Convolution powers in the operator-valued framework
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We consider the framework of an operator-valued noncommutative probability space over a unital <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show how for a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -valued distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> one can define convolution powers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript squared-plus eta"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⊞ </mml:mo> <mml:mi> η </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\boxplus \eta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with respect to free additive convolution) and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript multiset-union eta"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⊎ </mml:mo> <mml:mi> η </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\uplus \eta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with respect to Boolean convolution), where the exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta"> <mml:semantics> <mml:mi> η </mml:mi> <mml:annotation encoding="application/x-tex">\eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a suitably chosen linear map from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , instead of being a nonnegative real number. More precisely, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript multiset-union eta"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⊎ </mml:mo> <mml:mi> η </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\uplus \eta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always defined when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta"> <mml:semantics> <mml:mi> η </mml:mi> <mml:annotation encoding="application/x-tex">\eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is completely positive, while <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Superscript squared-plus eta"> <mml:semantics> <mml:msup> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⊞ </mml:mo> <mml:mi> η </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mu ^{\boxplus \eta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always defined when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta minus 1"> <mml:semantics> <mml:mrow> <mml:mi>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| 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 | 0.000 |
| 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 it