GENERAL FORMS FOR MINIMAL SPECTRAL VALUES FOR A CLASS OF QUADRATIC PISOT NUMBERS
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
This paper studies the spectrum that results when all height one polynomials are evaluated at a Pisot number. This continues the research theme initiated by Erdős, Joó and Komornik in 1990. Of particular interest is the minimal non-zero value of this spectrum. Formally, this value is denoted as l1(q), and this definition is extended to all height m polynomials as ≔ɛɛɛɛɛlm(q) ≔ inf(|y|:y=ɛ0+ɛ1q1+…+ɛnqn, ɛi∈Z, |ɛi| ⩽ m, y≠0). A recent result in 2000, of Komornik, Loreti and Pedicini gives a complete description of lm(q) when q is the Golden ratio. This paper extends this result to include all unit quadratic Pisot numbers. A main theorem is as follows. THEOREM. Let q be a quadratic Pisot number that satisfies a polynomial of the form p(x) = x2−ax ± 1, with conjugate r. Let q have convergents {Ck/Dk} and let k be the maximal integer such that |Dkr−Ck|⩽m11−|r|; then A value related to l(q) is a(q), the minimal non-zero value when all ±1 polynomials are evaluated at q. Formally, this is An open question concerning how often a(q) = l(q) is also answered in this paper.
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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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