Why this work is in the frame
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Bibliographic record
Abstract
We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and the binomials that define V and W have the same binomial exponents, then the defining equations of V⋆W can be computed explicitly and directly from the defining equations of V and W. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal IG of a graph G and the toric ideal IH of a subgraph H of G. We also derive results about algebraic invariants of Hadamard products: assuming V and W are binomial with the same exponents, we show that deg(V⋆W)=deg(V)=deg(W) and dim(V⋆W)=dim(V)=dim(W). Finally, given any (not necessarily binomial) projective variety V and a point p∈Pn∖V(x0x1⋯xn), subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points q that satisfy p⋆V=q⋆V.
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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 | 0.000 |
| 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 | 0.000 |
| Open science | 0.000 | 0.000 |
| 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