrows all point right and six pointing left, then choose six columns whose vertical arrows point up and six pointing down. Every such choice obeys the ice rule and has zero flux.

### Computational notes

- The row-and-column construction gives C(12,6)^2=853776 distinct zero-flux configurations, providing a certified positive lower bound.
- The torus has 144 vertices and 288 edges. A direct orientation scan has 2^288 states, while the row transfer has 4096 boundary masks before flux reduction.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `square-ice-twelve-torus-zero-flux-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.