MINIMAL ZERO-SUM SEQUENCES OF LENGTH FOUR OVER FINITE CYCLIC GROUPS II
Bibliographic record
Abstract
Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n1g)⋅…⋅(nlg) where g ∈ G and n1, …, nl ∈ [1, ord (g)], and the index ind (S) of S is defined to be the minimum of (n1+⋯+nl)/ ord (g) over all possible g ∈ G such that 〈g〉 = G. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd (|G|, 6) = 1 has index 1. In this paper, we show that if G = 〈g〉 is a cyclic group with order of a product of two prime powers and gcd (|G|, 6) = 1, then every minimal zero-sum sequence S of the form S = (g)(n2g)(n3g)(n4g) has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of G is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".