Why this work is in the frame
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Bibliographic record
Abstract
Let G be a finite group. A G-variety X is an algebraic variety with a regular G-action; X is faithful if every 1 6 = g ∈ G acts non-trivially. I will refer to a dominant G-equivariant rational (respectively, regular) map of faithful G-varieties as a rational (respectively, regular) compression. All varieties, actions, vector spaces, maps, etc., are assumed to be defined over a fixed algebraically closed base field k of characteristic zero; all varieties are assumed to be irreducible. I would like to thank V. L. Popov for stimulating discussions and for helpful comments on an earlier draft of this note. 1. Essential dimension Let V be a faithful linear representation of G and let d be the minimal value of dim(X), where the minimum is taken over all rational compressions f: V 99K X. Note that (a) (see [1, Theorem 3.1] or [6, Theorem 3.4(b)]) d depends only on the group G and not on the choice of V, and (b) (cf. [6, Proposition 7.1]) in the definition of d we may assume that X is a G-invariant subvariety of V, i.e., X is the closure of the image of a rational covariant f: V 99K V. The number d is called the essential dimension of G and is usually denoted by ed(G). This number has interesting connections with the algebraic form of Hilbert’s 13th problem, cohomological invariants, generic polynomials and other topics; these connections are described in [1] and [2]. The case where G = Sn is of particular interest. (The notion of essential dimension is also of interest in the context of algebraic groups; see [6] and [7].) Problem 1. Find ed(G) and, in particular, ed(Sn). The value of ed(G) is known if G is an abelian group; see [1, Theorem 6.1]. For symmetric groups, ed(Sn) ≥ [n/2]; this is proved, in different ways,
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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 it