Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally compact SIN group, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a weight on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is diagonally bounded with bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a dispersed set. We show that the topological centre of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L upper U upper C left-parenthesis w Superscript negative 1 Baseline right-parenthesis Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>L</mml:mi> <mml:mi>U</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{LUC}(w^{-1})^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis w right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M(w)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and there are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left floor upper K right floor plus 1"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⌊ </mml:mo> <mml:mi>K</mml:mi> <mml:mo fence="false" stretchy="false"> ⌋ </mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lfloor K \rfloor + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> many points in the spectrum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L upper U upper C left-parenthesis w Superscript negative 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>L</mml:mi> <mml:mi>U</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{LUC}(w^{-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are determining for the topological centre (DTC). As corollaries, we obtain DTC results as well for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1 left-parenthesis w right-parenthesis Superscript asterisk asterisk"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>w</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">L_1(w)^{**}</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 C 0 left-parenthesis w Superscript negative 1 Baseline right-parenthesis Superscript up-tack"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊥ </mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">C_0(w^{-1})^\perp</mml:annotation> </mml:semantics>
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| 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".