Characteristic Points of Recursive Systems
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Bibliographic record
Abstract
Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems. Given an irreducible non-negative power series system with $m$ equations, let $\rho$ be the radius of convergence of the solution power series and let $\pmb{\tau}$ be the values of the solution series evaluated at $\rho$. The main results of the paper include: (a) the set of characteristic points form an antichain in ${\mathbb R}^{m+1}$, (b) given a characteristic point $(a,\mathbf{b})$, (i) the spectral radius of the Jacobian of $\pmb \gamma$ at $(a, \mathbf{b})$ is $\ge 1$, and (ii) it is $=1$ iff $(a,\mathbf{b}) = (\rho,\pmb{\tau})$, (c) if $(\rho,\pmb{\tau})$ is a characteristic point, then (i) $\rho$ is the largest $a$ for $(a,\mathbf{b})$ a characteristic point, and (ii) a characteristic point $(a,\mathbf{b})$ with $a=\rho$ is the extreme point $(\rho,\pmb{\tau})$.
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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) | 0.000 | 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 | 0.001 |
| 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