Chain-k Mappings: A Combinatorial and Spectral Approach to Analyzing Complete Mappings
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Bibliographic record
Abstract
This paper investigates bijections on a group G that arise from products of the form f(x)g(x), a problem that is centrally connected to the concept of a complete mapping. We introduce the notion of a “chain-k mapping” to analyze the structure of certain full-cycle permutations and explore the relationship between complete mappings and the spectral properties of their associated permutation matrices. Key results include a proof that the cyclic group Z/nZ admits a complete mapping if and only if n is odd, and a characterization of the cycle structures of permutations σ for which the maps id+σ and id−σ are automorphisms of a specific vector subspace. This work establishes a connection between combinatorial group theory and the eigenvalue theory of permutation matrices, viewed through the novel lens of “(-1)-elliptic elements”. We will begin with the motivation that led to this study, and look at problems including elliptic elements and complete mappings over finite fields.
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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 | 0.006 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.001 | 0.012 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| 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