The maximal function and conditional square function control the variation: An elementary proof
Bibliographic record
Abstract
In this note we prove the following good- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> inequality, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> λ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda > 0</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="delta element-of left-parenthesis 0 comma one half right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> δ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta \in \big (0, \frac {1}{2} \big )</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 C dot nu left-brace upper V Subscript r Baseline left-parenthesis f right-parenthesis greater-than 3 lamda semicolon script upper M left-parenthesis f right-parenthesis greater-than delta lamda right-brace less-than-or-equal-to nu left-brace s left-parenthesis f right-parenthesis greater-than delta lamda right-brace plus StartFraction delta squared Over left-parenthesis r minus 2 right-parenthesis squared EndFraction dot nu left-brace upper V Subscript r Baseline left-parenthesis f right-parenthesis greater-than lamda right-brace comma"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo> ⋅ </mml:mo> <mml:mi> ν </mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mn>3</mml:mn> <mml:mi> λ </mml:mi> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mi> δ </mml:mi> <mml:mi> λ </mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo> ≤ </mml:mo> <mml:mi> ν </mml:mi> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mi> δ </mml:mi> <mml:mi> λ </mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>+</mml:mo> <mml:mfrac> <mml:msup> <mml:mi> δ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo> ⋅ </mml:mo> <mml:mi> ν </mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mi> λ </mml:mi> <mml:mstyle scriptl
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| 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".