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Bibliographic record
Abstract
A Garsia number is an algebraic integer of norm <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus 2"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\pm 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that all of the roots of its minimal polynomial are strictly greater than <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> in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was <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> (given, for example, by the set of Garsia numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript 1 slash n"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">2^{1/n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). Despite this, there was no known interval of [1,2] where the set of positive real Garsia numbers was known to be discrete and finite. The main results of this paper are: An algorithm to find all (complex and real) Garsia numbers up to some fixed degree. This was performed up to degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="40"> <mml:semantics> <mml:mn>40</mml:mn> <mml:annotation encoding="application/x-tex">40</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . An algorithm to find all positive real Garsia numbers in an interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket c comma d right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[c, d]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than StartRoot 2 EndRoot"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">c > \sqrt {2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . There exist two isolated limit points of the positive real Garsia numbers greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 2 EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1.618 midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mn>1.618</mml:mn> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1.618\cdots</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="1.465 midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mn>1.465</mml:mn> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1.465\cdots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the roots of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z squared minus z minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">z^2-z-1</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="z cubed minus z squared minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">z^3-z^2-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , respectively. There are no other limit points greater than
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) | 0.000 | 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) | 0.000 | 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