Characterizations and models for the $C_{1,r}$ class and quantum annulus
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Bibliographic record
Abstract
Abstract For fixed $0<r<1$ , let $A_r=\{z \in \mathbb {C} : r<|z|<1\}$ be the annulus with boundary $\partial \overline {A}_r=\mathbb {T} \cup r\mathbb {T}$ , where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$ . An operator having $\overline {A}_r$ as a spectral set is called an $A_r$ - contraction . Also, a normal operator with its spectrum lying in the boundary $\partial \overline {A}_r$ is called an $A_r$ -unitary . The $C_{1,r}$ class was introduced by Bello and Yakubovich in the following way: $$\begin{align*}C_{1, r}=\{T: T \ \text{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \end{align*}$$ McCullough and Pascoe defined the quantum annulus $\mathbb Q \mathbb A_r$ by $$\begin{align*}\mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \end{align*}$$ If $\mathcal A_r$ denotes the set of all $A_r$ -contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$ . We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$ .
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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 | 0.001 | 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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