Local existence of 𝒦-sets, projective tensor products, and Arens regularity for 𝒜(ℰ₁+…+ℰ_{𝓃})
Bibliographic record
Abstract
<bold>Theorem.</bold> <italic> If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 1 comma ellipsis comma upper X Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X_{1},\dots ,X_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y 1 subset-of-or-equal-to upper X 1 comma ellipsis comma upper Y Subscript n Baseline subset-of-or-equal-to upper X Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Y_{1}\subseteq X_{1},\dots ,Y_{n}\subseteq X_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that </italic> (i) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Subscript j"> <mml:semantics> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Y_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <italic>is either a Kronecker set or</italic> (ii) <italic> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript j Baseline greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_{j}\ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Subscript j"> <mml:semantics> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Y_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a translate of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript p Sub Subscript j"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">K_{p_{j}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -set all of whose elements have order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript j"> <mml:semantics> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">p_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and </italic> (iii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis upper Y 1 plus midline-horizontal-ellipsis plus upper Y Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.008 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".