A Weaker Smoothness Criterion for the Inverse Function Theorem, the Intermediate Value Theorem, and the Mean Value Theorem in a non-Archimedean Setting
Bibliographic record
Abstract
We introduce a class of so-called very weakly locally uniformly differentiable (VWLUD) functions at a point of a general non-Archimedean ordered field extension of the real numbers, $$\mathcal{N}$$ , which is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. This new class of functions is defined by a significantly weaker criterion than that of the class of weakly locally uniformly differentiable (WLUD) functions studied in [1], which is nonetheless sufficient for a slight variation of the inverse function theorem and intermediate value theorem. Similarly, a weaker second order criterion is derived from the previously studied WLUD $$^2$$ condition for twice-differentiable functions. We show that VWLUD $$^2\,$$ functions at a point of $$\mathcal{N}$$ satisfy the mean value theorem in an interval around that point.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| 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 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".