Bibliographic record
Abstract
A k -uniform, d -regular instance of EXACT COVER is a family of m sets F_{n,d,k} = \{ S_j \subseteq \{1,\ldots,n\} \} , where each subset has size k and each 1 \le i \le n is contained in d of the S_j . It is satisfiable if there is a subset T \subseteq \{1,\ldots,n\} such that |T \cap S_j|=1 for all j . Alternately, we can consider it a d -regular instance of POSITIVE 1-IN- k SAT, i.e., a Boolean formula with m clauses and n variables where each clause contains k variables and demands that exactly one of them is true. We determine the satisfiability threshold for random instances of this type with k > 2 . Letting d^\star = \frac{\ln k}{(k-1)(- \ln (1-1/k))} + 1 \, , we show that F_{n,d,k} is satisfiable with high probability if d < d^\star and unsatisfiable with high probability if d > d^\star . We do this with a simple application of the first and second moment methods, boosting the probability of satisfiability below d^\star to 1-o(1) using the small subgraph conditioning method.
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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 | 0.004 | 0.027 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
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".