Periodic Coefficients and Random Fibonacci Sequences
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Bibliographic record
Abstract
The random Fibonacci sequence is defined by $t_1 = t_2 = 1$ and $t_n = \pm t_{n-1} + t_{n-2}$, for $n \geq 3$, where each $\pm$ sign is chosen at random with probability $P(+) = P(-) = \frac{1}{2}$. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate $1.13198824\ldots$. We will consider what happens to random Fibonacci sequences when we remove the randomness; specifically, we will choose coefficients which belong to the set $\{1, -1\}$ and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length.
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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 | 0.000 |
| 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