Bibliographic record
Abstract
Blocks are unit convergence between two consecutive odd numbers formed according to the three x plus one conjecture rules. The left odd number is the left hook, L, and the right odd number is the right hook, R. They include even numbers between their boundaries. They are divided into families (F1 = 5, 11, 17, … & F2 = 1, 7, 11, … & F3 = 3, 9, 15, …) and groups based on their group length (The number of the middle-even numbers between the two hooks). Blocks are taken individually and placed beside each other, similar to the domino tiles play, which, by their formulation, satisfies the conjecture rules. Formed chains reach number one in the convergence mode or continue generating odd positive numbers infinitely according to the generation mode. The final convergence to number one is reached because these blocks have all the positive integers included as left hooks (L1, L2, L3), and all the F1 and F2 odd positive numbers are included as right hooks (R1 and R2). Block rules mandate that a single left hook produces only one right hook. Accordingly, no looping or entanglement (Joining and consequent splitting) between chain branches would occur. Statistics show that R cannot increase infinitely. Repeated oscillation up and down without reaching number one would also violate the statistics. Statistics reveal that blocks of various lengths have a strict occurrence and repetition sequence along the positive integer series. Block lengths can extend infinitely, and each block length repeats its occurrence infinitely. In the generation mode, blocks are attached in reverse order to the conjecture/convergence rules. According to the rules, all positive integers can be generated starting from number one. Multiple sequences and clusters of specific block lengths occur according to specific rules and cannot continue infinitely.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.042 | 0.007 |
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".