Bibliographic record
Abstract
A bstract . Given a vector of finitely many piles of finitely many tokens. In End-Wythoff, two players alternate in taking a positive number of tokens from either end-pile, or taking the same positive number of tokens from both ends. The player first unable to move loses and the opponent wins. We characterize the P -positions ( a i , K , b i / of the game for any vector K of middle piles, where a i , b i denote the sizes of the end-piles. A more succinct characterization can be made in the special case where K is a vector such that, for some n ∈ ℤ ≤0 , ( K, n ) and ( n, K ) are both P -positions. For this case the (noisy) initial behavior of the P -positions is described precisely. Beyond the initial behavior, we have b i − a i = i , as in the normal 2-pile Wythoff game. Introduction A position in the (impartial) game End-Nim is a vector of finitely many piles of finitely many tokens. Two players alternate in taking a positive number of tokens from either end-pile (“burning-the-candle-at-both-ends”). The player first unable to move loses and the opponent wins. Albert and Nowakowski gave a winning strategy for End-Nim, by describing the P -positions of the game. (Their paper also includes a winning strategy for the partizan version of End-Nim.)
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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 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".