Codex may have computationally resolved the open Q26 queen-domination case. Seeking independent reproduction

A queen dominates its own square and every square sharing its row, column, or diagonal. The question is whether 13 queens can dominate all 676 squares of a 26 by 26 board.

Using known structural constraints, every hypothetical 13-queen solution falls into one of 156 canonical row-and-column parity profiles. For each profile, I enumerated the compatible empty-row and empty-column inventories and ran an exact SAT check on every surviving inventory. All returned unsatisfiable. Separate internal reruns and reviews found no omitted profile or apparent overconstraint.

Combined with the known 14-queen construction, this indicates that the minimum is 14.

Important caveat: the disposable search implementation and complete logs were not retained, nor was a formal SAT proof certificate generated. Therefore, this should currently be treated as a preliminary exhaustive computational result, not an independently reproducible proof.

Here is the basic reduction.

Assume that 13 queens dominate a 26 by 26 board.

Let X be the empty rows and Y the empty columns. Every square outside X times Y is automatically covered by a queen in its row or column. Therefore, only the “empty core” X times Y needs diagonal coverage.

For a queen at row r and column c, record its diagonal labels:

d = r - c s = r + c

A core square at (x,y) is covered exactly when x - y equals one of the d labels or x + y equals one of the s labels.

Known structural results imply that any 13-queen solution must split 6 and 7 between the checkerboard colors, occupy either 12 or 13 rows, and occupy either 12 or 13 columns.

Up to transposing the board, this leaves three types:

13 occupied rows and 13 occupied columns 12 occupied rows and 13 occupied columns 12 occupied rows and 12 occupied columns

Symmetry and parity reduce these to 156 canonical profiles: 16, 91, and 49 respectively. The search found no valid placement in any profile.

Background and structural results: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v29i2p50

Current known values and the 14-queen construction: https://oeis.org/A075458/a075458.txt

Computation performed with ChatGPT/Codex under my direction; not yet externally reproduced.