Mersenne Block Dynamics: A Framework for the Collatz Conjecture
Bibliographic record
Abstract
This paper introduces Mersenne Block Dynamics, a structural framework for analyzing the accelerated Collatz or Syracuse map on odd integers. The approach decomposes orbits based on the 2-adic valuation of the successor of an odd integer, effectively measuring the length of the trailing run of ones in its binary expansion, termed the Mersenne tail. This decomposition partitions the dynamics into deterministic blocks where the tail length decreases by exactly one bit at each step, creating a rigid wedge pattern in the binary representation. The framework defines a coarse-grained block map that transitions directly between the starts of successive blocks, isolating all arithmetic complexity into a specific exit exponent. The study derives explicit closed-form transition identities and exact time-scale bookkeeping for these block jumps. Furthermore, it establishes that the block length and exit parameters follow independent geometric distributions in terms of natural density. Under a heuristic assumption of orbit mixing, this intrinsic statistical model predicts a net negative expected logarithmic drift, recovering the classical probabilistic prediction for the Collatz conjecture within a precise structural coordinate system.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.003 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".