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Bibliographic record
Abstract
We examine sequences of polynomials with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet plus 1 comma negative 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{+1,-1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coefficients constructed using the iterations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis right-arrow p left-parenthesis x right-parenthesis plus-or-minus x Superscript d plus 1 Baseline p Superscript asterisk Baseline left-parenthesis negative x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ± </mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x)\rightarrow p(x)\pm x^{d+ 1}p^{*}(-x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">p^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the reciprocal polynomial of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p 0 equals 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_{0}=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> these generate the Rudin-Shapiro polynomials. We show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 4"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 4"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence. We determine all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p 0"> <mml:semantics> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">p_{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p 0"> <mml:semantics> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </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.
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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.002 | 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