The graph of a family of functions over quadratic extensions of finite fields
Bibliographic record
Abstract
Brochero and Teixeira (2023) [4] showed the behavior of the functional graph of f a ( X ) = X q + 1 + a X 2 over quadratic extensions of finite fields explicitly for a ∈ { 1 , − 1 } . In this article, we create a family of functions using repeated iterations of the function f a ( X ) and taking values of a ∈ { 1 , − 1 } in each iteration. Let α be an n -sequence of values for a , taken over { 1 , − 1 } , and f α ( X ) be the resulting function. We present the form of f α ( X ) and use it to derive a closed formula for the number and length of cycles present in the functional graph of f α ( X ) . We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.
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How this classification was reachedexpand
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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".