Dynamics of the Fibonacci Order of Appearance Map
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
The order of appearancez(n) of a positive integer n in the Fibonacci sequence is defined as the smallest positive integer j such that n divides the jth Fibonacci number. We prove that for every k≥0 there exist infinitely many integers that reach a fixed point of z after applying exactly k iterations of z. In addition, we show that, if x is a fixed point of z greater than 5, then there exist infinitely many integers whose orbits under z reach x. We also give a new proof of the theorem stating that every positive integer reaches a fixed point of z after a finite number of iterations.This paper began when the first and fourth named authors were looking through papers of the Fibonacci Quarterly in a search for a research project for the 2023 SMALL REU at Williams College. We gratefully dedicate this article to Curtis Cooper for his quarter-century of excellent stewardship as Editor-in-Chief of the Fibonacci Quarterly.
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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 | 0.000 |
| 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