On a Lebesgue-like Measure on the Levi-Civita Space $$\mathcal{R} ^j$$
Bibliographic record
Abstract
In a previous paper [2], we developed a new Lebesgue-like measure on the Levi-Civita field $$\mathcal{R}$$ that proved to be a strict improvement over the previously defined S-measure defined in [13, 9]. Nevertheless, we were only at first able to define such a measure for the one dimensional case leaving the case for higher dimensions as an open-ended question to be further researched. In another paper [15], the authors developed a generalization of the S-measure into higher dimensions using simplexes as their basic building blocks instead of boxes as simplexes proved to be more suitable for the topological structure of the Levi-Civita field $$\mathcal{R}$$ . However, the resulting measure naturally inherited the same limitations that the original S-measure on $$\mathcal{R}$$ had. In this new paper, we expand the same characterization given in [2] for the one-dimensional S-measurable sets to the S-measurable sets in $$\mathcal{R}^j$$ as defined in [15] and develop our own generalization to higher dimensions for the measure given in [2].
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".