No two non-real conjugates of a Pisot number have the same imaginary part
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Bibliographic record
Abstract
We show that the number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha equals left-parenthesis 1 plus NestedStartRoot 3 plus 2 StartRoot 5 EndRoot NestedEndRoot right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> </mml:msqrt> <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">\alpha =(1+\sqrt {3+2\sqrt {5}})/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with minimal polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript 4 Baseline minus 2 x cubed plus x minus 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x^4-2x^3+x-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the only Pisot number whose four distinct conjugates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 1 comma alpha 2 comma alpha 3 comma alpha 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _1,\alpha _2,\alpha _3,\alpha _4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the additive relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 1 plus alpha 2 equals alpha 3 plus alpha 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _1+\alpha _2=\alpha _3+\alpha _4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 1 equals alpha 2 plus alpha 3 plus alpha 4"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha _1 = \alpha _2 + \alpha _3+\alpha _4</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="alpha 1 plus alpha 2 plus alpha 3 plus alpha 4 equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi> α </mml:mi>
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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