# P3698: Maximum total one-dimensional persistence in a six-square image

- ID: `P3698`
- Reference: `six-grid-max-total-one-persistence`
- Page: https://theoremdb.org/statements/P3698
- 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/six-grid-max-total-one-persistence?representation=markdown
- Record maturity: Reviewed problem

## The problem

Add the \(36\) closed unit squares of a \(6\times6\) board one at a time in some order, and let \(X_k\) be the cubical complex after \(k\) additions. Determine \(P_6=\max\sum_{k=1}^{36}\dim_{\mathbb F_2}H_1(X_k;\mathbb F_2)\). Is \(P_6>91\)?

### Remarks

- **Remark.** Every ordering is a permutation of the 36 labeled cells.
- **Remark.** The sum equals the total lifetime of all one-dimensional bars in this tie-free filtration.

### What counts as a solution

- Give an ordering with total at least 92, or certify by exhaustive search that every ordering has total at most 91.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

A randomized ordering already reaches 91, making the next integer a precise search target. Each prefix Betti number is cheap to verify.

- For each prefix, compute beta_1 exactly from cubical boundary ranks or from beta_0 minus Euler characteristic. Pixel-complement connectivity conventions can give wrong answers at diagonal contacts.
- Branch on early cells up to dihedral symmetry. An upper bound should account for the number and remaining lifetime of possible holes.

- Recorded example: One 91-order, using row-major cell indices 0 through 35, is 5,30,20,29,25,1,15,10,13,4,2,18,8,24,34,27,9,22,35,23,3,17,6,32,31,14,26,12,0,11,28,16,33,7,21,19.

### Computational notes

- Exact cubical Euler-characteristic checks give prefix beta_1 values 0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,4,4,4,4,5,6,7,7,6,5,5,5,5,4,3,3,2,1,0 for the displayed order; their sum is 91.
- A seeded random sweep of 50000 orders found no larger total before local-swap search was interrupted.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `six-grid-max-total-one-persistence`, 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.