The linear bound for Haar multiplier paraproducts
Bibliographic record
Abstract
We study the natural resolution of the conjugated Haar multiplier <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript sigma"> <mml:semantics> <mml:msub> <mml:mi>T</mml:mi> <mml:mi> σ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">T_\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> : <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript w Sub Superscript one half Subscript Baseline upper T Subscript sigma Baseline upper M Subscript w Sub Superscript negative one half Subscript Baseline equals left-parenthesis sans-serif upper P Subscript ModifyingAbove w Sub Superscript one half Subscript With caret Superscript left-parenthesis 0 comma 1 right-parenthesis Baseline plus sans-serif upper P Subscript ModifyingAbove w Sub Superscript one half Subscript With caret Superscript left-parenthesis 1 comma 0 right-parenthesis Baseline plus sans-serif upper P Subscript mathematical left-angle w Sub Superscript one half Subscript mathematical right-angle Superscript left-parenthesis 0 comma 0 right-parenthesis Baseline right-parenthesis upper T Subscript sigma Baseline left-parenthesis sans-serif upper P Subscript ModifyingAbove w Sub Superscript negative one half Subscript With caret Superscript left-parenthesis 0 comma 1 right-parenthesis Baseline plus sans-serif upper P Subscript ModifyingAbove w Sub Superscript negative one half Subscript With caret Superscript left-parenthesis 1 comma 0 right-parenthesis Baseline plus sans-serif upper P Subscript mathematical left-angle w Sub Superscript negative one half Subscript mathematical right-angle Superscript left-parenthesis 0 comma 0 right-parenthesis Baseline right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> σ </mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mspace width="negativethinmathspace"/> <mml:mo>=</mml:mo> <mml:mspace width="negativethinmathspace"/> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:msub>
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.
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 | 0.004 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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; both teacher heads agree on what is shown here.
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".