Critical sets of inertias for matrix patterns
Why this work is in the frame
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Bibliographic record
Abstract
An n by n zero-nonzero (resp. sign) pattern is a matrix with entries in {*, 0} (resp. {+, −, 0}), where * denotes a nonzero real number. If allows all (n + 1)(n + 2)/2 possible inertias, then is inertially arbitrary. A nonempty, proper subset S of the set of all possible inertias is a critical set of inertias of order n provided an n by n pattern is inertially arbitrary whenever allows all inertias in S, and S is a minimal critical set if no proper subset of S is a critical set. This new concept of critical sets of inertias facilitates the identification of inertially arbitrary patterns. Minimal critical sets of inertias for irreducible zero-nonzero patterns of orders n = 2, 3, 4 are identified. Using these critical sets of inertias, it is shown that a direct sum of two non-inertially-arbitrary irreducible zero-nonzero patterns of orders ≤ 4 is not inertially arbitrary. However, it is shown that there exists a reducible inertially arbitrary zero-nonzero pattern that is a direct sum of non-inertially-arbitrary irreducible zero-nonzero patterns of order 5. Minimal critical sets of inertias for irreducible sign patterns of orders n = 2, 3 are also identified. It is shown that in general critical sets of inertias for irreducible sign patterns are different from those of irreducible zero-nonzero patterns.
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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