A SAT+CAS Method for Enumerating Williamson Matrices of Even Order
Bibliographic record
Abstract
We present for the first time an exhaustive enumeration of Williamson matrices of even order n < 65. The search method relies on the novel SAT+CAS paradigm of coupling SAT solvers with computer algebra systems so as to take advantage of the advances made in both the field of satisfiability checking and the field of symbolic computation. Additionally, we use a programmatic SAT solver which allows conflict clauses to be learned programmatically, through a piece of code specifically tailored to the domain area. Prior to our work, Williamson matrices had only been enumerated for odd orders n < 60, so our work increases the bounds that Williamson matrices have been enumerated up to and provides the first enumeration of Williamson matrices of even order. Our results show that Williamson matrices of even order tend to be much more abundant than those of odd orders. In particular, Williamson matrices exist for every even order n < 65 but do not exist in orders 35, 47, 53, and 59.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.016 | 0.004 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".