Integrals, partitions, and cellular automata
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Bibliographic record
Abstract
We prove that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline StartFraction minus log f left-parenthesis x right-parenthesis Over x EndFraction d x equals StartFraction pi squared Over 3 a b EndFraction comma"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mfrac> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:mfrac> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:msup> <mml:mi> π </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*}\int _0^1\frac {-\log f(x)}xdx=\frac {\pi ^2}{3ab},\end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the decreasing function that satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript a Baseline minus f Superscript b Baseline equals x Superscript a Baseline minus x Superscript b"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>b</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f^a-f^b=x^a-x^b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than a greater-than b"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0>a>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an integer and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b equals a plus 1"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b=a+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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