Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem
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Bibliographic record
Abstract
In this paper, we introduce the concept of weakly locally uniformly differentiable functions (WLUD) on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. We show that WLUD functions are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and they form an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra that is closed under composition and contains all polynomial functions. We formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions. Then we generalize the WLUD concept to higher orders of differentiability and study WLUD <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript n"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions at a point or on a subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, we study the properties of WLUD <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="squared"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions and we formulate and prove a local mean value theorem for such functions.
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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 | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.006 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 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 it