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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="upper F left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 0 Overscript upper N Endscripts a Subscript n Baseline z Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">F(z)=\sum _{n=0}^N a_n z^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a polynomial with complex coefficients and roots <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 1"> <mml:semantics> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\alpha _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="alpha Subscript upper N"> <mml:semantics> <mml:msub> <mml:mi> α </mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\alpha _N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper F double-vertical-bar Subscript p"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\|F\|_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm over the unit circle, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper F double-vertical-bar Subscript p"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\|F\|_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote Mahler’s measure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Gonçalves’ inequality asserts that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column double-vertical-bar upper F double-vertical-bar Subscript 2 2nd Column a m p semicolon greater-than-or-equal-to StartAbsoluteValue a Subscript upper N Baseline EndAbsoluteValue left-parenthesis product Underscript n equals 1 Overscript upper N Endscripts max left-brace right-brace comma 1 comma vertical-bar vertical-bar alpha n 2 plus product Underscript n equals 1 Overscript upper N Endscripts min left-brace right-brace comma 1 comma vertical-bar vertical-bar alpha n 2 right-parenthesis Superscript 1 slash 2 Baseline 3rd Column a m p semicolon equals double-vertical-bar upper F double-vertical-bar Subscript 0 Baseline left-parenthesis 1 plus StartFraction StartAbsoluteValue a 0 a Subscript upper N Baseline EndAbsoluteValue squared Over double-vertical-bar upper F double-vertical-bar Superscript 4 Baseline EndFraction right-parenthesis Superscript 1 slash 2 Baseline period EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>F</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mn>2</mml:mn> </mml:msub> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo> ≥ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:munderover> <mml:mo> ∏
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 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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