Some properties of invariants for Cohen-Macaulay local rings
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Bibliographic record
Abstract
Let (R,m,k) be a Cohen-Macaulay local ring with canonical module ω. In this note, we study how relations between index(R) and e(R) and between index(R) and gll(R) are preserved when factoring out regular sequences and localizing at prime ideals. In particular, we give conditions for when index(R)=e(R) implies index(R)=index(R¯)=e(R¯) and when index(R)=gll(R) implies index(R)=index(R¯)=gll(R¯), where R¯=R/xR and x∈m is a regular sequence. When R is an abstract hypersurface and k is infinite, we also show that the inequality index(Rp)≤index(R) holds for each unramified prime ideal p∈Spec(R). Finally, we prove the inequalities index(Rp)≤gll(Rp)≤index(Rp)+t−1 for height one prime ideals p∈Spec(R) that satisfy pRp⊆trR(ω)Rp and contain an element in their tth symbolic power p(t) which is regular on ⊕i=0∞p(i)/p(i+1).
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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