Bibliographic record
Abstract
Throughout this paper, the word “circle” means a circle or a straight line. We are always assuming that the space R is equipped with a fixed “standard” Euclidean inner product. A collection of curves in R passing through 0 is said to be a simple bundle of curves if no two of them are tangent at 0. A simple bundle of curves is called rectifiable if there exists a germ of diffeomorphism in a neighborhood of the origin that sends all curves from this bundle to straight lines. Rectifiable bundles of curves appear, for example, in Riemannian geometry — the set of geodesics passing through a given point is rectifiable. A. G. Khovanskii proved in [1] that a rectifiable simple bundle of more than 6 circles on plane necessarily pass through some point different from the origin. F. A. Izadi [2] generalized Khovanskii’s arguments to dimension 3. A rectifiable simple bundle of circles in R containing sufficiently many circles in general position must pass through some other common point. In dimension 4, this is not true. The simplest counterexample is a family of circles that are obtained from straight lines by some complex projective transformation (with respect to some identification R = C such that the multiplication by i is an orthogonal operator). It turns out that in dimension 4 there is a large family of transformations that round lines (i.e., take them to circles). To construct such a family, fix a quaternionic structure on R compatible with the Euclidean structure. If A and B are some affine maps, then the map x 7→ A(x)−1B(x) rounds lines (the multiplication and the inverse are in the sense of quaternions). Such transformations will be called (left) quaternionic fractional transformations. Right quaternionic fractional transformations AB−1 also round ∗Partially supported by RFBR 99-01-00245 and CRDF RM1-2086
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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".