Limiting behavior of Rudin–Shapiro sequence autocorrelations
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis r 0 comma r 1 comma midline-horizontal-ellipsis comma r Subscript 2 Sub Superscript m Subscript minus 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(r_0, r_1, \cdots , r_{2^m-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th Rudin–Shapiro sequence defined by <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript i Baseline equals left-parenthesis negative 1 right-parenthesis Superscript reverse-solidus number-sign of pairs of consecutive ones in the binary expansion of i"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mtext>\# of pairs of consecutive ones in the binary expansion of </mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">r_{i} = (-1)^{\text {\# of pairs of consecutive ones in the binary expansion of $i$}}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript m Baseline left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_m(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be its aperiodic autocorrelation at shift <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we study the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript 1"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^1</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="script l squared"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{2}</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="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moments of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript m Baseline left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_{m}(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 0 greater-than k less-than-or-equal-to x Endscripts left-parenthesis upper C Subscript m Baseline left-parenthesis k right-parenthesis right-parenthesis squared"> <mml:semantics> <mml:mrow
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| 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 it