Effective Inverse Spectral Problem for Rational Lax Matrices and Applications
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Bibliographic record
Abstract
We reconstruct a rational Lax matrix of size <it>R</it> + 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy-like kernel (a bi-differential of bi-weight (1 − ν, ν)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν, 1 − ν respectively. All objects are described in the most explicit terms using Theta functions. Via a sequence of “elementary twists”, we construct sequences of Lax matrices sharing the same spectral curve and polar structure and related by conjugations by rational matrices. Particular choices of elementary twists lead to construction of sequences of Lax matrices related to finite–band recurrence relations (i.e. difference operators) sharing the same shape. Recurrences of this kind are satisfied by several types of orthogonal and biorthogonal polynomials. The relevance of formulæ obtained to the study of the large degree asymptotics for these polynomials is indicated.
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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 | 0.005 | 0.002 |
| 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