<b>Curves and Regular Surfaces in</b> ℝ <sup>3</sup>
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
This chapter takes a different approach to the problem that can be loosely described as follows: a “smooth surface” is defined to be a topological subspace of R<sup>3</sup> that can be covered in a piecewise fashion by a collection of parametrized surfaces in such a way that the pieces “fit together nicely”. By definition, a regular surface is a patchwork of images of parametrized surfaces. The chapter shows that the existence of charts on regular surfaces makes it possible to answer questions about extended smoothness of maps on regular surfaces using methods developed for Euclidean smoothness. Having defined a regular surface and established some of its basic properties, the chapter also presents a rigorous definition of “tangent plane”. It defines four types of regular surfaces: open sets in regular surfaces, graphs of functions, surfaces of revolution, and level sets of functions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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