Shape, scale, and minimality of matrix ranges
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Bibliographic record
Abstract
We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely, the product of the smallest matrix convex sets over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript i Baseline subset-of-or-equal-to double-struck upper C Superscript d"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">K_i \subseteq \mathbb {C}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the smallest matrix convex set over the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript i"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Second, if a matrix convex set is given as the matrix range of an operator tuple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (not necessarily self-adjoint) contractions to commuting normal operators, both concretely and abstractly.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.003 |
| 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