Statistical Analysis of Descending Open Cycles of Collatz Function
Bibliographic record
Abstract
Collatz dynamic systems present a statistical space that can be studied rigorously. In a previous study, the author presented Collatz space in a unique dynamic numerical mode by tabulating a sequential correlation pattern of division by 2 of Collatz function’s even numbers until the numbers became odd with a consecutive occurrence, following an attribute of a 50:50 probability of division by 2 once (ascending behavior) as opposed to division by 2 more than once (descending behavior). In this paper, we describe the path of the Collatz function as sequences comprised of groups of the function’s iterates (open cycles) that end up with the first odd integer that is less than the starting odd integer. The descending behavior of the open cycles is attributed to a deterministic factor as observation of the cycles’ sequences shows. We do statistical analysis on 4 large samples of open cycles and orbits to 1. We define R(n) as the cycles’ deterministic variable defined as the ratio of division by 2 once to division by 2 more than once. We use statistical analysis to study the randomness of the orbits of the cycles’ starting odd positive integers as well as orbits to 1 up to 1,002,097,149.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| 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 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".