A SAT-based Resolution of Lam's Problem (SAT instances and certificates)
Bibliographic record
Abstract
This repository contains SAT instances and certificates accompanying the paper "A SAT-based Resolution of Lam's Problem" appearing at AAAI 2021. This paper developed a method to generate certificates proving the nonexistence of a word of weight 19 in the code generated by a projective plane of order ten. Together with previously computed certificates this solves Lam's Problem. The 'a1' archive contains a certificate showing that there are exactly 66 A1 matrices up to isomorphism. Run the provided check.sh script to verify the certificate. The 'a2' archive contains certificates showing that there are exactly 650,370 A2 matrices up to isomorphism. Run the provided check.sh script to verify the certificates. The 'main' archive contains precomputed SAT instances for each of the A2 matrices up to isomorphism and partial solutions of the SAT instances. The main certificates may be generated and verified by extracting the main archive into the weight19/main directory of the MathCheck2 repository for Lam's problem (available from bitbucket.org/cbright/mathcheck2) and running the driver.sh script. The final-step/solve.sh script verifies that no partial solution can be completed to a full incidence matrix of a projective plane of order ten.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.006 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.064 | 0.063 |
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".