Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We introduce a notion of parity for transversals, and use it to show that in Latin squares of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mspace width="0.277778em"/> <mml:mo form="prefix">mod</mml:mo> <mml:mspace width="0.277778em"/> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is the number of diagonals of a given Latin square that contain exactly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> different symbols. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denote the matrix obtained by deleting row <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> and column <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> </mml:math> from a parent matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . Define <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:math> to be the number of transversals in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , for some fixed Latin square <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> . We show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width="0.277778em"/> <mml:mo form="prefix">mod</mml:mo> <mml:mspace width="0.277778em"/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> . Also, if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> has odd order then the number of transversals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> equals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:math> mod 2. We conjecture that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="0.277778em"/> <mml:mo form="prefix">mod</mml:mo> <mml:mspace width="0.277778em"/> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> . In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -regular bipartite graph on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> vertices is divisible by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>4</mml:mn> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it