# P2640: Maximum determinant of a skew Seidel matrix of order 34

- ID: `P2640`
- Reference: `skew-seidel-maxdet-34`
- Page: https://theoremdb.org/statements/P2640
- Record maturity: Reviewed problem with recorded work

## Problem

Determine \(\max\det S\), where \(S\) ranges over the \(34\times34\) skew-symmetric matrices with zero diagonal and off-diagonal entries in \(\{-1,1\}\).

### Problem setup

- **Definition.** Skew-symmetric means S^T=-S.
- **Remark.** Such matrices encode tournaments; switching and relabeling preserve the determinant.

### What counts as a solution

- Provide a skew Seidel matrix attaining the maximum and an exact upper-bound certificate over all switching classes.

## Status

A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint. [1](#reference-1)

## Work

### Evidence for the current status

**Computation 1 (The maximum is between 35^16 and 6,876,227,375,063^2).** A principal submatrix of a skew conference matrix gives the lower endpoint, while the skew Ehlich-Wojtas bound and Pfaffian integrality give the upper endpoint.

Write \(D_{34}\) for the requested maximum. The certified bounds are
\[
5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625
 =35^{16}
 \leq D_{34}
 \leq 6{,}876{,}227{,}375{,}063^2
 =47{,}282{,}502{,}913{,}565{,}795{,}274{,}253{,}969.
\]
The lower endpoint is attained by the explicit order-34 matrix in `ssm34-artifact-gs36-principal-minor`. The upper endpoint follows for every matrix in the stated family. These sources and the computation do not determine equality at either endpoint, so the exact maximum remains open in this entry.

### Background and intake notes

Local search generates incumbents quickly, while proving optimality requires partitioned computation whose failed regions can be shared.

- Original intake status: A 2024 primary paper studies this maximum for general even order and supplies upper bounds. Its abstract does not state the exact order-34 value; a full table and citation check remains necessary.
- Canonicalize tournaments under vertex relabeling and switching. Determinants are squares of integer Pfaffians, which gives a strong exact consistency check.
- Trap: the Hadamard bound 33^17 ignores skew-symmetry and is far too loose. Floating-point log determinants are useful only for ranking.

- Recorded example: The best sampled matrix is encoded by the 561 upper-triangle signs in row-major order under seed 1534.

### Other known results

- **Computation 2** (reproduced): Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8. [2](#reference-2)
- **Proposition 1** (supported): The general order 2 modulo 4 bound gives 65 times 31^16, and the odd-square constraint rounds it down. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python reconstructs the published circulant blocks and checks every claimed identity with exact arithmetic. [2](#reference-2)

### Computational notes

- Among 20000 seeded random tournaments, the best floating-point score was replayed with exact Bareiss elimination. Its determinant was 1445221973536022889601=38016075199^2, confirming the required Pfaffian-square check.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `skew-seidel-maxdet-34`, 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>Sarah Klanderman, MurphyKate Montee, Andrzej Piotrowski, Alex Rice, and Bryan Shader, Determinants of Seidel Tournament Matrices, arXiv:2406.09697v1 (2024). Klanderman, Montee, Piotrowski, Rice, and Shader, Theorems 5.4 and 5.5, combined with the order-36 construction of Goethals and Seidel and the exact replay below https://arxiv.org/abs/2406.09697
   - Also cited at Klanderman et al., Theorem 5.4(a,d); Greaves and Suda, arXiv:1601.02769, Theorem 1.1 and the skew EW matrix discussion; Cayley's Pfaffian identity
   - preprint; primary source; arXiv:2406.09697v1; checked 2026-08-01
   - Source use: original_summary
   - Provides the skew Ehlich-Wojtas determinant bound and Pfaffian constraints used for the upper endpoint.
   - Source named by the research packet.
2. <a id="reference-2"></a>J. M. Goethals and J. J. Seidel, A skew Hadamard matrix of order 36, Journal of the Australian Mathematical Society 11(3) (1970), 343-344. Goethals and Seidel, A skew Hadamard matrix of order 36, Theorems 1 and 2, pages 343-344; exact replay in ssm34-artifact-gs36-principal-minor https://doi.org/10.1017/S144678870000673X
   - Also cited at Inline Python 3 standard-library replay of Goethals and Seidel, executed on 2026-07-25
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Gives the order-36 skew conference matrix whose principal submatrix attains determinant 35^16.