On a class of polynomials related to Barker sequences
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Bibliographic record
Abstract
For an odd integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we introduce the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L upper P Subscript n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}P_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Laurent polynomials <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis z right-parenthesis equals left-parenthesis n plus 1 right-parenthesis plus sigma-summation Underscript StartLayout 1st Row k equals 1 2nd Row k odd EndLayout Overscript n Endscripts c Subscript k Baseline left-parenthesis z Superscript k Baseline plus z Superscript negative k Baseline right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle scriptlevel="1"> <mml:mtable rowspacing="0.1em" columnspacing="0em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>k</mml:mi> <mml:mtext> odd</mml:mtext> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { odd}}}^{n}c_k (z^k+z^{-k}),</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> with all coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript k"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">c_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P element-of script upper L upper P Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">P \in \mathcal {L}P_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have large Mahler measures, namely, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper P right-parenthesis greater-than left-parenthesis n plus 1 right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">M(P) > (
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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