INTERMEDIATE AND LIMITING BEHAVIOR OF POWERS OF SOME CIRCULANT MATRICES
Bibliographic record
Abstract
Let A be an arbitrary circulant stochastic matrix, and let x 0 be a vector. An canonical form is derived for A k (as k !1 ) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with tran- sition matrix A 0 , and by analyzing how A acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of A k is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence xk = A k x0 is accomplished by utilizing congruential matrix identities and (0,1)-matrices to determine the minimum 2-adic order of the co- ordinates of xk through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeat- edly applied simultaneously to each term of a finite sequence when read cyclically.
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How this classification was reachedexpand
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".